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</ul> </li> <li id="toc-Exploratory_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exploratory_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Exploratory analysis</span> </div> </a> <ul id="toc-Exploratory_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Estimation,_filtering,_and_smoothing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimation,_filtering,_and_smoothing"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Estimation, filtering, and smoothing</span> </div> </a> <ul id="toc-Estimation,_filtering,_and_smoothing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curve_fitting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curve_fitting"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Curve fitting</span> </div> </a> <ul id="toc-Curve_fitting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Function_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Function_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Function approximation</span> </div> </a> <ul id="toc-Function_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prediction_and_forecasting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prediction_and_forecasting"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Prediction and forecasting</span> </div> </a> <ul id="toc-Prediction_and_forecasting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classification"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Classification</span> </div> </a> <ul id="toc-Classification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Segmentation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Segmentation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Segmentation</span> </div> </a> <ul id="toc-Segmentation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Clustering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Clustering"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Clustering</span> </div> </a> <ul id="toc-Clustering-sublist" class="vector-toc-list"> <li id="toc-Subsequence_time_series_clustering" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Subsequence_time_series_clustering"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9.1</span> <span>Subsequence time series clustering</span> </div> </a> <ul id="toc-Subsequence_time_series_clustering-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Models" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Models"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Models</span> </div> </a> <button aria-controls="toc-Models-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 Models subsection</span> </button> <ul id="toc-Models-sublist" class="vector-toc-list"> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Notation</span> </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conditions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Conditions</span> </div> </a> <ul id="toc-Conditions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tools" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tools"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Tools</span> </div> </a> <ul id="toc-Tools-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measures"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Measures</span> </div> </a> <ul id="toc-Measures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Visualization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Visualization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Visualization</span> </div> </a> <button aria-controls="toc-Visualization-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 Visualization subsection</span> </button> <ul id="toc-Visualization-sublist" class="vector-toc-list"> <li id="toc-Overlapping_charts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Overlapping_charts"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Overlapping charts</span> </div> </a> <ul id="toc-Overