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<span class="vector-toc-numb">4.1</span> <span>First component</span> </div> </a> <ul id="toc-First_component-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_components" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_components"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Further components</span> </div> </a> <ul id="toc-Further_components-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Covariances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Covariances"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Covariances</span> </div> </a> <ul id="toc-Covariances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimensionality_reduction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimensionality_reduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Dimensionality reduction</span> </div> </a> <ul id="toc-Dimensionality_reduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Singular_value_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Singular_value_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Singular value decomposition</span> </div> </a> <ul id="toc-Singular_value_decomposition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_considerations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_considerations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Further considerations</span> </div> </a> <ul id="toc-Further_considerations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Table_of_symbols_and_abbreviations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Table_of_symbols_and_abbreviations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Table of symbols and abbreviations</span> </div> </a> <ul id="toc-Table_of_symbols_and_abbreviations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_and_limitations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties_and_limitations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Properties and limitations</span> </div> </a> <button aria-controls="toc-Properties_and_limitations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties and limitations subsection</span> </button> <ul id="toc-Properties_and_limitations-sublist" class="vector-toc-list"> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limitations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Limitations</span> </div> </a> <ul id="toc-Limitations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-PCA_and_information_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#PCA_and_information_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>PCA and information theory</span> </div> </a> <ul id="toc-PCA_and_information_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Computation_using_the_covariance_method" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Computation_using_the_covariance_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Computation using the covariance method</span> </div> </a> <ul id="toc-Computation_using_the_covariance_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_using_the_covariance_method" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derivation_using_the_covariance_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Derivation using the covariance method</span> </div> </a> <ul id="toc-Derivation_using_the_covariance_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Covariance-free_computation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Covariance-free_computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Covariance-free computation</span> </div> </a> <button aria-controls="toc-Covariance-free_computation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Covariance-free computation subsection</span> </button> <ul id="toc-Covariance-free_computation-sublist" class="vector-toc-list"> <li id="toc-Iterative_computation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Iterative_computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Iterative computation</span> </div> </a> <ul id="toc-Iterative_computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_NIPALS_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_NIPALS_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>The NIPALS method</span> </div> </a> <ul id="toc-The_NIPALS_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Online/sequential_estimation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Online/sequential_estimation"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>Online/sequential estimation</span> </div> </a> <ul id="toc-Online/sequential_estimation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Qualitative_variables" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Qualitative_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Qualitative variables</span> </div> </a> <ul id="toc-Qualitative_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Intelligence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intelligence"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>Intelligence</span> </div> </a> <ul id="toc-Intelligence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Residential_differentiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Residential_differentiation"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.2</span> <span>Residential differentiation</span> </div> </a> <ul id="toc-Residential_differentiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Development_indexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Development_indexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.3</span> <span>Development indexes</span> </div> </a> <ul id="toc-Development_indexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Population_genetics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Population_genetics"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.4</span> <span>Population genetics</span> </div> </a> <ul id="toc-Population_genetics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Market_research_and_indexes_of_attitude" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Market_research_and_indexes_of_attitude"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.5</span> <span>Market research and indexes of attitude</span> </div> </a> <ul id="toc-Market_research_and_indexes_of_attitude-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantitative_finance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantitative_finance"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.6</span> <span>Quantitative finance</span> </div> </a> <ul id="toc-Quantitative_finance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Neuroscience" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Neuroscience"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.7</span> <span>Neuroscience</span> </div> </a> <ul id="toc-Neuroscience-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_with_other_methods" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_with_other_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Relation with other methods</span> </div> </a> <button aria-controls="toc-Relation_with_other_methods-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relation with other methods subsection</span> </button> <ul id="toc-Relation_with_other_methods-sublist" class="vector-toc-list"> <li id="toc-Correspondence_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Correspondence_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.1</span> <span>Correspondence analysis</span> </div> </a> <ul id="toc-Correspondence_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Factor_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Factor_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.2</span> <span>Factor analysis</span> </div> </a> <ul id="toc-Factor_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-K-means_clustering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#K-means_clustering"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.3</span> <span><span>K</span>-means clustering</span> </div> </a> <ul id="toc-K-means_clustering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-negative_matrix_factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-negative_matrix_factorization"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.4</span> <span>Non-negative matrix factorization</span> </div> </a> <ul id="toc-Non-negative_matrix_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Iconography_of_correlations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Iconography_of_correlations"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.5</span> <span>Iconography of correlations</span> </div> </a> <ul id="toc-Iconography_of_correlations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Generalizations</span> </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Generalizations subsection</span> </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Sparse_PCA" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sparse_PCA"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.1</span> <span>Sparse PCA</span> </div> </a> <ul id="toc-Sparse_PCA-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonlinear_PCA" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonlinear_PCA"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.2</span> <span>Nonlinear PCA</span> </div> </a> <ul id="toc-Nonlinear_PCA-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Robust_PCA" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Robust_PCA"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.3</span> <span>Robust PCA</span> </div> </a> <ul id="toc-Robust_PCA-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Similar_techniques" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Similar_techniques"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Similar techniques</span> </div> </a> <button aria-controls="toc-Similar_techniques-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Similar techniques subsection</span> </button> <ul id="toc-Similar_techniques-sublist" class="vector-toc-list"> <li id="toc-Independent_component_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Independent_component_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">15.1</span> <span>Independent component analysis</span> </div> </a> <ul id="toc-Independent_component_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Network_component_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Network_component_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">15.2</span> <span>Network component analysis</span> </div> </a> <ul id="toc-Network_component_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discriminant_analysis_of_principal_components" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discriminant_analysis_of_principal_components"> <div class="vector-toc-text"> <span class="vector-toc-numb">15.3</span> <span>Discriminant analysis of principal components</span> </div> </a> <ul id="toc-Discriminant_analysis_of_principal_components-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Directional_component_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Directional_component_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">15.4</span> <span>Directional component analysis</span> </div> </a> <ul id="toc-Directional_component_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Software/source_code" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Software/source_code"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>Software/source code</span> </div> </a> <ul id="toc-Software/source_code-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">18</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">19</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">20</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label 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Available in 34 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-34" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">34 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%A7%D9%84%D8%B9%D9%86%D8%B5%D8%B1_%D8%A7%D9%84%D8%B1%D8%A6%D9%8A%D8%B3%D9%8A" title="تحليل العنصر الرئيسي – Arabic" lang="ar" hreflang="ar" data-title="تحليل العنصر الرئيسي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Anal%C3%ADs_de_componentes_principales" title="Analís de componentes principales – Asturian" lang="ast" hreflang="ast" data-title="Analís de componentes principales" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C3%BA-s%C3%AAng-h%C5%ABn_hun-sek" title="Chú-sêng-hūn hun-sek – Minnan" lang="nan" hreflang="nan" data-title="Chú-sêng-hūn hun-sek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_de_components_principals" title="Anàlisi de components principals – Catalan" lang="ca" hreflang="ca" data-title="Anàlisi de components principals" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Anal%C3%BDza_hlavn%C3%ADch_komponent" title="Analýza hlavních komponent – Czech" lang="cs" hreflang="cs" data-title="Analýza hlavních komponent" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hauptkomponentenanalyse" title="Hauptkomponentenanalyse – German" lang="de" hreflang="de" data-title="Hauptkomponentenanalyse" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Peakomponentide_anal%C3%BC%C3%BCs" title="Peakomponentide analüüs – Estonian" lang="et" hreflang="et" data-title="Peakomponentide analüüs" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_de_componentes_principales" title="Análisis de componentes principales – Spanish" lang="es" hreflang="es" data-title="Análisis de componentes principales" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Analizo_al_precipaj_konsisteroj" title="Analizo al precipaj konsisteroj – Esperanto" lang="eo" hreflang="eo" data-title="Analizo al precipaj konsisteroj" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Osagai_nagusien_analisi" title="Osagai nagusien analisi – Basque" lang="eu" hreflang="eu" data-title="Osagai nagusien analisi" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%DB%8C%D9%84_%D9%85%D8%A4%D9%84%D9%81%D9%87%E2%80%8C%D9%87%D8%A7%DB%8C_%D8%A7%D8%B5%D9%84%DB%8C" title="تحلیل مؤلفه‌های اصلی – Persian" lang="fa" hreflang="fa" data-title="تحلیل مؤلفه‌های اصلی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_en_composantes_principales" title="Analyse en composantes principales – French" lang="fr" hreflang="fr" data-title="Analyse en composantes principales" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/An%C3%A1lise_de_compo%C3%B1entes_principais" title="Análise de compoñentes principais – Galician" lang="gl" hreflang="gl" data-title="Análise de compoñentes principais" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A3%BC%EC%84%B1%EB%B6%84_%EB%B6%84%EC%84%9D" title="주성분 분석 – Korean" lang="ko" hreflang="ko" data-title="주성분 분석" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_komponen_utama" title="Analisis komponen utama – Indonesian" lang="id" hreflang="id" data-title="Analisis komponen utama" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Meginhlutagreining" title="Meginhlutagreining – Icelandic" lang="is" hreflang="is" data-title="Meginhlutagreining" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Analisi_delle_componenti_principali" title="Analisi delle componenti principali – Italian" lang="it" hreflang="it" data-title="Analisi delle componenti principali" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%99%D7%AA%D7%95%D7%97_%D7%A8%D7%9B%D7%99%D7%91%D7%99%D7%9D_%D7%A2%D7%99%D7%A7%D7%A8%D7%99%D7%99%D7%9D" title="ניתוח רכיבים עיקריים – Hebrew" lang="he" hreflang="he" data-title="ניתוח רכיבים עיקריים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/F%C5%91komponens-anal%C3%ADzis" title="Főkomponens-analízis – Hungarian" lang="hu" hreflang="hu" data-title="Főkomponens-analízis" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hoofdcomponentenanalyse" title="Hoofdcomponentenanalyse – Dutch" lang="nl" hreflang="nl" data-title="Hoofdcomponentenanalyse" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%BB%E6%88%90%E5%88%86%E5%88%86%E6%9E%90" title="主成分分析 – Japanese" lang="ja" hreflang="ja" data-title="主成分分析" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_g%C5%82%C3%B3wnych_sk%C5%82adowych" title="Analiza głównych składowych – Polish" lang="pl" hreflang="pl" data-title="Analiza głównych składowych" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/An%C3%A1lise_de_componentes_principais" title="Análise de componentes principais – Portuguese" lang="pt" hreflang="pt" data-title="Análise de componentes principais" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D1%8B%D1%85_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%82" title="Метод главных компонент – Russian" lang="ru" hreflang="ru" data-title="Метод главных компонент" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Anal%C3%BDza_hlavn%C3%BDch_komponentov" title="Analýza hlavných komponentov – Slovak" lang="sk" hreflang="sk" data-title="Analýza hlavných komponentov" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Analiza_glavnih_komponenti" title="Analiza glavnih komponenti – Serbian" lang="sr" hreflang="sr" data-title="Analiza glavnih komponenti" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/P%C3%A4%C3%A4komponenttianalyysi" title="Pääkomponenttianalyysi – Finnish" lang="fi" hreflang="fi" data-title="Pääkomponenttianalyysi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Principalkomponentanalys" title="Principalkomponentanalys – Swedish" lang="sv" hreflang="sv" data-title="Principalkomponentanalys" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Analisis_ng_pangunahing_bahagi" title="Analisis ng pangunahing bahagi – Tagalog" lang="tl" hreflang="tl" data-title="Analisis ng pangunahing bahagi" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Temel_bile%C5%9Fen_analizi" title="Temel bileşen analizi – Turkish" lang="tr" hreflang="tr" data-title="Temel bileşen analizi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%B3%D0%BE%D0%BB%D0%BE%D0%B2%D0%BD%D0%B8%D1%85_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%82" title="Метод головних компонент – Ukrainian" lang="uk" hreflang="uk" data-title="Метод головних компонент" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_ph%C3%A2n_t%C3%ADch_th%C3%A0nh_ph%E1%BA%A7n_ch%C3%ADnh" title="Phép phân tích thành phần chính – Vietnamese" lang="vi" hreflang="vi" data-title="Phép phân tích thành phần chính" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Method of data analysis</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:GaussianScatterPCA.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/GaussianScatterPCA.svg/290px-GaussianScatterPCA.svg.png" decoding="async" width="290" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/GaussianScatterPCA.svg/435px-GaussianScatterPCA.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/GaussianScatterPCA.svg/580px-GaussianScatterPCA.svg.png 2x" data-file-width="720" data-file-height="720" /></a><figcaption>PCA of a <a href="/wiki/Multivariate_Gaussian_distribution" class="mw-redirect" title="Multivariate Gaussian distribution">multivariate Gaussian distribution</a> centered at (1,3) with a standard deviation of 3 in roughly the (0.866, 0.5) direction and of 1 in the orthogonal direction. The vectors shown are the <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvectors</a> of the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the mean.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1244144826">.mw-parser-output .machine-learning-list-title{background-color:#ddddff}html.skin-theme-clientpref-night .mw-parser-output .machine-learning-list-title{background-color:#222}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .machine-learning-list-title{background-color:#222}}</style> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output 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.mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a><br />and <a href="/wiki/Data_mining" title="Data mining">data mining</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Paradigms</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">Semi-supervised learning</a></li> <li><a href="/wiki/Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Meta-learning_(computer_science)" title="Meta-learning (computer science)">Meta-learning</a></li> <li><a href="/wiki/Online_machine_learning" title="Online machine learning">Online learning</a></li> <li><a href="/wiki/Batch_learning" class="mw-redirect" title="Batch learning">Batch learning</a></li> <li><a href="/wiki/Curriculum_learning" title="Curriculum learning">Curriculum learning</a></li> <li><a href="/wiki/Rule-based_machine_learning" title="Rule-based machine learning">Rule-based learning</a></li> <li><a href="/wiki/Neuro-symbolic_AI" title="Neuro-symbolic AI">Neuro-symbolic AI</a></li> <li><a href="/wiki/Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">Neuromorphic engineering</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Problems</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Generative_model" title="Generative model">Generative modeling</a></li> <li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></li> <li><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></li> <li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li> <li><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></li> <li><a href="/wiki/Data_cleaning" class="mw-redirect" title="Data cleaning">Data cleaning</a></li> <li><a href="/wiki/Automated_machine_learning" title="Automated machine learning">AutoML</a></li> <li><a href="/wiki/Association_rule_learning" title="Association rule learning">Association rules</a></li> <li><a href="/wiki/Semantic_analysis_(machine_learning)" title="Semantic analysis (machine learning)">Semantic analysis</a></li> <li><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></li> <li><a href="/wiki/Feature_engineering" title="Feature engineering">Feature engineering</a></li> <li><a href="/wiki/Feature_learning" title="Feature learning">Feature learning</a></li> <li><a href="/wiki/Learning_to_rank" title="Learning to rank">Learning to rank</a></li> <li><a href="/wiki/Grammar_induction" title="Grammar induction">Grammar induction</a></li> <li><a href="/wiki/Ontology_learning" title="Ontology learning">Ontology learning</a></li> <li><a href="/wiki/Multimodal_learning" title="Multimodal learning">Multimodal learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a><br /><span class="nobold"><span style="font-size:85%;">(<b><a href="/wiki/Statistical_classification" title="Statistical classification">classification</a></b> • <b><a href="/wiki/Regression_analysis" title="Regression analysis">regression</a></b>)</span></span> </div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Apprenticeship_learning" title="Apprenticeship learning">Apprenticeship learning</a></li> <li><a href="/wiki/Decision_tree_learning" title="Decision tree learning">Decision trees</a></li> <li><a href="/wiki/Ensemble_learning" title="Ensemble learning">Ensembles</a> <ul><li><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bagging</a></li> <li><a href="/wiki/Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a></li> <li><a href="/wiki/Random_forest" title="Random forest">Random forest</a></li></ul></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes</a></li> <li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural networks</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Perceptron" title="Perceptron">Perceptron</a></li> <li><a href="/wiki/Relevance_vector_machine" title="Relevance vector machine">Relevance vector machine (RVM)</a></li> <li><a href="/wiki/Support_vector_machine" title="Support vector machine">Support vector machine (SVM)</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/BIRCH" title="BIRCH">BIRCH</a></li> <li><a href="/wiki/CURE_algorithm" title="CURE algorithm">CURE</a></li> <li><a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">Hierarchical</a></li> <li><a href="/wiki/K-means_clustering" title="K-means clustering"><i>k</i>-means</a></li> <li><a href="/wiki/Fuzzy_clustering" title="Fuzzy clustering">Fuzzy</a></li> <li><a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">Expectation–maximization (EM)</a></li> <li><br /><a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a></li> <li><a href="/wiki/OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a></li> <li><a href="/wiki/Mean_shift" title="Mean shift">Mean shift</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">CCA</a></li> <li><a href="/wiki/Independent_component_analysis" title="Independent component analysis">ICA</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">LDA</a></li> <li><a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">NMF</a></li> <li><a class="mw-selflink selflink">PCA</a></li> <li><a href="/wiki/Proper_generalized_decomposition" title="Proper generalized decomposition">PGD</a></li> <li><a href="/wiki/T-distributed_stochastic_neighbor_embedding" title="T-distributed stochastic neighbor embedding">t-SNE</a></li> <li><a href="/wiki/Sparse_dictionary_learning" title="Sparse dictionary learning">SDL</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Graphical_model" title="Graphical model">Graphical models</a> <ul><li><a href="/wiki/Bayesian_network" title="Bayesian network">Bayes net</a></li> <li><a href="/wiki/Conditional_random_field" title="Conditional random field">Conditional random field</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Random_sample_consensus" title="Random sample consensus">RANSAC</a></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a href="/wiki/Local_outlier_factor" title="Local outlier factor">Local outlier factor</a></li> <li><a href="/wiki/Isolation_forest" title="Isolation forest">Isolation forest</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural network</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Autoencoder" title="Autoencoder">Autoencoder</a></li> <li><a href="/wiki/Deep_learning" title="Deep learning">Deep learning</a></li> <li><a href="/wiki/Feedforward_neural_network" title="Feedforward neural network">Feedforward neural network</a></li> <li><a href="/wiki/Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network</a> <ul><li><a href="/wiki/Long_short-term_memory" title="Long short-term memory">LSTM</a></li> <li><a href="/wiki/Gated_recurrent_unit" title="Gated recurrent unit">GRU</a></li> <li><a href="/wiki/Echo_state_network" title="Echo state network">ESN</a></li> <li><a href="/wiki/Reservoir_computing" title="Reservoir computing">reservoir computing</a></li></ul></li> <li><a href="/wiki/Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a> <ul><li><a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted</a></li></ul></li> <li><a href="/wiki/Generative_adversarial_network" title="Generative adversarial network">GAN</a></li> <li><a href="/wiki/Diffusion_model" title="Diffusion model">Diffusion model</a></li> <li><a href="/wiki/Self-organizing_map" title="Self-organizing map">SOM</a></li> <li><a href="/wiki/Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network</a> <ul><li><a href="/wiki/U-Net" title="U-Net">U-Net</a></li> <li><a href="/wiki/LeNet" title="LeNet">LeNet</a></li> <li><a href="/wiki/AlexNet" title="AlexNet">AlexNet</a></li> <li><a href="/wiki/DeepDream" title="DeepDream">DeepDream</a></li></ul></li> <li><a href="/wiki/Neural_radiance_field" title="Neural radiance field">Neural radiance field</a></li> <li><a href="/wiki/Transformer_(machine_learning_model)" class="mw-redirect" title="Transformer (machine learning model)">Transformer</a> <ul><li><a href="/wiki/Vision_transformer" title="Vision transformer">Vision</a></li></ul></li> <li><a href="/wiki/Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li> <li><a href="/wiki/Spiking_neural_network" title="Spiking neural network">Spiking neural network</a></li> <li><a href="/wiki/Memtransistor" title="Memtransistor">Memtransistor</a></li> <li><a href="/wiki/Electrochemical_RAM" title="Electrochemical RAM">Electrochemical RAM</a> (ECRAM)</li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Q-learning" title="Q-learning">Q-learning</a></li> <li><a href="/wiki/State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li> <li><a href="/wiki/Temporal_difference_learning" title="Temporal difference learning">Temporal difference (TD)</a></li> <li><a href="/wiki/Multi-agent_reinforcement_learning" title="Multi-agent reinforcement learning">Multi-agent</a> <ul><li><a href="/wiki/Self-play_(reinforcement_learning_technique)" class="mw-redirect" title="Self-play (reinforcement learning technique)">Self-play</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Learning with humans</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Active_learning_(machine_learning)" title="Active learning (machine learning)">Active learning</a></li> <li><a href="/wiki/Crowdsourcing" title="Crowdsourcing">Crowdsourcing</a></li> <li><a href="/wiki/Human-in-the-loop" title="Human-in-the-loop">Human-in-the-loop</a></li> <li><a href="/wiki/Reinforcement_learning_from_human_feedback" title="Reinforcement learning from human feedback">RLHF</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Model diagnostics</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li> <li><a href="/wiki/Learning_curve_(machine_learning)" title="Learning curve (machine learning)">Learning curve</a></li> <li><a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">ROC curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Mathematical foundations</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Kernel_machines" class="mw-redirect" title="Kernel machines">Kernel machines</a></li> <li><a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li> <li><a href="/wiki/Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li> <li><a href="/wiki/Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></li> <li><a href="/wiki/Occam_learning" title="Occam learning">Occam learning</a></li> <li><a href="/wiki/Probably_approximately_correct_learning" title="Probably approximately correct learning">PAC learning</a></li> <li><a href="/wiki/Statistical_learning_theory" title="Statistical learning theory">Statistical learning</a></li> <li><a href="/wiki/Vapnik%E2%80%93Chervonenkis_theory" title="Vapnik–Chervonenkis theory">VC theory</a></li> <li><a href="/wiki/Topological_deep_learning" title="Topological deep learning">Topological deep learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Journals and conferences</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/ECML_PKDD" title="ECML PKDD">ECML PKDD</a></li> <li><a href="/wiki/Conference_on_Neural_Information_Processing_Systems" title="Conference on Neural Information Processing Systems">NeurIPS</a></li> <li><a href="/wiki/International_Conference_on_Machine_Learning" title="International Conference on Machine Learning">ICML</a></li> <li><a href="/wiki/International_Conference_on_Learning_Representations" title="International Conference on Learning Representations">ICLR</a></li> <li><a href="/wiki/International_Joint_Conference_on_Artificial_Intelligence" title="International Joint Conference on Artificial Intelligence">IJCAI</a></li> <li><a href="/wiki/Machine_Learning_(journal)" title="Machine Learning (journal)">ML</a></li> <li><a href="/wiki/Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">JMLR</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Related articles</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Glossary of artificial intelligence</a></li> <li><a href="/wiki/List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a> <ul><li><a href="/wiki/List_of_datasets_in_computer_vision_and_image_processing" title="List of datasets in computer vision and image processing">List of datasets in computer vision and image processing</a></li></ul></li> <li><a href="/wiki/Outline_of_machine_learning" title="Outline of machine learning">Outline of machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Machine_learning" title="Template:Machine learning"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Machine_learning" title="Template talk:Machine learning"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Machine_learning" title="Special:EditPage/Template:Machine learning"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Principal component analysis</b> (<b>PCA</b>) is a <a href="/wiki/Linear_map" title="Linear map">linear</a> <a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">dimensionality reduction</a> technique with applications in <a href="/wiki/Exploratory_data_analysis" title="Exploratory data analysis">exploratory data analysis</a>, visualization and <a href="/wiki/Data_Preprocessing" class="mw-redirect" title="Data Preprocessing">data preprocessing</a>. </p><p>The data is <a href="/wiki/Linear_map" title="Linear map">linearly transformed</a> onto a new <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> such that the directions (principal components) capturing the largest variation in the data can be easily identified. </p><p>The <b>principal components</b> of a collection of points in a <a href="/wiki/Real_coordinate_space" title="Real coordinate space">real coordinate space</a> are a sequence of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> <a href="/wiki/Unit_vector" title="Unit vector">unit vectors</a>, where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th vector is the direction of a line that best fits the data while being <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> to the first <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2ca5c639f26340e0e80f5883cc93a00254513c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.805ex; height:2.343ex;" alt="{\displaystyle i-1}"></span> vectors. Here, a best-fitting line is defined as one that minimizes the average squared <a href="/wiki/Perpendicular_distance" title="Perpendicular distance">perpendicular</a> <a href="/wiki/Distance_from_a_point_to_a_line" title="Distance from a point to a line">distance from the points to the line</a>. These directions (i.e., principal components) constitute an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> in which different individual dimensions of the data are <a href="/wiki/Linear_correlation" class="mw-redirect" title="Linear correlation">linearly uncorrelated</a>. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Principal component analysis has applications in many fields such as <a href="/wiki/Population_genetics" title="Population genetics">population genetics</a>, <a href="/wiki/Microbiome" title="Microbiome">microbiome</a> studies, and <a href="/wiki/Atmospheric_science" title="Atmospheric science">atmospheric science</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When performing PCA, the first principal component of a set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an <a href="/wiki/Linear_independence" title="Linear independence">independent set</a>. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th principal component can be taken as a direction orthogonal to the first <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2ca5c639f26340e0e80f5883cc93a00254513c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.805ex; height:2.343ex;" alt="{\displaystyle i-1}"></span> principal components that maximizes the variance of the projected data. </p><p>For either objective, it can be shown that the principal components are <a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvectors</a> of the data's <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a>. Thus, the principal components are often computed by <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">eigendecomposition</a> of the data covariance matrix or <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to <a href="/wiki/Factor_analysis" title="Factor analysis">factor analysis</a>. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to <a href="/wiki/Canonical_correlation" title="Canonical correlation">canonical correlation analysis (CCA)</a>. CCA defines coordinate systems that optimally describe the <a href="/wiki/Cross-covariance" title="Cross-covariance">cross-covariance</a> between two datasets while PCA defines a new <a href="/wiki/Orthogonal_coordinate_system" class="mw-redirect" title="Orthogonal coordinate system">orthogonal coordinate system</a> that optimally describes variance in a single dataset.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-mark2017_5-0" class="reference"><a href="#cite_note-mark2017-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-l1tucker_6-0" class="reference"><a href="#cite_note-l1tucker-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Robust_statistics" title="Robust statistics">Robust</a> and <a href="/wiki/Lp_space" title="Lp space">L1-norm</a>-based variants of standard PCA have also been proposed.<sup id="cite_ref-mark2014_7-0" class="reference"><a href="#cite_note-mark2014-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-l1tucker_6-1" class="reference"><a href="#cite_note-l1tucker-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>PCA was invented in 1901 by <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> as an analogue of the <a href="/wiki/Principal_axis_theorem" title="Principal axis theorem">principal axis theorem</a> in mechanics; it was later independently developed and named by <a href="/wiki/Harold_Hotelling" title="Harold Hotelling">Harold Hotelling</a> in the 1930s.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Depending on the field of application, it is also named the discrete <a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève</a> transform (KLT) in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, the <a href="/wiki/Harold_Hotelling" title="Harold Hotelling">Hotelling</a> transform in multivariate quality control, <a href="/wiki/Proper_orthogonal_decomposition" title="Proper orthogonal decomposition">proper orthogonal decomposition</a> (POD) in mechanical engineering, <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> (SVD) of <b>X</b> (invented in the last quarter of the 19th century<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>), <a href="/wiki/Eigendecomposition" class="mw-redirect" title="Eigendecomposition">eigenvalue decomposition</a> (EVD) of <b>X</b><sup>T</sup><b>X</b> in linear algebra, <a href="/wiki/Factor_analysis" title="Factor analysis">factor analysis</a> (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's <i>Principal Component Analysis</i>),<sup id="cite_ref-Jolliffe2002_13-0" class="reference"><a href="#cite_note-Jolliffe2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Eckart%E2%80%93Young_theorem" class="mw-redirect" title="Eckart–Young theorem">Eckart–Young theorem</a> (Harman, 1960), or <a href="/wiki/Empirical_orthogonal_functions" title="Empirical orthogonal functions">empirical orthogonal functions</a> (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral decomposition</a> in noise and vibration, and <a href="/wiki/Mode_shape" class="mw-redirect" title="Mode shape">empirical modal analysis</a> in structural dynamics. </p> <div class="mw-heading mw-heading2"><h2 id="Intuition">Intuition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=3" title="Edit section: Intuition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SCREE_plot.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/SCREE_plot.jpg/220px-SCREE_plot.jpg" decoding="async" width="220" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/SCREE_plot.jpg/330px-SCREE_plot.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/SCREE_plot.jpg/440px-SCREE_plot.jpg 2x" data-file-width="624" data-file-height="372" /></a><figcaption>A scree plot that is meant to help interpret the PCA and decide how many components to retain. The start of the bend in the line (point of inflexion or "knee") should indicate how many components are retained, hence in this example, three factors should be retained.</figcaption></figure> <p>PCA can be thought of as fitting a <i>p</i>-dimensional <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a> to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small. </p><p>To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must <a href="/wiki/Normalization_(statistics)" title="Normalization (statistics)">normalize</a> each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues. </p><p><a href="/wiki/Biplot" title="Biplot">Biplots</a> and <a href="/wiki/Scree_plot" title="Scree plot">scree plots</a> (degree of <a href="/wiki/Explained_variance" class="mw-redirect" title="Explained variance">explained variance</a>) are used to interpret findings of the PCA. </p> <div class="mw-heading mw-heading2"><h2 id="Details">Details</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=4" title="Edit section: Details"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>PCA is defined as an <a href="/wiki/Orthogonal_transformation" title="Orthogonal transformation">orthogonal</a> <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> on a real <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a> that transforms the data to a new <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on.<sup id="cite_ref-Jolliffe2002_13-1" class="reference"><a href="#cite_note-Jolliffe2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Consider an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}"></span> data <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>, <b>X</b>, with column-wise zero <a href="/wiki/Empirical_mean" class="mw-redirect" title="Empirical mean">empirical mean</a> (the sample mean of each column has been shifted to zero), where each of the <i>n</i> rows represents a different repetition of the experiment, and each of the <i>p</i> columns gives a particular kind of feature (say, the results from a particular sensor). </p><p>Mathematically, the transformation is defined by a set of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> of <i>p</i>-dimensional vectors of weights or coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2245bb26441ba4d846e01432a96b9636249a152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.195ex; height:3.176ex;" alt="{\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}}"></span> that map each row vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd191f28280e2911d96c0215133be088f7106ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.428ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}}"></span> of <b>X</b> to a new vector of principal component <i>scores</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8050ef005553dd900b6cbc190a802e9a37442b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.739ex; height:3.176ex;" alt="{\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}}"></span>, given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mspace width="2em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mspace width="2em" /> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2446cfb8f55eda2da4e5330ee2bcf24412967ae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:56.429ex; height:3.009ex;" alt="{\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l}"></span></dd></dl> <p>in such a way that the individual variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},\dots ,t_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1},\dots ,t_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f861e46c43925d0782bac685220724051ba4fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.634ex; height:2.343ex;" alt="{\displaystyle t_{1},\dots ,t_{l}}"></span> of <b>t</b> considered over the data set successively inherit the maximum possible variance from <b>X</b>, with each coefficient vector <b>w</b> constrained to be a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> is usually selected to be strictly less than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> to reduce dimensionality). </p><p>The above may equivalently be written in matrix form as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3533d0f1998a93db21b1d627328eed10cd6a65b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.74ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} }"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0a3bf2582c483a36b8508ed4d78883f0f5c7de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.621ex; height:3.009ex;" alt="{\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1867120f28e358087d4439f00928c48e3ee927e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.914ex; height:3.343ex;" alt="{\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f65c743cf19f46c1ab84f85fd6843d1af48227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:12.569ex; height:3.343ex;" alt="{\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="First_component">First component</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=5" title="Edit section: First component"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to maximize variance, the first weight vector <b>w</b><sub>(1)</sub> thus has to satisfy </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff156c1b42d71f80bb9fb3cf97c7d700e49169e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.134ex; height:7.509ex;" alt="{\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}}"></span></dd></dl> <p>Equivalently, writing this in matrix form gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mrow> <mo>{</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">w</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>}</mo> </mrow> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">w</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7343496fe1ddd39a196a95ca598203399e7c7418" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.449ex; height:5.509ex;" alt="{\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}}"></span></dd></dl> <p>Since <b>w</b><sub>(1)</sub> has been defined to be a unit vector, it equivalently also satisfies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">w</mi> </mrow> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5a458dbb1d0da3fdc5c42598003e52a293d444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.627ex; height:6.343ex;" alt="{\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}}"></span></dd></dl> <p>The quantity to be maximised can be recognised as a <a href="/wiki/Rayleigh_quotient" title="Rayleigh quotient">Rayleigh quotient</a>. A standard result for a <a href="/wiki/Positive_semidefinite_matrix" class="mw-redirect" title="Positive semidefinite matrix">positive semidefinite matrix</a> such as <b>X</b><sup>T</sup><b>X</b> is that the quotient's maximum possible value is the largest <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> of the matrix, which occurs when <i><b>w</b></i> is the corresponding <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvector</a>. </p><p>With <b>w</b><sub>(1)</sub> found, the first principal component of a data vector <b>x</b><sub>(<i>i</i>)</sub> can then be given as a score <i>t</i><sub>1(<i>i</i>)</sub> = <b>x</b><sub>(<i>i</i>)</sub> ⋅ <b>w</b><sub>(1)</sub> in the transformed co-ordinates, or as the corresponding vector in the original variables, {<b>x</b><sub>(<i>i</i>)</sub> ⋅ <b>w</b><sub>(1)</sub>} <b>w</b><sub>(1)</sub>. </p> <div class="mw-heading mw-heading3"><h3 id="Further_components">Further components</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=6" title="Edit section: Further components"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>k</i>-th component can be found by subtracting the first <i>k</i> − 1 principal components from <b>X</b>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {X}} _{k}=\mathbf {X} -\sum _{s=1}^{k-1}\mathbf {X} \mathbf {w} _{(s)}\mathbf {w} _{(s)}^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">X</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {X}} _{k}=\mathbf {X} -\sum _{s=1}^{k-1}\mathbf {X} \mathbf {w} _{(s)}\mathbf {w} _{(s)}^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be55d5f0212efebea48ace7f71f001fc3ee4fb05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.256ex; height:7.343ex;" alt="{\displaystyle \mathbf {\hat {X}} _{k}=\mathbf {X} -\sum _{s=1}^{k-1}\mathbf {X} \mathbf {w} _{(s)}\mathbf {w} _{(s)}^{\mathsf {T}}}"></span></dd></dl> <p>and then finding the weight vector which extracts the maximum variance from this new data matrix </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{(k)}=\mathop {\operatorname {arg\,max} } _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {\hat {X}} _{k}\mathbf {w} \right\|^{2}\right\}=\arg \max \left\{{\tfrac {\mathbf {w} ^{\mathsf {T}}\mathbf {\hat {X}} _{k}^{\mathsf {T}}\mathbf {\hat {X}} _{k}\mathbf {w} }{\mathbf {w} ^{T}\mathbf {w} }}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo>⁡</mo> <mrow> <mo>{</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">X</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>}</mo> </mrow> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">X</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">X</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{(k)}=\mathop {\operatorname {arg\,max} } _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {\hat {X}} _{k}\mathbf {w} \right\|^{2}\right\}=\arg \max \left\{{\tfrac {\mathbf {w} ^{\mathsf {T}}\mathbf {\hat {X}} _{k}^{\mathsf {T}}\mathbf {\hat {X}} _{k}\mathbf {w} }{\mathbf {w} ^{T}\mathbf {w} }}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eeefffd225dd97db95cdeb09d3e33e1120b8f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.216ex; height:6.843ex;" alt="{\displaystyle \mathbf {w} _{(k)}=\mathop {\operatorname {arg\,max} } _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {\hat {X}} _{k}\mathbf {w} \right\|^{2}\right\}=\arg \max \left\{{\tfrac {\mathbf {w} ^{\mathsf {T}}\mathbf {\hat {X}} _{k}^{\mathsf {T}}\mathbf {\hat {X}} _{k}\mathbf {w} }{\mathbf {w} ^{T}\mathbf {w} }}\right\}}"></span></dd></dl> <p>It turns out that this gives the remaining eigenvectors of <b>X</b><sup>T</sup><b>X</b>, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the weight vectors are eigenvectors of <b>X</b><sup>T</sup><b>X</b>. </p><p>The <i>k</i>-th principal component of a data vector <b>x</b><sub>(<i>i</i>)</sub> can therefore be given as a score <i>t</i><sub><i>k</i>(<i>i</i>)</sub> = <b>x</b><sub>(<i>i</i>)</sub> ⋅ <b>w</b><sub>(<i>k</i>)</sub> in the transformed coordinates, or as the corresponding vector in the space of the original variables, {<b>x</b><sub>(<i>i</i>)</sub> ⋅ <b>w</b><sub>(<i>k</i>)</sub>} <b>w</b><sub>(<i>k</i>)</sub>, where <b>w</b><sub>(<i>k</i>)</sub> is the <i>k</i>th eigenvector of <b>X</b><sup>T</sup><b>X</b>. </p><p>The full principal components decomposition of <b>X</b> can therefore be given as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3533d0f1998a93db21b1d627328eed10cd6a65b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.74ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} }"></span></dd></dl> <p>where <b>W</b> is a <i>p</i>-by-<i>p</i> matrix of weights whose columns are the eigenvectors of <b>X</b><sup>T</sup><b>X</b>. The transpose of <b>W</b> is sometimes called the <a href="/wiki/Whitening_transformation" title="Whitening transformation">whitening or sphering transformation</a>. Columns of <b>W</b> multiplied by the square root of corresponding eigenvalues, that is, eigenvectors scaled up by the variances, are called <i>loadings</i> in PCA or in Factor analysis. </p> <div class="mw-heading mw-heading3"><h3 id="Covariances">Covariances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=7" title="Edit section: Covariances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>X</b><sup>T</sup><b>X</b> itself can be recognized as proportional to the empirical sample <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> of the dataset <b>X<sup>T</sup></b>.<sup id="cite_ref-Jolliffe2002_13-2" class="reference"><a href="#cite_note-Jolliffe2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 30–31">: 30–31 </span></sup> </p><p>The sample covariance <i>Q</i> between two of the different principal components over the dataset is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}Q(\mathrm {PC} _{(j)},\mathrm {PC} _{(k)})&\propto (\mathbf {X} \mathbf {w} _{(j)})^{\mathsf {T}}(\mathbf {X} \mathbf {w} _{(k)})\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {X} \mathbf {w} _{(k)}\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\lambda _{(k)}\mathbf {w} _{(k)}\\&=\lambda _{(k)}\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {w} _{(k)}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>Q</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>∝</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Q(\mathrm {PC} _{(j)},\mathrm {PC} _{(k)})&\propto (\mathbf {X} \mathbf {w} _{(j)})^{\mathsf {T}}(\mathbf {X} \mathbf {w} _{(k)})\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {X} \mathbf {w} _{(k)}\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\lambda _{(k)}\mathbf {w} _{(k)}\\&=\lambda _{(k)}\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {w} _{(k)}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eaea8acfaeb76b5531d1cde92b323f60f4d0d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; margin-top: -0.202ex; width:37.039ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}Q(\mathrm {PC} _{(j)},\mathrm {PC} _{(k)})&\propto (\mathbf {X} \mathbf {w} _{(j)})^{\mathsf {T}}(\mathbf {X} \mathbf {w} _{(k)})\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {X} \mathbf {w} _{(k)}\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\lambda _{(k)}\mathbf {w} _{(k)}\\&=\lambda _{(k)}\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {w} _{(k)}\end{aligned}}}"></span></dd></dl> <p>where the eigenvalue property of <b>w</b><sub>(<i>k</i>)</sub> has been used to move from line 2 to line 3. However eigenvectors <b>w</b><sub>(<i>j</i>)</sub> and <b>w</b><sub>(<i>k</i>)</sub> corresponding to eigenvalues of a symmetric matrix are orthogonal (if the eigenvalues are different), or can be orthogonalised (if the vectors happen to share an equal repeated value). The product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset. </p><p>Another way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix. </p><p>In matrix form, the empirical covariance matrix for the original variables can be written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} \propto \mathbf {X} ^{\mathsf {T}}\mathbf {X} =\mathbf {W} \mathbf {\Lambda } \mathbf {W} ^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>∝</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Λ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} \propto \mathbf {X} ^{\mathsf {T}}\mathbf {X} =\mathbf {W} \mathbf {\Lambda } \mathbf {W} ^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3cc2f5703b3b5d406336a87681a538cd08f5ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.345ex; height:3.009ex;" alt="{\displaystyle \mathbf {Q} \propto \mathbf {X} ^{\mathsf {T}}\mathbf {X} =\mathbf {W} \mathbf {\Lambda } \mathbf {W} ^{\mathsf {T}}}"></span></dd></dl> <p>The empirical covariance matrix between the principal components becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} ^{\mathsf {T}}\mathbf {Q} \mathbf {W} \propto \mathbf {W} ^{\mathsf {T}}\mathbf {W} \,\mathbf {\Lambda } \,\mathbf {W} ^{\mathsf {T}}\mathbf {W} =\mathbf {\Lambda } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mo>∝</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Λ</mi> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Λ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {W} ^{\mathsf {T}}\mathbf {Q} \mathbf {W} \propto \mathbf {W} ^{\mathsf {T}}\mathbf {W} \,\mathbf {\Lambda } \,\mathbf {W} ^{\mathsf {T}}\mathbf {W} =\mathbf {\Lambda } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17e8d1f7a0f4527d49e30f2615ee2440e2256781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.356ex; height:3.009ex;" alt="{\displaystyle \mathbf {W} ^{\mathsf {T}}\mathbf {Q} \mathbf {W} \propto \mathbf {W} ^{\mathsf {T}}\mathbf {W} \,\mathbf {\Lambda } \,\mathbf {W} ^{\mathsf {T}}\mathbf {W} =\mathbf {\Lambda } }"></span></dd></dl> <p>where <b>Λ</b> is the diagonal matrix of eigenvalues <i>λ</i><sub>(<i>k</i>)</sub> of <b>X</b><sup>T</sup><b>X</b>. <i>λ</i><sub>(<i>k</i>)</sub> is equal to the sum of the squares over the dataset associated with each component <i>k</i>, that is, <i>λ</i><sub>(<i>k</i>)</sub> = Σ<sub><i>i</i></sub> <i>t</i><sub><i>k</i></sub><sup>2</sup><sub>(<i>i</i>)</sub> = Σ<sub><i>i</i></sub> (<b>x</b><sub>(<i>i</i>)</sub> ⋅ <b>w</b><sub>(<i>k</i>)</sub>)<sup>2</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Dimensionality_reduction">Dimensionality reduction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=8" title="Edit section: Dimensionality reduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The transformation <b>T</b> = <b>X</b> <b>W</b> maps a data vector <b>x</b><sub>(<i>i</i>)</sub> from an original space of <i>p</i> variables to a new space of <i>p</i> variables which are uncorrelated over the dataset. However, not all the principal components need to be kept. Keeping only the first <i>L</i> principal components, produced by using only the first <i>L</i> eigenvectors, gives the truncated transformation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} _{L}=\mathbf {X} \mathbf {W} _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} _{L}=\mathbf {X} \mathbf {W} _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3cbd7a3041a2b0404f28ad3cb2ff680991c1c8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.443ex; height:2.509ex;" alt="{\displaystyle \mathbf {T} _{L}=\mathbf {X} \mathbf {W} _{L}}"></span></dd></dl> <p>where the matrix <b>T</b><sub>L</sub> now has <i>n</i> rows but only <i>L</i> columns. In other words, PCA learns a linear transformation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=W_{L}^{\mathsf {T}}x,x\in \mathbb {R} ^{p},t\in \mathbb {R} ^{L},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msubsup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=W_{L}^{\mathsf {T}}x,x\in \mathbb {R} ^{p},t\in \mathbb {R} ^{L},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18a022988b5ca6e8c1834e7f2c1d0aa87b5a376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.459ex; height:3.343ex;" alt="{\displaystyle t=W_{L}^{\mathsf {T}}x,x\in \mathbb {R} ^{p},t\in \mathbb {R} ^{L},}"></span> where the columns of <span class="texhtml"><i>p</i> × <i>L</i></span> matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e379c3906bbeab443aad12d02dafe33d679c38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.545ex; height:2.509ex;" alt="{\displaystyle W_{L}}"></span> form an orthogonal basis for the <i>L</i> features (the components of representation <i>t</i>) that are decorrelated.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> By construction, of all the transformed data matrices with only <i>L</i> columns, this score matrix maximises the variance in the original data that has been preserved, while minimising the total squared reconstruction error <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {T} \mathbf {W} ^{T}-\mathbf {T} _{L}\mathbf {W} _{L}^{T}\|_{2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {T} \mathbf {W} ^{T}-\mathbf {T} _{L}\mathbf {W} _{L}^{T}\|_{2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/406fd6d90d1959d718164d5389b4b23b33776563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.593ex; height:3.343ex;" alt="{\displaystyle \|\mathbf {T} \mathbf {W} ^{T}-\mathbf {T} _{L}\mathbf {W} _{L}^{T}\|_{2}^{2}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {X} -\mathbf {X} _{L}\|_{2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {X} -\mathbf {X} _{L}\|_{2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f860f93f74a49b5330042de306f661584aeb2ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.61ex; height:3.176ex;" alt="{\displaystyle \|\mathbf {X} -\mathbf {X} _{L}\|_{2}^{2}}"></span>. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PCA_of_Haplogroup_J_using_37_STRs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/PCA_of_Haplogroup_J_using_37_STRs.png/220px-PCA_of_Haplogroup_J_using_37_STRs.png" decoding="async" width="220" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/PCA_of_Haplogroup_J_using_37_STRs.png/330px-PCA_of_Haplogroup_J_using_37_STRs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/PCA_of_Haplogroup_J_using_37_STRs.png/440px-PCA_of_Haplogroup_J_using_37_STRs.png 2x" data-file-width="683" data-file-height="564" /></a><figcaption>A principal components analysis scatterplot of <a href="/wiki/Y-STR" title="Y-STR">Y-STR</a> <a href="/wiki/Haplotype" title="Haplotype">haplotypes</a> calculated from repeat-count values for 37 Y-chromosomal STR markers from 354 individuals.<br /> PCA has successfully found linear combinations of the markers that separate out different clusters corresponding to different lines of individuals' Y-chromosomal genetic descent.</figcaption></figure> <p>Such <a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">dimensionality reduction</a> can be a very useful step for visualising and processing high-dimensional datasets, while still retaining as much of the variance in the dataset as possible. For example, selecting <i>L</i> = 2 and keeping only the first two principal components finds the two-dimensional plane through the high-dimensional dataset in which the data is most spread out, so if the data contains <a href="/wiki/Cluster_analysis" title="Cluster analysis">clusters</a> these too may be most spread out, and therefore most visible to be plotted out in a two-dimensional diagram; whereas if two directions through the data (or two of the original variables) are chosen at random, the clusters may be much less spread apart from each other, and may in fact be much more likely to substantially overlay each other, making them indistinguishable. </p><p>Similarly, in <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>, the larger the number of <a href="/wiki/Explanatory_variable" class="mw-redirect" title="Explanatory variable">explanatory variables</a> allowed, the greater is the chance of <a href="/wiki/Overfitting" title="Overfitting">overfitting</a> the model, producing conclusions that fail to generalise to other datasets. One approach, especially when there are strong correlations between different possible explanatory variables, is to reduce them to a few principal components and then run the regression against them, a method called <a href="/wiki/Principal_component_regression" title="Principal component regression">principal component regression</a>. </p><p>Dimensionality reduction may also be appropriate when the variables in a dataset are noisy. If each column of the dataset contains independent identically distributed Gaussian noise, then the columns of <b>T</b> will also contain similarly identically distributed Gaussian noise (such a distribution is invariant under the effects of the matrix <b>W</b>, which can be thought of as a high-dimensional rotation of the co-ordinate axes). However, with more of the total variance concentrated in the first few principal components compared to the same noise variance, the proportionate effect of the noise is less—the first few components achieve a higher <a href="/wiki/Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a>. PCA thus can have the effect of concentrating much of the signal into the first few principal components, which can usefully be captured by dimensionality reduction; while the later principal components may be dominated by noise, and so disposed of without great loss. If the dataset is not too large, the significance of the principal components can be tested using <a href="/wiki/Bootstrapping_(statistics)#Parametric_bootstrap" title="Bootstrapping (statistics)">parametric bootstrap</a>, as an aid in determining how many principal components to retain.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Singular_value_decomposition">Singular value decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=9" title="Edit section: Singular value decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></div> <p>The principal components transformation can also be associated with another matrix factorization, the <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> (SVD) of <b>X</b>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c985d509782890595e6d63d4436e87af80b6fec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.258ex; height:2.676ex;" alt="{\displaystyle \mathbf {X} =\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{T}}"></span></dd></dl> <p>Here <b>Σ</b> is an <i>n</i>-by-<i>p</i> <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">rectangular diagonal matrix</a> of positive numbers <i>σ</i><sub>(<i>k</i>)</sub>, called the singular values of <b>X</b>; <b>U</b> is an <i>n</i>-by-<i>n</i> matrix, the columns of which are orthogonal unit vectors of length <i>n</i> called the left singular vectors of <b>X</b>; and <b>W</b> is a <i>p</i>-by-<i>p</i> matrix whose columns are orthogonal unit vectors of length <i>p</i> and called the right singular vectors of <b>X</b>. </p><p>In terms of this factorization, the matrix <b>X</b><sup>T</sup><b>X</b> can be written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {X} ^{T}\mathbf {X} &=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {U} ^{\mathsf {T}}\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\hat {\Sigma }} ^{2}\mathbf {W} ^{\mathsf {T}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">Σ</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {X} ^{T}\mathbf {X} &=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {U} ^{\mathsf {T}}\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\hat {\Sigma }} ^{2}\mathbf {W} ^{\mathsf {T}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0686f238c9dc8c2afd70ad1034c81e7de06439c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:26.834ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {X} ^{T}\mathbf {X} &=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {U} ^{\mathsf {T}}\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\hat {\Sigma }} ^{2}\mathbf {W} ^{\mathsf {T}}\end{aligned}}}"></span></dd></dl> <p>where <b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {\Sigma }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">Σ</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {\Sigma }} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48a9ce1fa79f846accfe03843fc72e118a2559da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {\Sigma }} }"></span></b> is the square diagonal matrix with the singular values of <b>X </b>and the excess zeros chopped off that satisfies<b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {{\hat {\Sigma }}^{2}} =\mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">Σ</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {{\hat {\Sigma }}^{2}} =\mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed16bfaee8a9f431165715449985d1b1ef627b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.421ex; height:3.343ex;" alt="{\displaystyle \mathbf {{\hat {\Sigma }}^{2}} =\mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } }"></span></b>. Comparison with the eigenvector factorization of <b>X</b><sup>T</sup><b>X</b> establishes that the right singular vectors <b>W</b> of <b>X</b> are equivalent to the eigenvectors of <b>X</b><sup>T</sup><b>X</b>, while the singular values <i>σ</i><sub>(<i>k</i>)</sub> of <b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }"></span></b> are equal to the square-root of the eigenvalues <i>λ</i><sub>(<i>k</i>)</sub> of <b>X</b><sup>T</sup><b>X</b>. </p><p>Using the singular value decomposition the score matrix <b>T</b> can be written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {T} &=\mathbf {X} \mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {T} &=\mathbf {X} \mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b452ad843af2b8001a3c2a18bc4725d26227c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:16.574ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {T} &=\mathbf {X} \mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \end{aligned}}}"></span></dd></dl> <p>so each column of <b>T</b> is given by one of the left singular vectors of <b>X</b> multiplied by the corresponding singular value. This form is also the <a href="/wiki/Polar_decomposition" title="Polar decomposition">polar decomposition</a> of <b>T</b>. </p><p>Efficient algorithms exist to calculate the SVD of <b>X</b> without having to form the matrix <b>X</b><sup>T</sup><b>X</b>, so computing the SVD is now the standard way to calculate a principal components analysis from a data matrix,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> unless only a handful of components are required. </p><p>As with the eigen-decomposition, a truncated <span class="texhtml"><i>n</i> × <i>L</i></span> score matrix <b>T</b><sub>L</sub> can be obtained by considering only the first L largest singular values and their singular vectors: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} _{L}=\mathbf {U} _{L}\mathbf {\Sigma } _{L}=\mathbf {X} \mathbf {W} _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} _{L}=\mathbf {U} _{L}\mathbf {\Sigma } _{L}=\mathbf {X} \mathbf {W} _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/171ae4f48fd4107aadc315011429a3e5959046ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.232ex; height:2.509ex;" alt="{\displaystyle \mathbf {T} _{L}=\mathbf {U} _{L}\mathbf {\Sigma } _{L}=\mathbf {X} \mathbf {W} _{L}}"></span></dd></dl> <p>The truncation of a matrix <b>M</b> or <b>T</b> using a truncated singular value decomposition in this way produces a truncated matrix that is the nearest possible matrix of <a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> <i>L</i> to the original matrix, in the sense of the difference between the two having the smallest possible <a href="/wiki/Frobenius_norm" class="mw-redirect" title="Frobenius norm">Frobenius norm</a>, a result known as the <a href="/wiki/Low-rank_approximation#Proof_of_Eckart–Young–Mirsky_theorem_(for_Frobenius_norm)" title="Low-rank approximation">Eckart–Young theorem</a> [1936]. </p> <div class="mw-heading mw-heading2"><h2 id="Further_considerations">Further considerations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=10" title="Edit section: Further considerations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The singular values (in <b>Σ</b>) are the square roots of the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the matrix <b>X</b><sup>T</sup><b>X</b>. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is associated with each eigenvector. The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean. PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions. The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information (see <a href="/wiki/Principle_Component_Analysis#PCA_and_information_theory" class="mw-redirect" title="Principle Component Analysis">below</a>). PCA is often used in this manner for <a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">dimensionality reduction</a>. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" (as defined above). This advantage, however, comes at the price of greater computational requirements if compared, for example, and when applicable, to the <a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">discrete cosine transform</a>, and in particular to the DCT-II which is simply known as the "DCT". <a href="/wiki/Nonlinear_dimensionality_reduction" title="Nonlinear dimensionality reduction">Nonlinear dimensionality reduction</a> techniques tend to be more computationally demanding than PCA. </p><p>PCA is sensitive to the scaling of the variables. If we have just two variables and they have the same <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a> and are completely correlated, then the PCA will entail a rotation by 45° and the "weights" (they are the cosines of rotation) for the two variables with respect to the principal component will be equal. But if we multiply all values of the first variable by 100, then the first principal component will be almost the same as that variable, with a small contribution from the other variable, whereas the second component will be almost aligned with the second original variable. This means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis. (Different results would be obtained if one used Fahrenheit rather than Celsius for example.) Pearson's original paper was entitled "On Lines and Planes of Closest Fit to Systems of Points in Space" – "in space" implies physical Euclidean space where such concerns do not arise. One way of making the PCA less arbitrary is to use variables scaled so as to have unit variance, by standardizing the data and hence use the autocorrelation matrix instead of the autocovariance matrix as a basis for PCA. However, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance. </p><p>Mean subtraction (a.k.a. "mean centering") is necessary for performing classical PCA to ensure that the first principal component describes the direction of maximum variance. If mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the data. A mean of zero is needed for finding a basis that minimizes the <a href="/wiki/Minimum_mean_square_error" title="Minimum mean square error">mean square error</a> of the approximation of the data.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Mean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations. Correlations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: <i>Pearson Product-Moment Correlation</i>). Also see the article by Kromrey & Foster-Johnson (1998) on <i>"Mean-centering in Moderated Regression: Much Ado About Nothing"</i>. Since <a href="/wiki/Covariance_matrix#Relation_to_the_correlation_matrix" title="Covariance matrix">covariances are correlations of normalized variables</a> (<a href="/wiki/Standard_score#Calculation" title="Standard score">Z- or standard-scores</a>) a PCA based on the correlation matrix of <b>X</b> is <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equal</a> to a PCA based on the covariance matrix of <b>Z</b>, the standardized version of <b>X</b>. </p><p>PCA is a popular primary technique in <a href="/wiki/Pattern_recognition" title="Pattern recognition">pattern recognition</a>. It is not, however, optimized for class separability.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> However, it has been used to quantify the distance between two or more classes by calculating center of mass for each class in principal component space and reporting Euclidean distance between center of mass of two or more classes.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">linear discriminant analysis</a> is an alternative which is optimized for class separability. </p> <div class="mw-heading mw-heading2"><h2 id="Table_of_symbols_and_abbreviations">Table of symbols and abbreviations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=11" title="Edit section: Table of symbols and abbreviations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th>Symbol </th> <th>Meaning </th> <th>Dimensions </th> <th>Indices </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =[X_{ij}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =[X_{ij}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4daaf2305609bbd2142de16bbb6cb774218d22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.813ex; height:3.009ex;" alt="{\displaystyle \mathbf {X} =[X_{ij}]}"></span> </td> <td>data matrix, consisting of the set of all data vectors, one vector per row </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1\ldots n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1\ldots n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802b10ee4c4a608cfb603b9da10e1e105888dae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.956ex; height:2.176ex;" alt="{\displaystyle i=1\ldots n}"></span> <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> </td> <td>the number of row vectors in the data set </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4bf91a527dc01af9ef6ace81199becf1308e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}"></span> </td> <td><i>scalar</i> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> </td> <td>the number of elements in each row vector (dimension) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4bf91a527dc01af9ef6ace81199becf1308e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}"></span> </td> <td><i>scalar</i> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> </td> <td>the number of dimensions in the dimensionally reduced subspace, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq L\leq p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>L</mi> <mo>≤</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq L\leq p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9301ddade3fc9beed0e13f943862f3bda085e655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.112ex; height:2.509ex;" alt="{\displaystyle 1\leq L\leq p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4bf91a527dc01af9ef6ace81199becf1308e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}"></span> </td> <td><i>scalar</i> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =[u_{j}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =[u_{j}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95df093fae62b2ab6dfa4527c69d1cbd3b9d553f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.117ex; height:3.009ex;" alt="{\displaystyle \mathbf {u} =[u_{j}]}"></span> </td> <td>vector of empirical <a href="/wiki/Mean" title="Mean">means</a>, one mean for each column <i>j</i> of the data matrix </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b3ff9128b8bc9ccf1c3b9a3ba1d253b95f5754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p\times 1}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} =[s_{j}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} =[s_{j}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf19191b22b96e131d03b5b3d1ce7dde1cc877ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.448ex; height:3.009ex;" alt="{\displaystyle \mathbf {s} =[s_{j}]}"></span> </td> <td>vector of empirical <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviations</a>, one standard deviation for each column <i>j</i> of the data matrix </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b3ff9128b8bc9ccf1c3b9a3ba1d253b95f5754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p\times 1}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {h} =[h_{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {h} =[h_{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ef2db0dd2d098b4893574eca8b44d04fde0653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.016ex; height:2.843ex;" alt="{\displaystyle \mathbf {h} =[h_{i}]}"></span> </td> <td>vector of all 1's </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bce5f6a6d0d32834484048c16f3b39f9c23d076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle 1\times n}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1\ldots n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1\ldots n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802b10ee4c4a608cfb603b9da10e1e105888dae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.956ex; height:2.176ex;" alt="{\displaystyle i=1\ldots n}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =[B_{ij}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =[B_{ij}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8b8dd810252420486677821841d9c24c7cf58d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.534ex; height:3.009ex;" alt="{\displaystyle \mathbf {B} =[B_{ij}]}"></span> </td> <td><a href="/wiki/Standard_deviation" title="Standard deviation">deviations</a> from the mean of each column <i>j</i> of the data matrix </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1\ldots n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1\ldots n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802b10ee4c4a608cfb603b9da10e1e105888dae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.956ex; height:2.176ex;" alt="{\displaystyle i=1\ldots n}"></span> <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} =[Z_{ij}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} =[Z_{ij}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/924bacc6ac643a5165b78f62aa36ec85f3e94cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.091ex; height:3.009ex;" alt="{\displaystyle \mathbf {Z} =[Z_{ij}]}"></span> </td> <td><a href="/wiki/Z-score" class="mw-redirect" title="Z-score">z-scores</a>, computed using the mean and standard deviation for each column <i>j</i> of the data matrix </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1\ldots n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1\ldots n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802b10ee4c4a608cfb603b9da10e1e105888dae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.956ex; height:2.176ex;" alt="{\displaystyle i=1\ldots n}"></span> <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} =[C_{jj'}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <msup> <mi>j</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} =[C_{jj'}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1652c6b60c22140c190acfa8f49084ccc4ca8f9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.104ex; height:3.176ex;" alt="{\displaystyle \mathbf {C} =[C_{jj'}]}"></span> </td> <td><a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e6de741a3aa8176f5a487de8e8f602aa75c5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.269ex; height:2.009ex;" alt="{\displaystyle p\times p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j'=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j'=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4c3648460a685356d8ba2c78d128463102a5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.597ex; height:2.843ex;" alt="{\displaystyle j'=1\ldots p}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} =[R_{jj'}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <msup> <mi>j</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} =[R_{jj'}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecc703ef34aa1c05d4b86462435d9eef3dbadde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.278ex; height:3.176ex;" alt="{\displaystyle \mathbf {R} =[R_{jj'}]}"></span> </td> <td><a href="/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">correlation matrix</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e6de741a3aa8176f5a487de8e8f602aa75c5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.269ex; height:2.009ex;" alt="{\displaystyle p\times p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j'=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j'=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4c3648460a685356d8ba2c78d128463102a5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.597ex; height:2.843ex;" alt="{\displaystyle j'=1\ldots p}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} =[V_{jj'}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <msup> <mi>j</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} =[V_{jj'}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebedde6d7a291b8a304447f9467f85c6325d4755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.886ex; height:3.176ex;" alt="{\displaystyle \mathbf {V} =[V_{jj'}]}"></span> </td> <td>matrix consisting of the set of all <a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvectors</a> of <b>C</b>, one eigenvector per column </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e6de741a3aa8176f5a487de8e8f602aa75c5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.269ex; height:2.009ex;" alt="{\displaystyle p\times p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j'=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j'=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4c3648460a685356d8ba2c78d128463102a5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.597ex; height:2.843ex;" alt="{\displaystyle j'=1\ldots p}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} =[D_{jj'}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <msup> <mi>j</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} =[D_{jj'}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a835ec95447679384c75da973ebdb23fcb0d56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.485ex; height:3.176ex;" alt="{\displaystyle \mathbf {D} =[D_{jj'}]}"></span> </td> <td><a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> consisting of the set of all <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> of <b>C</b> along its <a href="/wiki/Principal_diagonal" class="mw-redirect" title="Principal diagonal">principal diagonal</a>, and 0 for all other elements ( note <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Lambda } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Λ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\Lambda } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baacea0858757143db3b530f015a72d1224b44fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.873ex; height:2.176ex;" alt="{\displaystyle \mathbf {\Lambda } }"></span> used above ) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e6de741a3aa8176f5a487de8e8f602aa75c5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.269ex; height:2.009ex;" alt="{\displaystyle p\times p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j'=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j'=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4c3648460a685356d8ba2c78d128463102a5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.597ex; height:2.843ex;" alt="{\displaystyle j'=1\ldots p}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} =[W_{jl}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {W} =[W_{jl}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c3a7bd748ee5e1b0ebbddc41e61fa6d16d26a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.748ex; height:3.009ex;" alt="{\displaystyle \mathbf {W} =[W_{jl}]}"></span> </td> <td>matrix of basis vectors, one vector per column, where each basis vector is one of the eigenvectors of <b>C</b>, and where the vectors in <b>W</b> are a sub-set of those in <b>V</b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/421f277e37454673b290462b1a4d7c83e6a6da1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.682ex; height:2.509ex;" alt="{\displaystyle p\times L}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\ldots p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\ldots p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63773ee6dfc3225a40703460419a739da8966cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.913ex; height:2.509ex;" alt="{\displaystyle j=1\ldots p}"></span> <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=1\ldots L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=1\ldots L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc810a89f47bd6745a75d2e7d4e1288383d2130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.034ex; height:2.176ex;" alt="{\displaystyle l=1\ldots L}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} =[T_{il}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =[T_{il}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684b2536b0cdea539fc13d7d4b92b40557727609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.899ex; height:2.843ex;" alt="{\displaystyle \mathbf {T} =[T_{il}]}"></span> </td> <td>matrix consisting of <i>n</i> row vectors, where each vector is the projection of the corresponding data vector from matrix <b>X</b> onto the basis vectors contained in the columns of matrix <b>W</b>. </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f977ed3af3b9fdd1df56ca880201dc55257f5487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.818ex; height:2.176ex;" alt="{\displaystyle n\times L}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1\ldots n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1\ldots n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802b10ee4c4a608cfb603b9da10e1e105888dae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.956ex; height:2.176ex;" alt="{\displaystyle i=1\ldots n}"></span> <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=1\ldots L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=1\ldots L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc810a89f47bd6745a75d2e7d4e1288383d2130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.034ex; height:2.176ex;" alt="{\displaystyle l=1\ldots L}"></span> </td></tr></tbody></table> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Properties_and_limitations">Properties and limitations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=12" title="Edit section: Properties and limitations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=13" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some properties of PCA include:<sup id="cite_ref-Jolliffe2002_13-3" class="reference"><a href="#cite_note-Jolliffe2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2020)">page needed</span></a></i>]</sup> </p> <dl><dd><big><b><i>Property 1</i>:</b></big> For any integer <i>q</i>, 1 ≤ <i>q</i> ≤ <i>p</i>, consider the orthogonal <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\mathbf {B'} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mo>′</mo> </msup> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\mathbf {B'} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f96b2fc01e4498d0032535ba4091a530e90fb09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.169ex; height:2.843ex;" alt="{\displaystyle y=\mathbf {B'} x}"></span></dd></dl></dd> <dd>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> is a <i>q-element</i> vector and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B'} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B'} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3a60af7af9565be68f1da12a7591fbbb5bd1c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.586ex; height:2.509ex;" alt="{\displaystyle \mathbf {B'} }"></span> is a (<i>q</i> × <i>p</i>) matrix, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Sigma } _{y}=\mathbf {B'} \mathbf {\Sigma } \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\Sigma } _{y}=\mathbf {B'} \mathbf {\Sigma } \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c18c25c286a8202ce8040b828329e3b753371a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.497ex; height:3.176ex;" alt="{\displaystyle \mathbf {\Sigma } _{y}=\mathbf {B'} \mathbf {\Sigma } \mathbf {B} }"></span> be the <a href="/wiki/Variance" title="Variance">variance</a>-<a href="/wiki/Covariance" title="Covariance">covariance</a> matrix for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. Then the trace of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Sigma } _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\Sigma } _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ede5f608c1c218c37ba30ff7cc0f704f22a2b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.981ex; height:2.843ex;" alt="{\displaystyle \mathbf {\Sigma } _{y}}"></span>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f637033e1a3054767f851664810ce8867818dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.606ex; height:3.009ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})}"></span>, is maximized by taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mathbf {A} _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mathbf {A} _{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd6de918583ffbdbda469ad0cfbe42b07af91de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.008ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} =\mathbf {A} _{q}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170a6dcb5a8231b8c462f7661c857ed16a900fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.008ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} _{q}}"></span> consists of the first <i>q</i> columns of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {B'} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {B'} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9063631d2aa2609825e9d04c4219782348598531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.49ex; height:3.009ex;" alt="{\displaystyle (\mathbf {B'} }"></span> is the transpose of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14499b13268e357926fbde72cf8f2461250c0575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.806ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} )}"></span>. (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> is not defined here)</dd></dl> <dl><dd><big><b><i>Property 2</i>:</b></big> Consider again the <a href="/wiki/Orthonormal_transformation" class="mw-redirect" title="Orthonormal transformation">orthonormal transformation</a> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\mathbf {B'} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mo>′</mo> </msup> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\mathbf {B'} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f96b2fc01e4498d0032535ba4091a530e90fb09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.169ex; height:2.843ex;" alt="{\displaystyle y=\mathbf {B'} x}"></span></dd></dl></dd> <dd>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,\mathbf {B} ,\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,\mathbf {B} ,\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1af8f1709ae8cd3acd8606ca6c1b228d87fb6664" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.318ex; height:2.509ex;" alt="{\displaystyle x,\mathbf {B} ,\mathbf {A} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Sigma } _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\Sigma } _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ede5f608c1c218c37ba30ff7cc0f704f22a2b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.981ex; height:2.843ex;" alt="{\displaystyle \mathbf {\Sigma } _{y}}"></span> defined as before. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f637033e1a3054767f851664810ce8867818dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.606ex; height:3.009ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})}"></span> is minimized by taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mathbf {A} _{q}^{*},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mathbf {A} _{q}^{*},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d72c7a3a60b0970312df8883cea821722f3c9fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.72ex; height:3.176ex;" alt="{\displaystyle \mathbf {B} =\mathbf {A} _{q}^{*},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} _{q}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{q}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f519c1a96c1998906f7b9eef0de96c38a8531a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.074ex; height:3.176ex;" alt="{\displaystyle \mathbf {A} _{q}^{*}}"></span> consists of the last <i>q</i> columns of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>.</dd></dl> <p>The statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs. Because these last PCs have variances as small as possible they are useful in their own right. They can help to detect unsuspected near-constant linear relationships between the elements of <span class="texhtml mvar" style="font-style:italic;">x</span>, and they may also be useful in <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a>, in selecting a subset of variables from <span class="texhtml mvar" style="font-style:italic;">x</span>, and in outlier detection. </p> <dl><dd><big><b><i>Property 3</i>:</b></big> (Spectral decomposition of <span class="texhtml"><b>Σ</b></span>) <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Sigma } =\lambda _{1}\alpha _{1}\alpha _{1}'+\cdots +\lambda _{p}\alpha _{p}\alpha _{p}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\Sigma } =\lambda _{1}\alpha _{1}\alpha _{1}'+\cdots +\lambda _{p}\alpha _{p}\alpha _{p}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fafe15138fc3db5a368d45e0873e984090256aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.435ex; height:2.843ex;" alt="{\displaystyle \mathbf {\Sigma } =\lambda _{1}\alpha _{1}\alpha _{1}'+\cdots +\lambda _{p}\alpha _{p}\alpha _{p}'}"></span></dd></dl></dd></dl> <p>Before we look at its usage, we first look at <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> elements, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} (x_{j})=\sum _{k=1}^{P}\lambda _{k}\alpha _{kj}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (x_{j})=\sum _{k=1}^{P}\lambda _{k}\alpha _{kj}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cac51884981f8c2ef1f57e28d939936f931112c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.404ex; height:7.343ex;" alt="{\displaystyle \operatorname {Var} (x_{j})=\sum _{k=1}^{P}\lambda _{k}\alpha _{kj}^{2}}"></span></dd></dl> <p>Then, perhaps the main statistical implication of the result is that not only can we decompose the combined variances of all the elements of <span class="texhtml mvar" style="font-style:italic;">x</span> into decreasing contributions due to each PC, but we can also decompose the whole <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> into contributions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b9118a2c1615a64c1809fa49d69bc6a41a7196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.597ex; height:2.843ex;" alt="{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}"></span> from each PC. Although not strictly decreasing, the elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b9118a2c1615a64c1809fa49d69bc6a41a7196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.597ex; height:2.843ex;" alt="{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}"></span> will tend to become smaller as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> increases, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b9118a2c1615a64c1809fa49d69bc6a41a7196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.597ex; height:2.843ex;" alt="{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}"></span> is nonincreasing for increasing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>, whereas the elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717fd9a74d91add9739563c16ac357ce33924860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.009ex;" alt="{\displaystyle \alpha _{k}}"></span> tend to stay about the same size because of the normalization constraints: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{k}'\alpha _{k}=1,k=1,\dots ,p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{k}'\alpha _{k}=1,k=1,\dots ,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15ca48f9704ceb90bdf489d3afb28ae1abb28a3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.267ex; height:2.843ex;" alt="{\displaystyle \alpha _{k}'\alpha _{k}=1,k=1,\dots ,p}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Limitations">Limitations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=14" title="Edit section: Limitations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As noted above, the results of PCA depend on the scaling of the variables. This can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.<sup id="cite_ref-Leznik_20-0" class="reference"><a href="#cite_note-Leznik-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The applicability of PCA as described above is limited by certain (tacit) assumptions<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> made in its derivation. In particular, PCA can capture linear correlations between the features but fails when this assumption is violated (see Figure 6a in the reference). In some cases, coordinate transformations can restore the linearity assumption and PCA can then be applied (see <a href="/wiki/Kernel_principal_component_analysis" title="Kernel principal component analysis">kernel PCA</a>). </p><p>Another limitation is the mean-removal process before constructing the covariance matrix for PCA. In fields such as astronomy, all the signals are non-negative, and the mean-removal process will force the mean of some astrophysical exposures to be zero, which consequently creates unphysical negative fluxes,<sup id="cite_ref-soummer12_22-0" class="reference"><a href="#cite_note-soummer12-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> and forward modeling has to be performed to recover the true magnitude of the signals.<sup id="cite_ref-pueyo16_23-0" class="reference"><a href="#cite_note-pueyo16-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> As an alternative method, <a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">non-negative matrix factorization</a> focusing only on the non-negative elements in the matrices is well-suited for astrophysical observations.<sup id="cite_ref-blantonRoweis07_24-0" class="reference"><a href="#cite_note-blantonRoweis07-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-zhu16_25-0" class="reference"><a href="#cite_note-zhu16-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ren18_26-0" class="reference"><a href="#cite_note-ren18-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> See more at <a href="#Non-negative_matrix_factorization">the relation between PCA and non-negative matrix factorization</a>. </p><p>PCA is at a disadvantage if the data has not been standardized before applying the algorithm to it. PCA transforms the original data into data that is relevant to the principal components of that data, which means that the new data variables cannot be interpreted in the same ways that the originals were. They are linear interpretations of the original variables. Also, if PCA is not performed properly, there is a high likelihood of information loss.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>PCA relies on a linear model. If a dataset has a pattern hidden inside it that is nonlinear, then PCA can actually steer the analysis in the complete opposite direction of progress.<sup id="cite_ref-abbott_28-0" class="reference"><a href="#cite_note-abbott-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (June 2021)">page needed</span></a></i>]</sup> Researchers at Kansas State University discovered that the sampling error in their experiments impacted the bias of PCA results. "If the number of subjects or blocks is smaller than 30, and/or the researcher is interested in PC's beyond the first, it may be better to first correct for the serial correlation, before PCA is conducted".<sup id="cite_ref-jiang_29-0" class="reference"><a href="#cite_note-jiang-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The researchers at Kansas State also found that PCA could be "seriously biased if the autocorrelation structure of the data is not correctly handled".<sup id="cite_ref-jiang_29-1" class="reference"><a href="#cite_note-jiang-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="PCA_and_information_theory">PCA and information theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=15" title="Edit section: PCA and information theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dimensionality reduction results in a loss of information, in general. PCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models. </p><p>Under the assumption that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\mathbf {s} +\mathbf {n} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\mathbf {s} +\mathbf {n} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e208576cdd9184ecfe654a75e6aae8ea760a250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.538ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} =\mathbf {s} +\mathbf {n} ,}"></span></dd></dl> <p>that is, that the data vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span> is the sum of the desired information-bearing signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644ae690160e658898a141e568a7fb0ee6040004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }"></span> and a noise signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"></span> one can show that PCA can be optimal for dimensionality reduction, from an information-theoretic point-of-view. </p><p>In particular, Linsker showed that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644ae690160e658898a141e568a7fb0ee6040004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }"></span> is Gaussian and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"></span> is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the <a href="/wiki/Mutual_information" title="Mutual information">mutual information</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\mathbf {y} ;\mathbf {s} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\mathbf {y} ;\mathbf {s} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40eb1146cefd2cd629f256a63ce28a61f7f862d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.482ex; height:2.843ex;" alt="{\displaystyle I(\mathbf {y} ;\mathbf {s} )}"></span> between the desired information <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644ae690160e658898a141e568a7fb0ee6040004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }"></span> and the dimensionality-reduced output <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} =\mathbf {W} _{L}^{T}\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =\mathbf {W} _{L}^{T}\mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad07884737ff1501a0136652d2bc80f33ab50eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.072ex; height:3.176ex;" alt="{\displaystyle \mathbf {y} =\mathbf {W} _{L}^{T}\mathbf {x} }"></span>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (that is, the components of the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"></span> are <a href="/wiki/Iid" class="mw-redirect" title="Iid">iid</a>), but the information-bearing signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644ae690160e658898a141e568a7fb0ee6040004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }"></span> is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the <i>information loss</i>, which is defined as<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\mathbf {x} ;\mathbf {s} )-I(\mathbf {y} ;\mathbf {s} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\mathbf {x} ;\mathbf {s} )-I(\mathbf {y} ;\mathbf {s} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f386141449ea7b1836a7fbe2fbd49609783effdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.45ex; height:2.843ex;" alt="{\displaystyle I(\mathbf {x} ;\mathbf {s} )-I(\mathbf {y} ;\mathbf {s} ).}"></span></dd></dl> <p>The optimality of PCA is also preserved if the noise <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"></span> is iid and at least more Gaussian (in terms of the <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a>) than the information-bearing signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644ae690160e658898a141e568a7fb0ee6040004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }"></span>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"></span> becomes dependent. </p> <div class="mw-heading mw-heading2"><h2 id="Computation_using_the_covariance_method">Computation using the covariance method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=16" title="Edit section: Computation using the covariance method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following is a detailed description of PCA using the covariance method<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> as opposed to the correlation method.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>The goal is to transform a given data set <b>X</b> of dimension <i>p</i> to an alternative data set <b>Y</b> of smaller dimension <i>L</i>. Equivalently, we are seeking to find the matrix <b>Y</b>, where <b>Y</b> is the <a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève</a> transform (KLT) of matrix <b>X</b>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Y} =\mathbb {KLT} \{\mathbf {X} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> <mi mathvariant="double-struck">L</mi> <mi mathvariant="double-struck">T</mi> </mrow> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} =\mathbb {KLT} \{\mathbf {X} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62cbcf4ec22d2863e88f770e407c552409c93741" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.371ex; height:2.843ex;" alt="{\displaystyle \mathbf {Y} =\mathbb {KLT} \{\mathbf {X} \}}"></span> </p> <ol> <li> <b>Organize the data set</b> <div class="paragraphbreak" style="margin-top:0.5em"></div> <p>Suppose you have data comprising a set of observations of <i>p</i> variables, and you want to reduce the data so that each observation can be described with only <i>L</i> variables, <i>L</i> < <i>p</i>. Suppose further, that the data are arranged as a set of <i>n</i> data vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf310f687fd3d2303a5ce91900eb01a9ddd41ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.592ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}}"></span> with each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d2ef3df60acdb53bdf90535264041fea7231cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{i}}"></span> representing a single grouped observation of the <i>p</i> variables. </p> <ul><li>Write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf310f687fd3d2303a5ce91900eb01a9ddd41ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.592ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}}"></span> as row vectors, each with <i>p</i> elements.</li> <li>Place the row vectors into a single matrix <b>X</b> of dimensions <i>n</i> × <i>p</i>.</li></ul> </li> <li> <b>Calculate the empirical mean</b> <ul><li>Find the empirical mean along each column <i>j</i> = 1, ..., <i>p</i>.</li> <li>Place the calculated mean values into an empirical mean vector <b>u</b> of dimensions <i>p</i> × 1. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{j}={\frac {1}{n}}\sum _{i=1}^{n}X_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{j}={\frac {1}{n}}\sum _{i=1}^{n}X_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86017c79a11d86c94815e74601cdb2d8cc3327e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.099ex; height:6.843ex;" alt="{\displaystyle u_{j}={\frac {1}{n}}\sum _{i=1}^{n}X_{ij}}"></span></li></ul> </li> <li> <b>Calculate the deviations from the mean</b> <div class="paragraphbreak" style="margin-top:0.5em"></div> <p>Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> Hence we proceed by centering the data as follows: </p> <ul><li>Subtract the empirical mean vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b166adee74f8c332c8697a3e03cb1f01e081a3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.875ex; height:2.676ex;" alt="{\displaystyle \mathbf {u} ^{T}}"></span> from each row of the data matrix <b>X</b>.</li> <li>Store mean-subtracted data in the <i>n</i> × <i>p</i> matrix <b>B</b>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mathbf {X} -\mathbf {h} \mathbf {u} ^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mathbf {X} -\mathbf {h} \mathbf {u} ^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2411abb22dee44c3572f57200531b7b0efd3cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.219ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} =\mathbf {X} -\mathbf {h} \mathbf {u} ^{T}}"></span> where <b>h</b> is an <span class="texhtml"><i>n</i> × 1</span> column vector of all 1s: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{i}=1\,\qquad \qquad {\text{for }}i=1,\ldots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{i}=1\,\qquad \qquad {\text{for }}i=1,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a3b4e3b9b4f235f95214c20a02db301b0c4ad0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.08ex; height:2.509ex;" alt="{\displaystyle h_{i}=1\,\qquad \qquad {\text{for }}i=1,\ldots ,n}"></span></li></ul> <p>In some applications, each variable (column of <b>B</b>) may also be scaled to have a variance equal to 1 (see <a href="/wiki/Z-score" class="mw-redirect" title="Z-score">Z-score</a>).<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> This step affects the calculated principal components, but makes them independent of the units used to measure the different variables. </p> </li> <li> <b>Find the covariance matrix</b> <ul><li>Find the <i>p</i> × <i>p</i> empirical <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> <b>C</b> from matrix <b>B</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} ={1 \over {n-1}}\mathbf {B} ^{*}\mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} ={1 \over {n-1}}\mathbf {B} ^{*}\mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/345032375bfceb3d3c2c72bbf98aa9bedab49ea5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.12ex; height:5.343ex;" alt="{\displaystyle \mathbf {C} ={1 \over {n-1}}\mathbf {B} ^{*}\mathbf {B} }"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></span> is the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> operator. If <b>B</b> consists entirely of real numbers, which is the case in many applications, the "conjugate transpose" is the same as the regular <a href="/wiki/Transpose" title="Transpose">transpose</a>.</li> <li>The reasoning behind using <span class="texhtml"><i>n</i> − 1</span> instead of <i>n</i> to calculate the covariance is <a href="/wiki/Bessel%27s_correction" title="Bessel's correction">Bessel's correction</a>.</li></ul> </li> <li> <b>Find the eigenvectors and eigenvalues of the covariance matrix</b> <ul><li>Compute the matrix <b>V</b> of <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> which <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizes</a> the covariance matrix <b>C</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} ^{-1}\mathbf {C} \mathbf {V} =\mathbf {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} ^{-1}\mathbf {C} \mathbf {V} =\mathbf {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cebe257c2bac0dcce240b9f6460513d0e2153f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.451ex; height:2.676ex;" alt="{\displaystyle \mathbf {V} ^{-1}\mathbf {C} \mathbf {V} =\mathbf {D} }"></span> where <b>D</b> is the <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> of <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of <b>C</b>. This step will typically involve the use of a computer-based algorithm for <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">computing eigenvectors and eigenvalues</a>. These algorithms are readily available as sub-components of most <a href="/wiki/Matrix_algebra" class="mw-redirect" title="Matrix algebra">matrix algebra</a> systems, such as <a href="/wiki/SAS_(software)" title="SAS (software)">SAS</a>,<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>, <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a>,<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>,<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> <a href="/wiki/SciPy" title="SciPy">SciPy</a>, <a href="/wiki/IDL_(programming_language)" title="IDL (programming language)">IDL</a> (<a href="/wiki/Interactive_Data_Language" class="mw-redirect" title="Interactive Data Language">Interactive Data Language</a>), or <a href="/wiki/GNU_Octave" title="GNU Octave">GNU Octave</a> as well as <a href="/wiki/OpenCV" title="OpenCV">OpenCV</a>.</li> <li>Matrix <b>D</b> will take the form of an <i>p</i> × <i>p</i> diagonal matrix, where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{k\ell }=\lambda _{k}\qquad {\text{for }}k=\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{k\ell }=\lambda _{k}\qquad {\text{for }}k=\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb45d20b98b0cc25cbd2502a6b878bb5e92e25f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.532ex; height:2.509ex;" alt="{\displaystyle D_{k\ell }=\lambda _{k}\qquad {\text{for }}k=\ell }"></span> is the <i>j</i>th eigenvalue of the covariance matrix <b>C</b>, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{k\ell }=0\qquad {\text{for }}k\neq \ell .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>k</mi> <mo>≠</mo> <mi>ℓ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{k\ell }=0\qquad {\text{for }}k\neq \ell .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e721de69bc847376dbef4ad834afc3b34dfa18ae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.898ex; height:2.676ex;" alt="{\displaystyle D_{k\ell }=0\qquad {\text{for }}k\neq \ell .}"></span></li> <li>Matrix <b>V</b>, also of dimension <i>p</i> × <i>p</i>, contains <i>p</i> column vectors, each of length <i>p</i>, which represent the <i>p</i> eigenvectors of the covariance matrix <b>C</b>.</li> <li>The eigenvalues and eigenvectors are ordered and paired. The <i>j</i>th eigenvalue corresponds to the <i>j</i>th eigenvector.</li> <li>Matrix <b>V</b> denotes the matrix of <i>right</i> eigenvectors (as opposed to <i>left</i> eigenvectors). In general, the matrix of right eigenvectors need <i>not</i> be the (conjugate) transpose of the matrix of left eigenvectors.</li></ul> </li> <li> <b>Rearrange the eigenvectors and eigenvalues</b> <ul><li>Sort the columns of the eigenvector matrix <b>V</b> and eigenvalue matrix <b>D</b> in order of <i>decreasing</i> eigenvalue.</li> <li>Make sure to maintain the correct pairings between the columns in each matrix.</li></ul> </li> <li> <b>Compute the cumulative energy content for each eigenvector</b> <ul><li>The eigenvalues represent the distribution of the source data's energy<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (March 2011)">clarification needed</span></a></i>]</sup> among each of the eigenvectors, where the eigenvectors form a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for the data. The cumulative energy content <i>g</i> for the <i>j</i>th eigenvector is the sum of the energy content across all of the eigenvalues from 1 through <i>j</i>:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2011)">citation needed</span></a></i>]</sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{j}=\sum _{k=1}^{j}D_{kk}\qquad {\text{for }}j=1,\dots ,p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{j}=\sum _{k=1}^{j}D_{kk}\qquad {\text{for }}j=1,\dots ,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea78635798ac85a824c5daee78fea1717446080a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.307ex; height:7.343ex;" alt="{\displaystyle g_{j}=\sum _{k=1}^{j}D_{kk}\qquad {\text{for }}j=1,\dots ,p}"></span></li></ul> </li> <li> <b>Select a subset of the eigenvectors as basis vectors</b> <ul><li>Save the first <i>L</i> columns of <b>V</b> as the <i>p</i> × <i>L</i> matrix <b>W</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{kl}=V_{k\ell }\qquad {\text{for }}k=1,\dots ,p\qquad \ell =1,\dots ,L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> <mspace width="2em" /> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{kl}=V_{k\ell }\qquad {\text{for }}k=1,\dots ,p\qquad \ell =1,\dots ,L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0729377713d52e8672d2eeb6ee2330f675868a68" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.469ex; height:2.509ex;" alt="{\displaystyle W_{kl}=V_{k\ell }\qquad {\text{for }}k=1,\dots ,p\qquad \ell =1,\dots ,L}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq L\leq p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>L</mi> <mo>≤</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq L\leq p.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cadef48a3da99060e2ebbbd4a893a21ff5fa068" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.758ex; height:2.509ex;" alt="{\displaystyle 1\leq L\leq p.}"></span></li> <li>Use the vector <b>g</b> as a guide in choosing an appropriate value for <i>L</i>. The goal is to choose a value of <i>L</i> as small as possible while achieving a reasonably high value of <i>g</i> on a percentage basis. For example, you may want to choose <i>L</i> so that the cumulative energy <i>g</i> is above a certain threshold, like 90 percent. In this case, choose the smallest value of <i>L</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {g_{L}}{g_{p}}}\geq 0.9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <mo>≥</mo> <mn>0.9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {g_{L}}{g_{p}}}\geq 0.9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b366a06f1ca0c86ca94f2027a5e7480fd25b5e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.367ex; height:5.509ex;" alt="{\displaystyle {\frac {g_{L}}{g_{p}}}\geq 0.9}"></span></li></ul> </li> <li> <b>Project the data onto the new basis</b> <ul><li>The projected data points are the rows of the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} =\mathbf {B} \cdot \mathbf {W} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =\mathbf {B} \cdot \mathbf {W} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b13edf569eec24807349124e6b3e72712b6a9d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.301ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} =\mathbf {B} \cdot \mathbf {W} }"></span></li></ul> That is, the first column of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9593e3b995a1b57c078873a5ea186c7012e1a5ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} }"></span> is the projection of the data points onto the first principal component, the second column is the projection onto the second principal component, etc. </li> </ol> <div class="mw-heading mw-heading2"><h2 id="Derivation_using_the_covariance_method">Derivation using the covariance method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=17" title="Edit section: Derivation using the covariance method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <b>X</b> be a <i>d</i>-dimensional random vector expressed as column vector. Without loss of generality, assume <b>X</b> has zero mean. </p><p>We want to find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\ast )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\ast )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75fef365d874a9437c0ce03c76ccb887334570fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (\ast )}"></span> a <span class="texhtml"><i>d</i> × <i>d</i></span> <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal transformation matrix</a> <b>P</b> so that <b>PX</b> has a diagonal covariance matrix (that is, <b>PX</b> is a random vector with all its distinct components pairwise uncorrelated). </p><p>A quick computation assuming <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> were unitary yields: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {cov} (PX)&=\operatorname {E} [PX~(PX)^{*}]\\&=\operatorname {E} [PX~X^{*}P^{*}]\\&=P\operatorname {E} [XX^{*}]P^{*}\\&=P\operatorname {cov} (X)P^{-1}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mi>X</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>P</mi> <mi>X</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>P</mi> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>P</mi> <mi>X</mi> <mtext> </mtext> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>P</mi> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">]</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>P</mi> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {cov} (PX)&=\operatorname {E} [PX~(PX)^{*}]\\&=\operatorname {E} [PX~X^{*}P^{*}]\\&=P\operatorname {E} [XX^{*}]P^{*}\\&=P\operatorname {cov} (X)P^{-1}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e4800248eafcc33b2c22c5613f06b0c2455faad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.579ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {cov} (PX)&=\operatorname {E} [PX~(PX)^{*}]\\&=\operatorname {E} [PX~X^{*}P^{*}]\\&=P\operatorname {E} [XX^{*}]P^{*}\\&=P\operatorname {cov} (X)P^{-1}\\\end{aligned}}}"></span></dd></dl> <p>Hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\ast )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\ast )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75fef365d874a9437c0ce03c76ccb887334570fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (\ast )}"></span> holds if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cov} (X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1c40a6656eb7d06ad54a1a3cc41bd60da525a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.212ex; height:2.843ex;" alt="{\displaystyle \operatorname {cov} (X)}"></span> were diagonalisable by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. </p><p>This is very constructive, as cov(<b>X</b>) is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix. </p> <div class="mw-heading mw-heading2"><h2 id="Covariance-free_computation">Covariance-free computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=18" title="Edit section: Covariance-free computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In practical implementations, especially with <a href="/wiki/High_dimensional_data" class="mw-redirect" title="High dimensional data">high dimensional data</a> (large <span class="texhtml mvar" style="font-style:italic;">p</span>), the naive covariance method is rarely used because it is not efficient due to high computational and memory costs of explicitly determining the covariance matrix. The covariance-free approach avoids the <span class="texhtml"><i>np</i><sup>2</sup></span> operations of explicitly calculating and storing the covariance matrix <span class="texhtml"><b>X<sup>T</sup>X</b></span>, instead utilizing one of <a href="/wiki/Matrix-free_methods" title="Matrix-free methods">matrix-free methods</a>, for example, based on the function evaluating the product <span class="texhtml"><b>X<sup>T</sup>(X r)</b></span> at the cost of <span class="texhtml">2<i>np</i></span> operations. </p> <div class="mw-heading mw-heading3"><h3 id="Iterative_computation">Iterative computation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=19" title="Edit section: Iterative computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One way to compute the first principal component efficiently<sup id="cite_ref-roweis_42-0" class="reference"><a href="#cite_note-roweis-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> is shown in the following pseudo-code, for a data matrix <span class="texhtml"><b>X</b></span> with zero mean, without ever computing its covariance matrix. </p> <pre><span class="texhtml"><b>r</b></span> = a random vector of length <span class="texhtml mvar" style="font-style:italic;">p</span> <b>r</b> = <b>r</b> / norm(<b>r</b>) do <span class="texhtml mvar" style="font-style:italic;">c</span> times: <span class="texhtml"><b>s</b> = 0</span> (a vector of length <span class="texhtml mvar" style="font-style:italic;">p</span>) <span class="nowrap">for each row <b>x</b> in <b>X</b></span> <span class="nowrap"><b>s</b> = <b>s</b> + (<b>x</b> ⋅ <b>r</b>) <b>x</b></span> <span class="nowrap">λ = <b>r</b><sup>T</sup><b>s</b></span> <span class="nowrap">// λ is the eigenvalue</span> <span class="nowrap">error = |λ ⋅ <b>r</b> − <b>s</b>|</span> <span class="nowrap"><b>r</b> = <b>s</b> / norm(<b>s</b>)</span> <span class="nowrap">exit if error < tolerance</span> return <span class="nowrap">λ, <b>r</b></span> </pre> <p>This <a href="/wiki/Power_iteration" title="Power iteration">power iteration</a> algorithm simply calculates the vector <span class="texhtml"><b>X<sup>T</sup>(X r)</b></span>, normalizes, and places the result back in <span class="texhtml"><b>r</b></span>. The eigenvalue is approximated by <span class="texhtml"><b>r<sup>T</sup> (X<sup>T</sup>X) r</b></span>, which is the <a href="/wiki/Rayleigh_quotient" title="Rayleigh quotient">Rayleigh quotient</a> on the unit vector <span class="texhtml"><b>r</b></span> for the covariance matrix <span class="texhtml"><b>X<sup>T</sup>X </b></span>. If the largest singular value is well separated from the next largest one, the vector <span class="texhtml"><b>r</b></span> gets close to the first principal component of <span class="texhtml"><b>X</b></span> within the number of iterations <span class="texhtml mvar" style="font-style:italic;">c</span>, which is small relative to <span class="texhtml mvar" style="font-style:italic;">p</span>, at the total cost <span class="texhtml"><i>2cnp</i></span>. The <a href="/wiki/Power_iteration" title="Power iteration">power iteration</a> convergence can be accelerated without noticeably sacrificing the small cost per iteration using more advanced <a href="/wiki/Matrix-free_methods" title="Matrix-free methods">matrix-free methods</a>, such as the <a href="/wiki/Lanczos_algorithm" title="Lanczos algorithm">Lanczos algorithm</a> or the Locally Optimal Block Preconditioned Conjugate Gradient (<a href="/wiki/LOBPCG" title="LOBPCG">LOBPCG</a>) method. </p><p>Subsequent principal components can be computed one-by-one via deflation or simultaneously as a block. In the former approach, imprecisions in already computed approximate principal components additively affect the accuracy of the subsequently computed principal components, thus increasing the error with every new computation. The latter approach in the block power method replaces single-vectors <span class="texhtml"><b>r</b></span> and <span class="texhtml"><b>s</b></span> with block-vectors, matrices <span class="texhtml"><b>R</b></span> and <span class="texhtml"><b>S</b></span>. Every column of <span class="texhtml"><b>R</b></span> approximates one of the leading principal components, while all columns are iterated simultaneously. The main calculation is evaluation of the product <span class="texhtml"><b>X<sup>T</sup>(X R)</b></span>. Implemented, for example, in <a href="/wiki/LOBPCG" title="LOBPCG">LOBPCG</a>, efficient blocking eliminates the accumulation of the errors, allows using high-level <a href="/wiki/BLAS" class="mw-redirect" title="BLAS">BLAS</a> matrix-matrix product functions, and typically leads to faster convergence, compared to the single-vector one-by-one technique. </p> <div class="mw-heading mw-heading3"><h3 id="The_NIPALS_method">The NIPALS method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=20" title="Edit section: The NIPALS method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Non-linear iterative partial least squares (NIPALS)</i> is a variant the classical <a href="/wiki/Power_iteration" title="Power iteration">power iteration</a> with matrix deflation by subtraction implemented for computing the first few components in a principal component or <a href="/wiki/Partial_least_squares" class="mw-redirect" title="Partial least squares">partial least squares</a> analysis. For very-high-dimensional datasets, such as those generated in the *omics sciences (for example, <a href="/wiki/Genomics" title="Genomics">genomics</a>, <a href="/wiki/Metabolomics" title="Metabolomics">metabolomics</a>) it is usually only necessary to compute the first few PCs. The <a href="/wiki/Non-linear_iterative_partial_least_squares" class="mw-redirect" title="Non-linear iterative partial least squares">non-linear iterative partial least squares</a> (NIPALS) algorithm updates iterative approximations to the leading scores and loadings <b>t</b><sub>1</sub> and <b>r</b><sub>1</sub><sup>T</sup> by the <a href="/wiki/Power_iteration" title="Power iteration">power iteration</a> multiplying on every iteration by <b>X</b> on the left and on the right, that is, calculation of the covariance matrix is avoided, just as in the matrix-free implementation of the power iterations to <span class="texhtml"><b>X<sup>T</sup>X</b></span>, based on the function evaluating the product <span class="texhtml"><b>X<sup>T</sup>(X r)</b> = <b>((X r)<sup>T</sup>X)<sup>T</sup></b></span>. </p><p>The matrix deflation by subtraction is performed by subtracting the outer product, <b>t</b><sub>1</sub><b>r</b><sub>1</sub><sup>T</sup> from <b>X</b> leaving the deflated residual matrix used to calculate the subsequent leading PCs.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> For large data matrices, or matrices that have a high degree of column collinearity, NIPALS suffers from loss of orthogonality of PCs due to machine precision <a href="/wiki/Round-off_errors" class="mw-redirect" title="Round-off errors">round-off errors</a> accumulated in each iteration and matrix deflation by subtraction.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> A <a href="/wiki/Gram%E2%80%93Schmidt" class="mw-redirect" title="Gram–Schmidt">Gram–Schmidt</a> re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> NIPALS reliance on single-vector multiplications cannot take advantage of high-level <a href="/wiki/BLAS" class="mw-redirect" title="BLAS">BLAS</a> and results in slow convergence for clustered leading singular values—both these deficiencies are resolved in more sophisticated matrix-free block solvers, such as the Locally Optimal Block Preconditioned Conjugate Gradient (<a href="/wiki/LOBPCG" title="LOBPCG">LOBPCG</a>) method. </p> <div class="mw-heading mw-heading3"><h3 id="Online/sequential_estimation"><span id="Online.2Fsequential_estimation"></span>Online/sequential estimation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=21" title="Edit section: Online/sequential estimation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In an "online" or "streaming" situation with data arriving piece by piece rather than being stored in a single batch, it is useful to make an estimate of the PCA projection that can be updated sequentially. This can be done efficiently, but requires different algorithms.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Qualitative_variables">Qualitative variables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=22" title="Edit section: Qualitative variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In PCA, it is common that we want to introduce qualitative variables as supplementary elements. For example, many quantitative variables have been measured on plants. For these plants, some qualitative variables are available as, for example, the species to which the plant belongs. These data were subjected to PCA for quantitative variables. When analyzing the results, it is natural to connect the principal components to the qualitative variable <i>species</i>. For this, the following results are produced. </p> <ul><li>Identification, on the factorial planes, of the different species, for example, using different colors.</li> <li>Representation, on the factorial planes, of the centers of gravity of plants belonging to the same species.</li> <li>For each center of gravity and each axis, p-value to judge the significance of the difference between the center of gravity and origin.</li></ul> <p>These results are what is called <i>introducing a qualitative variable as supplementary element</i>. This procedure is detailed in and Husson, Lê, & Pagès (2009) and Pagès (2013). Few software offer this option in an "automatic" way. This is the case of <a rel="nofollow" class="external text" href="http://www.coheris.com/produits/analytics/logiciel-data-mining/">SPAD</a> that historically, following the work of <a href="/wiki/Ludovic_Lebart" title="Ludovic Lebart">Ludovic Lebart</a>, was the first to propose this option, and the R package <a rel="nofollow" class="external text" href="http://factominer.free.fr/">FactoMineR</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=23" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Intelligence">Intelligence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=24" title="Edit section: Intelligence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The earliest application of factor analysis was in locating and measuring components of human intelligence. It was believed that intelligence had various uncorrelated components such as spatial intelligence, verbal intelligence, induction, deduction etc and that scores on these could be adduced by factor analysis from results on various tests, to give a single index known as the <a href="/wiki/Intelligence_quotient" title="Intelligence quotient">Intelligence Quotient</a> (IQ). The pioneering statistical psychologist <a href="/wiki/Charles_Spearman" title="Charles Spearman">Spearman</a> actually developed factor analysis in 1904 for his <a href="/wiki/Two-factor_theory_of_intelligence" title="Two-factor theory of intelligence">two-factor theory</a> of intelligence, adding a formal technique to the science of <a href="/wiki/Psychometrics" title="Psychometrics">psychometrics</a>. In 1924 <a href="/wiki/Louis_Leon_Thurstone" title="Louis Leon Thurstone">Thurstone</a> looked for 56 factors of intelligence, developing the notion of Mental Age. Standard IQ tests today are based on this early work.<sup id="cite_ref-Kaplan,_R.M._2010_47-0" class="reference"><a href="#cite_note-Kaplan,_R.M._2010-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Residential_differentiation">Residential differentiation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=25" title="Edit section: Residential differentiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1949, Shevky and Williams introduced the theory of <b>factorial ecology</b>, which dominated studies of residential differentiation from the 1950s to the 1970s.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> Neighbourhoods in a city were recognizable or could be distinguished from one another by various characteristics which could be reduced to three by factor analysis. These were known as 'social rank' (an index of occupational status), 'familism' or family size, and 'ethnicity'; Cluster analysis could then be applied to divide the city into clusters or precincts according to values of the three key factor variables. An extensive literature developed around factorial ecology in urban geography, but the approach went out of fashion after 1980 as being methodologically primitive and having little place in postmodern geographical paradigms. </p><p>One of the problems with factor analysis has always been finding convincing names for the various artificial factors. In 2000, Flood revived the factorial ecology approach to show that principal components analysis actually gave meaningful answers directly, without resorting to factor rotation. The principal components were actually dual variables or shadow prices of 'forces' pushing people together or apart in cities. The first component was 'accessibility', the classic trade-off between demand for travel and demand for space, around which classical urban economics is based. The next two components were 'disadvantage', which keeps people of similar status in separate neighbourhoods (mediated by planning), and ethnicity, where people of similar ethnic backgrounds try to co-locate.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p><p>About the same time, the Australian Bureau of Statistics defined distinct indexes of advantage and disadvantage taking the first principal component of sets of key variables that were thought to be important. These SEIFA indexes are regularly published for various jurisdictions, and are used frequently in spatial analysis.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Development_indexes">Development indexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=26" title="Edit section: Development indexes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>PCA can be used as a formal method for the development of indexes. As an alternative <a href="/wiki/Confirmatory_composite_analysis" title="Confirmatory composite analysis">confirmatory composite analysis</a> has been proposed to develop and assess indexes.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p><p>The City Development Index was developed by PCA from about 200 indicators of city outcomes in a 1996 survey of 254 global cities. The first principal component was subject to iterative regression, adding the original variables singly until about 90% of its variation was accounted for. The index ultimately used about 15 indicators but was a good predictor of many more variables. Its comparative value agreed very well with a subjective assessment of the condition of each city. The coefficients on items of infrastructure were roughly proportional to the average costs of providing the underlying services, suggesting the Index was actually a measure of effective physical and social investment in the city. </p><p>The country-level <a href="/wiki/Human_Development_Index" title="Human Development Index">Human Development Index</a> (HDI) from <a href="/wiki/United_Nations_Development_Programme" title="United Nations Development Programme">UNDP</a>, which has been published since 1990 and is very extensively used in development studies,<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> has very similar coefficients on similar indicators, strongly suggesting it was originally constructed using PCA. </p> <div class="mw-heading mw-heading3"><h3 id="Population_genetics">Population genetics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=27" title="Edit section: Population genetics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1978 <a href="/wiki/Luigi_Luca_Cavalli-Sforza" title="Luigi Luca Cavalli-Sforza">Cavalli-Sforza</a> and others pioneered the use of principal components analysis (PCA) to summarise data on variation in human gene frequencies across regions. The components showed distinctive patterns, including gradients and sinusoidal waves. They interpreted these patterns as resulting from specific ancient migration events. </p><p>Since then, PCA has been ubiquitous in population genetics, with thousands of papers using PCA as a display mechanism. Genetics varies largely according to proximity, so the first two principal components actually show spatial distribution and may be used to map the relative geographical location of different population groups, thereby showing individuals who have wandered from their original locations.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p><p>PCA in genetics has been technically controversial, in that the technique has been performed on discrete non-normal variables and often on binary allele markers. The lack of any measures of standard error in PCA are also an impediment to more consistent usage. In August 2022, the molecular biologist <a href="/wiki/Eran_Elhaik" title="Eran Elhaik">Eran Elhaik</a> published a theoretical paper in <a href="/wiki/Scientific_Reports" title="Scientific Reports">Scientific Reports</a> analyzing 12 PCA applications. He concluded that it was easy to manipulate the method, which, in his view, generated results that were 'erroneous, contradictory, and absurd.' Specifically, he argued, the results achieved in population genetics were characterized by cherry-picking and <a href="/wiki/Circular_reasoning" title="Circular reasoning">circular reasoning</a>.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Market_research_and_indexes_of_attitude">Market research and indexes of attitude</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=28" title="Edit section: Market research and indexes of attitude"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Market research has been an extensive user of PCA. It is used to develop customer satisfaction or customer loyalty scores for products, and with clustering, to develop market segments that may be targeted with advertising campaigns, in much the same way as factorial ecology will locate geographical areas with similar characteristics.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>PCA rapidly transforms large amounts of data into smaller, easier-to-digest variables that can be more rapidly and readily analyzed. In any consumer questionnaire, there are series of questions designed to elicit consumer attitudes, and principal components seek out latent variables underlying these attitudes. For example, the Oxford Internet Survey in 2013 asked 2000 people about their attitudes and beliefs, and from these analysts extracted four principal component dimensions, which they identified as 'escape', 'social networking', 'efficiency', and 'problem creating'.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p><p>Another example from Joe Flood in 2008 extracted an attitudinal index toward housing from 28 attitude questions in a national survey of 2697 households in Australia. The first principal component represented a general attitude toward property and home ownership. The index, or the attitude questions it embodied, could be fed into a General Linear Model of tenure choice. The strongest determinant of private renting by far was the attitude index, rather than income, marital status or household type.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quantitative_finance">Quantitative finance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=29" title="Edit section: Quantitative finance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Quantitative_finance" class="mw-redirect" title="Quantitative finance">quantitative finance</a>, PCA is used<sup id="cite_ref-Miller_58-0" class="reference"><a href="#cite_note-Miller-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> in <a href="/wiki/Financial_risk_management" title="Financial risk management">financial risk management</a>, and has been applied to <a href="/wiki/Financial_modeling#Quantitative_finance" title="Financial modeling">other problems</a> such as <a href="/wiki/Portfolio_optimization" title="Portfolio optimization">portfolio optimization</a>. </p><p>PCA is commonly used in problems involving <a href="/wiki/Fixed_income" title="Fixed income">fixed income</a> securities and <a href="/wiki/Bond_fund" title="Bond fund">portfolios</a>, and <a href="/wiki/Interest_rate_derivative" title="Interest rate derivative">interest rate derivatives</a>. Valuations here depend on the entire <a href="/wiki/Yield_curve" title="Yield curve">yield curve</a>, comprising numerous highly correlated instruments, and PCA is used to define a set of components or factors that explain rate movements,<sup id="cite_ref-Hull_59-0" class="reference"><a href="#cite_note-Hull-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> thereby facilitating the modelling. One common risk management application is to <a href="/wiki/Value_at_risk#Computation_methods" title="Value at risk">calculating value at risk</a>, VaR, applying PCA to the <a href="/wiki/Monte_Carlo_methods_in_finance" title="Monte Carlo methods in finance">Monte Carlo simulation</a>. <sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> Here, for each simulation-sample, the components are stressed, and rates, and <a href="/wiki/Monte_Carlo_methods_for_option_pricing#Methodology" title="Monte Carlo methods for option pricing">in turn option values</a>, are then reconstructed; with VaR calculated, finally, over the entire run. PCA is also used in <a href="/wiki/Hedge_(finance)" title="Hedge (finance)">hedging</a> exposure to <a href="/wiki/Interest_rate_risk" title="Interest rate risk">interest rate risk</a>, given <a href="/wiki/Key_rate_duration" class="mw-redirect" title="Key rate duration">partial durations</a> and other sensitivities. <sup id="cite_ref-Hull_59-1" class="reference"><a href="#cite_note-Hull-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> Under both, the first three, typically, principal components of the system are of interest (<a href="/wiki/Fixed-income_attribution#Modeling_the_yield_curve" title="Fixed-income attribution">representing</a> "shift", "twist", and "curvature"). These principal components are derived from an eigen-decomposition of the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> of <a href="/wiki/Yield_curve" title="Yield curve">yield</a> at predefined maturities; <sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> and where the <a href="/wiki/Variance" title="Variance">variance</a> of each component is its <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> (and as the components are <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a>, no correlation need be incorporated in subsequent modelling). </p><p>For <a href="/wiki/Equity_(finance)" title="Equity (finance)">equity</a>, an optimal portfolio is one where the <a href="/wiki/Expected_return" title="Expected return">expected return</a> is maximized for a given level of risk, or alternatively, where risk is minimized for a given return; see <a href="/wiki/Markowitz_model" title="Markowitz model">Markowitz model</a> for discussion. Thus, one approach is to reduce portfolio risk, where <a href="/wiki/Asset_allocation" title="Asset allocation">allocation strategies</a> are applied to the "principal portfolios" instead of the underlying <a href="/wiki/Capital_stock" class="mw-redirect" title="Capital stock">stocks</a>. A second approach is to enhance portfolio return, using the principal components to select companies' stocks with upside potential. <sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> PCA has also been used to understand relationships <sup id="cite_ref-Miller_58-1" class="reference"><a href="#cite_note-Miller-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> between international <a href="/wiki/Equity_market" class="mw-redirect" title="Equity market">equity markets</a>, and within markets between groups of companies in industries or <a href="/wiki/Stock_market_index#Types_of_indices_by_coverage" title="Stock market index">sectors</a>. </p><p>PCA may also be applied to <a href="/wiki/Stress_test_(financial)" title="Stress test (financial)">stress testing</a>,<sup id="cite_ref-IMF_64-0" class="reference"><a href="#cite_note-IMF-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> essentially an analysis of a bank's ability to endure <a href="/wiki/List_of_bank_stress_tests" title="List of bank stress tests">a hypothetical adverse economic scenario</a>. Its utility is in "distilling the information contained in [several] <a href="/wiki/Macroeconomic_model" title="Macroeconomic model">macroeconomic variables</a> into a more manageable data set, which can then [be used] for analysis."<sup id="cite_ref-IMF_64-1" class="reference"><a href="#cite_note-IMF-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> Here, the resulting factors are linked to e.g. interest rates – based on the largest elements of the factor's <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvector</a> – and it is then observed how a "shock" to each of the factors affects the implied assets of each of the banks. </p> <div class="mw-heading mw-heading3"><h3 id="Neuroscience">Neuroscience</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=30" title="Edit section: Neuroscience"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A variant of principal components analysis is used in <a href="/wiki/Neuroscience" title="Neuroscience">neuroscience</a> to identify the specific properties of a stimulus that increases a <a href="/wiki/Neuron" title="Neuron">neuron</a>'s probability of generating an <a href="/wiki/Action_potential" title="Action potential">action potential</a>.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-brenner00_66-0" class="reference"><a href="#cite_note-brenner00-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> This technique is known as <a href="/wiki/Spike-triggered_covariance" title="Spike-triggered covariance">spike-triggered covariance analysis</a>. In a typical application an experimenter presents a <a href="/wiki/White_noise" title="White noise">white noise</a> process as a stimulus (usually either as a sensory input to a test subject, or as a <a href="/wiki/Electric_current" title="Electric current">current</a> injected directly into the neuron) and records a train of action potentials, or spikes, produced by the neuron as a result. Presumably, certain features of the stimulus make the neuron more likely to spike. In order to extract these features, the experimenter calculates the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> of the <i>spike-triggered ensemble</i>, the set of all stimuli (defined and discretized over a finite time window, typically on the order of 100 ms) that immediately preceded a spike. The <a href="/wiki/Eigenvectors_and_eigenvalues" class="mw-redirect" title="Eigenvectors and eigenvalues">eigenvectors</a> of the difference between the spike-triggered covariance matrix and the covariance matrix of the <i>prior stimulus ensemble</i> (the set of all stimuli, defined over the same length time window) then indicate the directions in the <a href="/wiki/Vector_space" title="Vector space">space</a> of stimuli along which the variance of the spike-triggered ensemble differed the most from that of the prior stimulus ensemble. Specifically, the eigenvectors with the largest positive eigenvalues correspond to the directions along which the variance of the spike-triggered ensemble showed the largest positive change compared to the variance of the prior. Since these were the directions in which varying the stimulus led to a spike, they are often good approximations of the sought after relevant stimulus features. </p><p>In neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential. <a href="/wiki/Spike_sorting" title="Spike sorting">Spike sorting</a> is an important procedure because <a href="/wiki/Electrophysiology#Extracellular_recording" title="Electrophysiology">extracellular</a> recording techniques often pick up signals from more than one neuron. In spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs <a href="/wiki/Cluster_analysis" title="Cluster analysis">clustering analysis</a> to associate specific action potentials with individual neurons. </p><p>PCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles. It has been used in determining collective variables, that is, <a href="/wiki/Order_parameters" class="mw-redirect" title="Order parameters">order parameters</a>, during <a href="/wiki/Phase_transitions" class="mw-redirect" title="Phase transitions">phase transitions</a> in the brain.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relation_with_other_methods">Relation with other methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=31" title="Edit section: Relation with other methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Correspondence_analysis">Correspondence analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=32" title="Edit section: Correspondence analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Correspondence_analysis" title="Correspondence analysis">Correspondence analysis</a> (CA) was developed by <a href="/wiki/Jean-Paul_Benz%C3%A9cri" title="Jean-Paul Benzécri">Jean-Paul Benzécri</a><sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> and is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently. It is traditionally applied to <a href="/wiki/Contingency_tables" class="mw-redirect" title="Contingency tables">contingency tables</a>. CA decomposes the <a href="/wiki/Chi-squared_statistic" class="mw-redirect" title="Chi-squared statistic">chi-squared statistic</a> associated to this table into orthogonal factors.<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> Because CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not. Several variants of CA are available including <a href="/wiki/Detrended_correspondence_analysis" title="Detrended correspondence analysis">detrended correspondence analysis</a> and <a href="/wiki/Canonical_correspondence_analysis" title="Canonical correspondence analysis">canonical correspondence analysis</a>. One special extension is <a href="/wiki/Multiple_correspondence_analysis" title="Multiple correspondence analysis">multiple correspondence analysis</a>, which may be seen as the counterpart of principal component analysis for categorical data.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Factor_analysis">Factor analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=33" title="Edit section: Factor analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PCA_versus_Factor_Analysis.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/PCA_versus_Factor_Analysis.jpg/220px-PCA_versus_Factor_Analysis.jpg" decoding="async" width="220" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/PCA_versus_Factor_Analysis.jpg/330px-PCA_versus_Factor_Analysis.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/PCA_versus_Factor_Analysis.jpg/440px-PCA_versus_Factor_Analysis.jpg 2x" data-file-width="494" data-file-height="570" /></a><figcaption>The above picture is an example of the difference between PCA and Factor Analysis. In the top diagram the "factor" (e.g., career path) represents the three observed variables (e.g., doctor, lawyer, teacher) whereas in the bottom diagram the observed variables (e.g., pre-school teacher, middle school teacher, high school teacher) are reduced into the component of interest (e.g., teacher).</figcaption></figure> <p>Principal component analysis creates variables that are linear combinations of the original variables. The new variables have the property that the variables are all orthogonal. The PCA transformation can be helpful as a pre-processing step before clustering. PCA is a variance-focused approach seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable. PCA is generally preferred for purposes of data reduction (that is, translating variable space into optimal factor space) but not when the goal is to detect the latent construct or factors. </p><p><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a> is similar to principal component analysis, in that factor analysis also involves linear combinations of variables. Different from PCA, factor analysis is a correlation-focused approach seeking to reproduce the inter-correlations among variables, in which the factors "represent the common variance of variables, excluding unique variance".<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> In terms of the correlation matrix, this corresponds with focusing on explaining the off-diagonal terms (that is, shared co-variance), while PCA focuses on explaining the terms that sit on the diagonal. However, as a side result, when trying to reproduce the on-diagonal terms, PCA also tends to fit relatively well the off-diagonal correlations.<sup id="cite_ref-Jolliffe2002_13-4" class="reference"><a href="#cite_note-Jolliffe2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 158">: 158 </span></sup> Results given by PCA and factor analysis are very similar in most situations, but this is not always the case, and there are some problems where the results are significantly different. Factor analysis is generally used when the research purpose is detecting data structure (that is, latent constructs or factors) or <a href="/wiki/Causal_modeling" class="mw-redirect" title="Causal modeling">causal modeling</a>. If the factor model is incorrectly formulated or the assumptions are not met, then factor analysis will give erroneous results.<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="K-means_clustering"><span class="texhtml"><var>K</var></span>-means clustering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=34" title="Edit section: K-means clustering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It has been asserted that the relaxed solution of <a href="/wiki/K-means_clustering" title="K-means clustering"><span class="texhtml"><var>k</var></span>-means clustering</a>, specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace.<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> However, that PCA is a useful relaxation of <span class="texhtml"><var>k</var></span>-means clustering was not a new result,<sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Non-negative_matrix_factorization">Non-negative matrix factorization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=35" title="Edit section: Non-negative matrix factorization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Fractional_Residual_Variances_comparison,_PCA_and_NMF.pdf" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Fractional_Residual_Variances_comparison%2C_PCA_and_NMF.pdf/page1-500px-Fractional_Residual_Variances_comparison%2C_PCA_and_NMF.pdf.jpg" decoding="async" width="500" height="245" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Fractional_Residual_Variances_comparison%2C_PCA_and_NMF.pdf/page1-750px-Fractional_Residual_Variances_comparison%2C_PCA_and_NMF.pdf.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Fractional_Residual_Variances_comparison%2C_PCA_and_NMF.pdf/page1-1000px-Fractional_Residual_Variances_comparison%2C_PCA_and_NMF.pdf.jpg 2x" data-file-width="1185" data-file-height="581" /></a><figcaption>Fractional residual variance (FRV) plots for PCA and NMF;<sup id="cite_ref-ren18_26-1" class="reference"><a href="#cite_note-ren18-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> for PCA, the theoretical values are the contribution from the residual eigenvalues. In comparison, the FRV curves for PCA reaches a flat plateau where no signal are captured effectively; while the NMF FRV curves decline continuously, indicating a better ability to capture signal. The FRV curves for NMF also converges to higher levels than PCA, indicating the less-overfitting property of NMF.</figcaption></figure> <p><a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">Non-negative matrix factorization</a> (NMF) is a dimension reduction method where only non-negative elements in the matrices are used, which is therefore a promising method in astronomy,<sup id="cite_ref-blantonRoweis07_24-1" class="reference"><a href="#cite_note-blantonRoweis07-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-zhu16_25-1" class="reference"><a href="#cite_note-zhu16-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ren18_26-2" class="reference"><a href="#cite_note-ren18-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> in the sense that astrophysical signals are non-negative. The PCA components are orthogonal to each other, while the NMF components are all non-negative and therefore constructs a non-orthogonal basis. </p><p>In PCA, the contribution of each component is ranked based on the magnitude of its corresponding eigenvalue, which is equivalent to the fractional residual variance (FRV) in analyzing empirical data.<sup id="cite_ref-soummer12_22-1" class="reference"><a href="#cite_note-soummer12-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> For NMF, its components are ranked based only on the empirical FRV curves.<sup id="cite_ref-ren18_26-3" class="reference"><a href="#cite_note-ren18-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> The residual fractional eigenvalue plots, that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\sum _{i=1}^{k}\lambda _{i}{\Big /}\sum _{j=1}^{n}\lambda _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.623em" minsize="1.623em">/</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\sum _{i=1}^{k}\lambda _{i}{\Big /}\sum _{j=1}^{n}\lambda _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f399b73ab250afa7ac858c14ba78a70fa17b242e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.179ex; height:7.676ex;" alt="{\displaystyle 1-\sum _{i=1}^{k}\lambda _{i}{\Big /}\sum _{j=1}^{n}\lambda _{j}}"></span> as a function of component number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> given a total of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> components, for PCA have a flat plateau, where no data is captured to remove the quasi-static noise, then the curves drop quickly as an indication of over-fitting (random noise).<sup id="cite_ref-soummer12_22-2" class="reference"><a href="#cite_note-soummer12-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The FRV curves for NMF is decreasing continuously<sup id="cite_ref-ren18_26-4" class="reference"><a href="#cite_note-ren18-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> when the NMF components are constructed <a href="/wiki/Non-negative_matrix_factorization#Sequential_NMF" title="Non-negative matrix factorization">sequentially</a>,<sup id="cite_ref-zhu16_25-2" class="reference"><a href="#cite_note-zhu16-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> indicating the continuous capturing of quasi-static noise; then converge to higher levels than PCA,<sup id="cite_ref-ren18_26-5" class="reference"><a href="#cite_note-ren18-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> indicating the less over-fitting property of NMF. </p> <div class="mw-heading mw-heading3"><h3 id="Iconography_of_correlations">Iconography of correlations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=36" title="Edit section: Iconography of correlations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is often difficult to interpret the principal components when the data include many variables of various origins, or when some variables are qualitative. This leads the PCA user to a delicate elimination of several variables. If observations or variables have an excessive impact on the direction of the axes, they should be removed and then projected as supplementary elements. In addition, it is necessary to avoid interpreting the proximities between the points close to the center of the factorial plane. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:AirMerIconographyCorrelation.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/AirMerIconographyCorrelation.jpg/220px-AirMerIconographyCorrelation.jpg" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/AirMerIconographyCorrelation.jpg/330px-AirMerIconographyCorrelation.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/AirMerIconographyCorrelation.jpg/440px-AirMerIconographyCorrelation.jpg 2x" data-file-width="544" data-file-height="503" /></a><figcaption>Iconography of correlations – Geochemistry of marine aerosols</figcaption></figure> <p>The <a href="/wiki/Iconography_of_correlations" title="Iconography of correlations">iconography of correlations</a>, on the contrary, which is not a projection on a system of axes, does not have these drawbacks. We can therefore keep all the variables. </p><p>The principle of the diagram is to underline the "remarkable" correlations of the correlation matrix, by a solid line (positive correlation) or dotted line (negative correlation). </p><p>A strong correlation is not "remarkable" if it is not direct, but caused by the effect of a third variable. Conversely, weak correlations can be "remarkable". For example, if a variable Y depends on several independent variables, the correlations of Y with each of them are weak and yet "remarkable". </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=37" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Sparse_PCA">Sparse PCA</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=38" title="Edit section: Sparse PCA"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sparse_PCA" title="Sparse PCA">Sparse PCA</a></div> <p>A particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables. <a href="/wiki/Sparse_PCA" title="Sparse PCA">Sparse PCA</a> overcomes this disadvantage by finding linear combinations that contain just a few input variables. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by adding sparsity constraint on the input variables. Several approaches have been proposed, including </p> <ul><li>a regression framework,<sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup></li> <li>a convex relaxation/semidefinite programming framework,<sup id="cite_ref-SDP_78-0" class="reference"><a href="#cite_note-SDP-78"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup></li> <li>a generalized power method framework<sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup></li> <li>an alternating maximization framework<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup></li> <li>forward-backward greedy search and exact methods using branch-and-bound techniques,<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup></li> <li>Bayesian formulation framework.<sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup></li></ul> <p>The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies were recently reviewed in a survey paper.<sup id="cite_ref-83" class="reference"><a href="#cite_note-83"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Nonlinear_PCA">Nonlinear PCA</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=39" title="Edit section: Nonlinear PCA"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Elmap_breastcancer_wiki.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Elmap_breastcancer_wiki.png/300px-Elmap_breastcancer_wiki.png" decoding="async" width="300" height="353" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Elmap_breastcancer_wiki.png/450px-Elmap_breastcancer_wiki.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/84/Elmap_breastcancer_wiki.png 2x" data-file-width="561" data-file-height="661" /></a><figcaption> Linear PCA versus nonlinear Principal Manifolds<sup id="cite_ref-84" class="reference"><a href="#cite_note-84"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> for <a href="/wiki/Scientific_visualization" title="Scientific visualization">visualization</a> of <a href="/wiki/Breast_cancer" title="Breast cancer">breast cancer</a> <a href="/wiki/Microarray" title="Microarray">microarray</a> data: a) Configuration of nodes and 2D Principal Surface in the 3D PCA linear manifold. The dataset is curved and cannot be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The "basal" breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the <a href="/wiki/Elastic_map" title="Elastic map">elastic maps</a> algorithm. Data are available for public competition.<sup id="cite_ref-85" class="reference"><a href="#cite_note-85"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> Software is available for free non-commercial use.<sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Most of the modern methods for <a href="/wiki/Nonlinear_dimensionality_reduction" title="Nonlinear dimensionality reduction">nonlinear dimensionality reduction</a> find their theoretical and algorithmic roots in PCA or K-means. Pearson's original idea was to take a straight line (or plane) which will be "the best fit" to a set of data points. <a href="/wiki/Trevor_Hastie" title="Trevor Hastie">Trevor Hastie</a> expanded on this concept by proposing <b>Principal <a href="/wiki/Curve" title="Curve">curves</a></b><sup id="cite_ref-87" class="reference"><a href="#cite_note-87"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> as the natural extension for the geometric interpretation of PCA, which explicitly constructs a manifold for data <a href="/wiki/Approximation" title="Approximation">approximation</a> followed by <a href="/wiki/Projection_(mathematics)" title="Projection (mathematics)">projecting</a> the points onto it. See also the <a href="/wiki/Elastic_map" title="Elastic map">elastic map</a> algorithm and <a href="/wiki/Principal_geodesic_analysis" title="Principal geodesic analysis">principal geodesic analysis</a>.<sup id="cite_ref-88" class="reference"><a href="#cite_note-88"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> Another popular generalization is <a href="/wiki/Kernel_PCA" class="mw-redirect" title="Kernel PCA">kernel PCA</a>, which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel. </p><p>In <a href="/wiki/Multilinear_subspace_learning" title="Multilinear subspace learning">multilinear subspace learning</a>,<sup id="cite_ref-Vasilescu2003_89-0" class="reference"><a href="#cite_note-Vasilescu2003-89"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Vasilescu2002tensorfaces_90-0" class="reference"><a href="#cite_note-Vasilescu2002tensorfaces-90"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-MPCA-MICA2005_91-0" class="reference"><a href="#cite_note-MPCA-MICA2005-91"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup> PCA is generalized to <a href="/wiki/Multilinear_principal_component_analysis" title="Multilinear principal component analysis">multilinear PCA</a> (MPCA) that extracts features directly from tensor representations. MPCA is solved by performing PCA in each mode of the tensor iteratively. MPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non-negative MPCA and robust MPCA. </p><p><i>N</i>-way principal component analysis may be performed with models such as <a href="/wiki/Tucker_decomposition" title="Tucker decomposition">Tucker decomposition</a>, <a href="/wiki/PARAFAC" class="mw-redirect" title="PARAFAC">PARAFAC</a>, multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS. </p> <div class="mw-heading mw-heading3"><h3 id="Robust_PCA">Robust PCA</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=40" title="Edit section: Robust PCA"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While PCA finds the mathematically optimal method (as in minimizing the squared error), it is still sensitive to <a href="/wiki/Outlier" title="Outlier">outliers</a> in the data that produce large errors, something that the method tries to avoid in the first place. It is therefore common practice to remove outliers before computing PCA. However, in some contexts, outliers can be difficult to identify.<sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup> For example, in <a href="/wiki/Data_mining" title="Data mining">data mining</a> algorithms like <a href="/wiki/Correlation_clustering" title="Correlation clustering">correlation clustering</a>, the assignment of points to clusters and outliers is not known beforehand. A recently proposed generalization of PCA<sup id="cite_ref-93" class="reference"><a href="#cite_note-93"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup> based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy. </p><p>Outlier-resistant variants of PCA have also been proposed, based on L1-norm formulations (<a href="/wiki/L1-norm_principal_component_analysis" title="L1-norm principal component analysis">L1-PCA</a>).<sup id="cite_ref-mark2014_7-1" class="reference"><a href="#cite_note-mark2014-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-mark2017_5-1" class="reference"><a href="#cite_note-mark2017-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Robust_principal_component_analysis" title="Robust principal component analysis">Robust principal component analysis</a> (RPCA) via decomposition in low-rank and sparse matrices is a modification of PCA that works well with respect to grossly corrupted observations.<sup id="cite_ref-RPCA_94-0" class="reference"><a href="#cite_note-RPCA-94"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-RPCA-BOUWMANS_95-0" class="reference"><a href="#cite_note-RPCA-BOUWMANS-95"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-RPCA-BOUWMANS-COSREV_96-0" class="reference"><a href="#cite_note-RPCA-BOUWMANS-COSREV-96"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Similar_techniques">Similar techniques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=41" title="Edit section: Similar techniques"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Independent_component_analysis">Independent component analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=42" title="Edit section: Independent component analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Independent_component_analysis" title="Independent component analysis">Independent component analysis</a> (ICA) is directed to similar problems as principal component analysis, but finds additively separable components rather than successive approximations. </p> <div class="mw-heading mw-heading3"><h3 id="Network_component_analysis">Network component analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=43" title="Edit section: Network component analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>, it tries to decompose it into two matrices such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=AP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>A</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=AP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f74a670b31bcb899a0e44fb324834c40f3976b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.363ex; height:2.176ex;" alt="{\displaystyle E=AP}"></span>. A key difference from techniques such as PCA and ICA is that some of the entries of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> are constrained to be 0. Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied : </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> has full column rank</li> <li>Each column of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> must have at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe61101dd489eb8e1a974ab6c409190a76541bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.586ex; height:2.343ex;" alt="{\displaystyle L-1}"></span> zeroes where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is the number of columns of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> (or alternatively the number of rows of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>). The justification for this criterion is that if a node is removed from the regulatory layer along with all the output nodes connected to it, the result must still be characterized by a connectivity matrix with full column rank.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> must have full row rank.</li></ol> <p>then the decomposition is unique up to multiplication by a scalar.<sup id="cite_ref-97" class="reference"><a href="#cite_note-97"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Discriminant_analysis_of_principal_components">Discriminant analysis of principal components</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=44" title="Edit section: Discriminant analysis of principal components"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Discriminant analysis of principal components (DAPC) is a multivariate method used to identify and describe clusters of genetically related individuals. Genetic variation is partitioned into two components: variation between groups and within groups, and it maximizes the former. Linear discriminants are linear combinations of alleles which best separate the clusters. Alleles that most contribute to this discrimination are therefore those that are the most markedly different across groups. The contributions of alleles to the groupings identified by DAPC can allow identifying regions of the genome driving the genetic divergence among groups<sup id="cite_ref-98" class="reference"><a href="#cite_note-98"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup> In DAPC, data is first transformed using a principal components analysis (PCA) and subsequently clusters are identified using discriminant analysis (DA). </p><p>A DAPC can be realized on R using the package Adegenet. (more info: <a rel="nofollow" class="external text" href="https://adegenet.r-forge.r-project.org/">adegenet on the web</a>) </p> <div class="mw-heading mw-heading3"><h3 id="Directional_component_analysis">Directional component analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=45" title="Edit section: Directional component analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Directional_component_analysis" title="Directional component analysis">Directional component analysis</a> (DCA) is a method used in the atmospheric sciences for analysing multivariate datasets.<sup id="cite_ref-jewson_99-0" class="reference"><a href="#cite_note-jewson-99"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> Like PCA, it allows for dimension reduction, improved visualization and improved interpretability of large data-sets. Also like PCA, it is based on a covariance matrix derived from the input dataset. The difference between PCA and DCA is that DCA additionally requires the input of a vector direction, referred to as the impact. Whereas PCA maximises explained variance, DCA maximises probability density given impact. The motivation for DCA is to find components of a multivariate dataset that are both likely (measured using probability density) and important (measured using the impact). DCA has been used to find the most likely and most serious heat-wave patterns in weather prediction ensembles ,<sup id="cite_ref-scheretal_100-0" class="reference"><a href="#cite_note-scheretal-100"><span class="cite-bracket">[</span>100<span class="cite-bracket">]</span></a></sup> and the most likely and most impactful changes in rainfall due to climate change .<sup id="cite_ref-jewsonetal_101-0" class="reference"><a href="#cite_note-jewsonetal-101"><span class="cite-bracket">[</span>101<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Software/source_code"><span id="Software.2Fsource_code"></span>Software/source code</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=46" title="Edit section: Software/source code"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/ALGLIB" title="ALGLIB">ALGLIB</a> – a C++ and C# library that implements PCA and truncated PCA</li> <li><a href="/wiki/Analytica_(software)" title="Analytica (software)">Analytica</a> – The built-in EigenDecomp function computes principal components.</li> <li><a href="/wiki/ELKI" title="ELKI">ELKI</a> – includes PCA for projection, including robust variants of PCA, as well as PCA-based <a href="/wiki/Cluster_analysis" title="Cluster analysis">clustering algorithms</a>.</li> <li><a href="/wiki/Gretl" title="Gretl">Gretl</a> – principal component analysis can be performed either via the <code>pca</code> command or via the <code>princomp()</code> function.</li> <li><a href="/wiki/Julia_language" class="mw-redirect" title="Julia language">Julia</a> – Supports PCA with the <code>pca</code> function in the MultivariateStats package</li> <li><a href="/wiki/KNIME" title="KNIME">KNIME</a> – A java based nodal arranging software for Analysis, in this the nodes called PCA, PCA compute, PCA Apply, PCA inverse make it easily.</li> <li><a href="/wiki/Maple_(software)" title="Maple (software)">Maple (software)</a> – The PCA command is used to perform a principal component analysis on a set of data.</li> <li><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> – Implements principal component analysis with the PrincipalComponents command using both covariance and correlation methods.</li> <li><a rel="nofollow" class="external text" href="https://github.com/markrogoyski/math-php">MathPHP</a> – <a href="/wiki/PHP" title="PHP">PHP</a> mathematics library with support for PCA.</li> <li><a href="/wiki/MATLAB" title="MATLAB">MATLAB</a> – The SVD function is part of the basic system. In the Statistics Toolbox, the functions <code>princomp</code> and <code>pca</code> (R2012b) give the principal components, while the function <code>pcares</code> gives the residuals and reconstructed matrix for a low-rank PCA approximation.</li> <li><a href="/wiki/Matplotlib" title="Matplotlib">Matplotlib</a> – <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> library have a PCA package in the .mlab module.</li> <li><a href="/wiki/Mlpack" title="Mlpack">mlpack</a> – Provides an implementation of principal component analysis in <a href="/wiki/C%2B%2B" title="C++">C++</a>.</li> <li><a rel="nofollow" class="external text" href="https://github.com/mikerabat/mrmath">mrmath</a> – A high performance math library for <a href="/wiki/Delphi_(software)" title="Delphi (software)">Delphi</a> and <a href="/wiki/Free_Pascal" title="Free Pascal">FreePascal</a> can perform PCA; including robust variants.</li> <li><a href="/wiki/NAG_Numerical_Library" title="NAG Numerical Library">NAG Library</a> – Principal components analysis is implemented via the <code>g03aa</code> routine (available in both the Fortran versions of the Library).</li> <li><a href="/wiki/NMath" title="NMath">NMath</a> – Proprietary numerical library containing PCA for the <a href="/wiki/.NET_Framework" title=".NET Framework">.NET Framework</a>.</li> <li><a href="/wiki/GNU_Octave" title="GNU Octave">GNU Octave</a> – Free software computational environment mostly compatible with MATLAB, the function <code>princomp</code> gives the principal component.</li> <li><a href="/wiki/OpenCV" title="OpenCV">OpenCV</a></li> <li><a href="/wiki/Oracle_Database" title="Oracle Database">Oracle Database</a> 12c – Implemented via <code>DBMS_DATA_MINING.SVDS_SCORING_MODE</code> by specifying setting value <code>SVDS_SCORING_PCA</code></li> <li><a href="/wiki/Orange_(software)" title="Orange (software)">Orange (software)</a> – Integrates PCA in its visual programming environment. PCA displays a scree plot (degree of explained variance) where user can interactively select the number of principal components.</li> <li><a href="/wiki/Origin_(data_analysis_software)" title="Origin (data analysis software)">Origin</a> – Contains PCA in its Pro version.</li> <li><a href="/wiki/Qlucore" title="Qlucore">Qlucore</a> – Commercial software for analyzing multivariate data with instant response using PCA.</li> <li><a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> – <a href="/wiki/Free_software" title="Free software">Free</a> statistical package, the functions <code>princomp</code> and <code>prcomp</code> can be used for principal component analysis; <code>prcomp</code> uses <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> which generally gives better numerical accuracy. Some packages that implement PCA in R, include, but are not limited to: <code>ade4</code>, <code>vegan</code>, <code>ExPosition</code>, <code>dimRed</code>, and <code>FactoMineR</code>.</li> <li><a href="/wiki/SAS_(software)" title="SAS (software)">SAS</a> – Proprietary software; for example, see<sup id="cite_ref-102" class="reference"><a href="#cite_note-102"><span class="cite-bracket">[</span>102<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Scikit-learn" title="Scikit-learn">scikit-learn</a> – Python library for machine learning which contains PCA, Probabilistic PCA, Kernel PCA, Sparse PCA and other techniques in the decomposition module.</li> <li><a href="/wiki/Scilab" title="Scilab">Scilab</a> – Free and open-source, cross-platform numerical computational package, the function <code>princomp</code> computes principal component analysis, the function <code>pca</code> computes principal component analysis with standardized variables.</li> <li><a href="/wiki/SPSS" title="SPSS">SPSS</a> – Proprietary software most commonly used by social scientists for PCA, factor analysis and associated cluster analysis.</li> <li><a href="/wiki/Weka_(machine_learning)" class="mw-redirect" title="Weka (machine learning)">Weka</a> – Java library for machine learning which contains modules for computing principal components.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=47" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Correspondence_analysis" title="Correspondence analysis">Correspondence analysis</a> (for contingency tables)</li> <li><a href="/wiki/Multiple_correspondence_analysis" title="Multiple correspondence analysis">Multiple correspondence analysis</a> (for qualitative variables)</li> <li><a href="/wiki/Factor_analysis_of_mixed_data" title="Factor analysis of mixed data">Factor analysis of mixed data</a> (for quantitative <b>and</b> qualitative variables)</li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/CUR_matrix_approximation" title="CUR matrix approximation">CUR matrix approximation</a> (can replace of low-rank SVD approximation)</li> <li><a href="/wiki/Detrended_correspondence_analysis" title="Detrended correspondence analysis">Detrended correspondence analysis</a></li> <li><a href="/wiki/Directional_component_analysis" title="Directional component analysis">Directional component analysis</a></li> <li><a href="/wiki/Dynamic_mode_decomposition" title="Dynamic mode decomposition">Dynamic mode decomposition</a></li> <li><a href="/wiki/Eigenface" title="Eigenface">Eigenface</a></li> <li><a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">Expectation–maximization algorithm</a></li> <li><a href="https://en.wikiversity.org/wiki/Exploratory_factor_analysis" class="extiw" title="v:Exploratory factor analysis">Exploratory factor analysis</a> (Wikiversity)</li> <li><a href="/wiki/Factorial_code" title="Factorial code">Factorial code</a></li> <li><a href="/wiki/Functional_principal_component_analysis" title="Functional principal component analysis">Functional principal component analysis</a></li> <li><a href="/wiki/Geometric_data_analysis" title="Geometric data analysis">Geometric data analysis</a></li> <li><a href="/wiki/Independent_component_analysis" title="Independent component analysis">Independent component analysis</a></li> <li><a href="/wiki/Kernel_PCA" class="mw-redirect" title="Kernel PCA">Kernel PCA</a></li> <li><a href="/wiki/L1-norm_principal_component_analysis" title="L1-norm principal component analysis">L1-norm principal component analysis</a></li> <li><a href="/wiki/Low-rank_approximation" title="Low-rank approximation">Low-rank approximation</a></li> <li><a href="/wiki/Matrix_decomposition" title="Matrix decomposition">Matrix decomposition</a></li> <li><a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">Non-negative matrix factorization</a></li> <li><a href="/wiki/Nonlinear_dimensionality_reduction" title="Nonlinear dimensionality reduction">Nonlinear dimensionality reduction</a></li> <li><a href="/wiki/Oja%27s_rule" title="Oja's rule">Oja's rule</a></li> <li><a href="/wiki/Point_distribution_model" title="Point distribution model">Point distribution model</a> (PCA applied to morphometry and computer vision)</li> <li><a href="https://en.wikibooks.org/wiki/Statistics/Multivariate_Data_Analysis/Principal_Component_Analysis" class="extiw" title="b:Statistics/Multivariate Data Analysis/Principal Component Analysis">Principal component analysis</a> (Wikibooks)</li> <li><a href="/wiki/Principal_component_regression" title="Principal component regression">Principal component regression</a></li> <li><a href="/wiki/Singular_spectrum_analysis" title="Singular spectrum analysis">Singular spectrum analysis</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li> <li><a href="/wiki/Sparse_PCA" title="Sparse PCA">Sparse PCA</a></li> <li><a href="/wiki/Transform_coding" title="Transform coding">Transform coding</a></li> <li><a href="/wiki/Weighted_least_squares" title="Weighted least squares">Weighted least squares</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=48" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free 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September 1, 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">June 4,</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=i2tutorials&rft.atitle=What+are+the+Pros+and+cons+of+the+PCA%3F&rft.date=2019-09-01&rft_id=https%3A%2F%2Fwww.i2tutorials.com%2Fwhat-are-the-pros-and-cons-of-the-pca%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-abbott-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-abbott_28-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbbott2014" class="citation book cs1">Abbott, Dean (May 2014). <i>Applied Predictive Analytics</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781118727966" title="Special:BookSources/9781118727966"><bdi>9781118727966</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Predictive+Analytics&rft.pub=Wiley&rft.date=2014-05&rft.isbn=9781118727966&rft.aulast=Abbott&rft.aufirst=Dean&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-jiang-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-jiang_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-jiang_29-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJiangEskridge2000" class="citation journal cs1">Jiang, Hong; Eskridge, Kent M. (2000). <a rel="nofollow" class="external text" href="https://newprairiepress.org/agstatconference/2000/proceedings/13/">"Bias in Principal Components Analysis Due to Correlated Observations"</a>. <i>Conference on Applied Statistics in Agriculture</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4148%2F2475-7772.1247">10.4148/2475-7772.1247</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2475-7772">2475-7772</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Conference+on+Applied+Statistics+in+Agriculture&rft.atitle=Bias+in+Principal+Components+Analysis+Due+to+Correlated+Observations&rft.date=2000&rft_id=info%3Adoi%2F10.4148%2F2475-7772.1247&rft.issn=2475-7772&rft.aulast=Jiang&rft.aufirst=Hong&rft.au=Eskridge%2C+Kent+M.&rft_id=https%3A%2F%2Fnewprairiepress.org%2Fagstatconference%2F2000%2Fproceedings%2F13%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLinsker1988" class="citation journal cs1">Linsker, Ralph (March 1988). "Self-organization in a perceptual network". <i>IEEE Computer</i>. <b>21</b> (3): <span class="nowrap">105–</span>117. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F2.36">10.1109/2.36</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1527671">1527671</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Computer&rft.atitle=Self-organization+in+a+perceptual+network&rft.volume=21&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E105-%3C%2Fspan%3E117&rft.date=1988-03&rft_id=info%3Adoi%2F10.1109%2F2.36&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1527671%23id-name%3DS2CID&rft.aulast=Linsker&rft.aufirst=Ralph&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeco_&_Obradovic1996" class="citation book cs1">Deco & Obradovic (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=z4XTBwAAQBAJ"><i>An Information-Theoretic Approach to Neural Computing</i></a>. New York, NY: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781461240167" title="Special:BookSources/9781461240167"><bdi>9781461240167</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Information-Theoretic+Approach+to+Neural+Computing&rft.place=New+York%2C+NY&rft.pub=Springer&rft.date=1996&rft.isbn=9781461240167&rft.au=Deco+%26+Obradovic&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dz4XTBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlumbley1991" class="citation book cs1">Plumbley, Mark (1991). <i>Information theory and unsupervised neural networks</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+theory+and+unsupervised+neural+networks&rft.date=1991&rft.aulast=Plumbley&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span>Tech Note</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeigerKubin,_Gernot2013" class="citation journal cs1">Geiger, Bernhard; Kubin, Gernot (January 2013). "Signal Enhancement as Minimization of Relevant Information Loss". <i>Proc. ITG Conf. On Systems, Communication and Coding</i>. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1205.6935">1205.6935</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012arXiv1205.6935G">2012arXiv1205.6935G</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+ITG+Conf.+On+Systems%2C+Communication+and+Coding&rft.atitle=Signal+Enhancement+as+Minimization+of+Relevant+Information+Loss&rft.date=2013-01&rft_id=info%3Aarxiv%2F1205.6935&rft_id=info%3Abibcode%2F2012arXiv1205.6935G&rft.aulast=Geiger&rft.aufirst=Bernhard&rft.au=Kubin%2C+Gernot&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">See also the tutorial <a rel="nofollow" class="external text" href="http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf">here</a></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc552.htm">"Engineering Statistics Handbook Section 6.5.5.2"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">19 January</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Engineering+Statistics+Handbook+Section+6.5.5.2&rft_id=http%3A%2F%2Fwww.itl.nist.gov%2Fdiv898%2Fhandbook%2Fpmc%2Fsection5%2Fpmc552.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">A.A. Miranda, Y.-A. Le Borgne, and G. Bontempi. <a rel="nofollow" class="external text" href="http://www.ulb.ac.be/di/map/yleborgn/pub/NPL_PCA_07.pdf">New Routes from Minimal Approximation Error to Principal Components</a>, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbdi._H.Williams,_L.J.2010" class="citation journal cs1"><a href="/w/index.php?title=AbdiWilliams&action=edit&redlink=1" class="new" title="AbdiWilliams (page does not exist)">Abdi. H.</a> & Williams, L.J. (2010). 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href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeSarboHausmannKukitz2007" class="citation journal cs1">DeSarbo, Wayne; Hausmann, Robert; Kukitz, Jeffrey (2007). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/247623679">"Restricted principal components analysis for marketing research"</a>. <i>Journal of Marketing in Management</i>. <b>2</b>: <span class="nowrap">305–</span>328 – via ResearchGate.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Marketing+in+Management&rft.atitle=Restricted+principal+components+analysis+for+marketing+research&rft.volume=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E305-%3C%2Fspan%3E328&rft.date=2007&rft.aulast=DeSarbo&rft.aufirst=Wayne&rft.au=Hausmann%2C+Robert&rft.au=Kukitz%2C+Jeffrey&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F247623679&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuttonBlank2013" class="citation book cs1">Dutton, William H; Blank, Grant (2013). <a rel="nofollow" class="external text" href="http://oxis.oii.ox.ac.uk/wp-content/uploads/2014/11/OxIS-2013.pdf"><i>Cultures of the Internet: The Internet in Britain</i></a> <span class="cs1-format">(PDF)</span>. Oxford Internet Institute. p. 6.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cultures+of+the+Internet%3A+The+Internet+in+Britain&rft.pages=6&rft.pub=Oxford+Internet+Institute&rft.date=2013&rft.aulast=Dutton&rft.aufirst=William+H&rft.au=Blank%2C+Grant&rft_id=http%3A%2F%2Foxis.oii.ox.ac.uk%2Fwp-content%2Fuploads%2F2014%2F11%2FOxIS-2013.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlood2008" class="citation journal cs1">Flood, Joe (2008). <a rel="nofollow" class="external text" href="https://www.academia.edu/33218811">"Multinomial Analysis for Housing Careers Survey"</a>. <i>Paper to the European Network for Housing Research Conference, Dublin</i><span class="reference-accessdate">. Retrieved <span class="nowrap">6 May</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Paper+to+the+European+Network+for+Housing+Research+Conference%2C+Dublin&rft.atitle=Multinomial+Analysis+for+Housing+Careers+Survey&rft.date=2008&rft.aulast=Flood&rft.aufirst=Joe&rft_id=https%3A%2F%2Fwww.academia.edu%2F33218811&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> <li id="cite_note-Miller-58"><span class="mw-cite-backlink">^ <a href="#cite_ref-Miller_58-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Miller_58-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">See Ch. 9 in Michael B. Miller (2013). <i>Mathematics and Statistics for Financial Risk Management</i>, 2nd Edition. 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UCLA<span class="reference-accessdate">. Retrieved <span class="nowrap">29 May</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Institute+for+Digital+Research+and+Education&rft.atitle=Principal+Components+Analysis&rft_id=https%3A%2F%2Fstats.idre.ucla.edu%2Fsas%2Foutput%2Fprincipal-components-analysis%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=49" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Jackson, J.E. (1991). <i>A User's Guide to Principal Components</i> (Wiley).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJolliffe1986" class="citation book cs1">Jolliffe, I. T. (1986). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/principalcompone00joll_0/page/487"><i>Principal Component Analysis</i></a></span>. Springer Series in Statistics. Springer-Verlag. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/principalcompone00joll_0/page/487">487</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.149.8828">10.1.1.149.8828</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fb98835">10.1007/b98835</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95442-4" title="Special:BookSources/978-0-387-95442-4"><bdi>978-0-387-95442-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principal+Component+Analysis&rft.series=Springer+Series+in+Statistics&rft.pages=487&rft.pub=Springer-Verlag&rft.date=1986&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.149.8828%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1007%2Fb98835&rft.isbn=978-0-387-95442-4&rft.aulast=Jolliffe&rft.aufirst=I.+T.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprincipalcompone00joll_0%2Fpage%2F487&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJolliffe2002" class="citation book cs1">Jolliffe, I. T. (2002). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/b98835"><i>Principal Component Analysis</i></a>. Springer Series in Statistics. New York: Springer-Verlag. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fb98835">10.1007/b98835</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95442-4" title="Special:BookSources/978-0-387-95442-4"><bdi>978-0-387-95442-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principal+Component+Analysis&rft.place=New+York&rft.series=Springer+Series+in+Statistics&rft.pub=Springer-Verlag&rft.date=2002&rft_id=info%3Adoi%2F10.1007%2Fb98835&rft.isbn=978-0-387-95442-4&rft.aulast=Jolliffe&rft.aufirst=I.+T.&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2Fb98835&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrincipal+component+analysis" class="Z3988"></span></li> <li>Husson François, Lê Sébastien & Pagès Jérôme (2009). <i>Exploratory Multivariate Analysis by Example Using R</i>. Chapman & Hall/CRC The R Series, London. 224p. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-7535-0938-2" title="Special:BookSources/978-2-7535-0938-2">978-2-7535-0938-2</a></li> <li>Pagès Jérôme (2014). <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=_RIeBQAAQBAJ&q=%22principal+component+analysis%22">Multiple Factor Analysis by Example Using R</a></i>. Chapman & Hall/CRC The R Series London 272 p</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Principal_component_analysis&action=edit&section=50" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output 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title="commons:Category:Principal component analysis">Principal component analysis</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=UUxIXU_Ob6E"><span class="plainlinks">University of Copenhagen video by Rasmus Bro</span></a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ey2PE5xi9-A#t=2385"><span class="plainlinks">Stanford University video by Andrew Ng</span></a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a></li> <li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1404.1100">A Tutorial on Principal Component Analysis</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=BfTMmoDFXyE"><span class="plainlinks">A layman's introduction to principal component analysis</span></a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> (a video of less than 100 seconds.)</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=FgakZw6K1QQ"><span class="plainlinks">StatQuest: StatQuest: Principal Component Analysis (PCA), Step-by-Step</span></a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a></li> <li><a rel="nofollow" class="external text" href="https://stats.stackexchange.com/a/140579">Layman's explanation in making sense of principal component analysis, eigenvectors & eigenvalues</a> on <a href="/wiki/Stack_Overflow" title="Stack Overflow">Stack Overflow</a></li> <li>See also the list of <a href="#Software/source_code">Software implementations</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"></div><div role="navigation" class="navbox" aria-labelledby="Statistics636" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Statistics" title="Template:Statistics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistics" title="Template talk:Statistics"><abbr title="Discuss this 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title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection636" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference636" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis636" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis636" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a class="mw-selflink selflink">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" 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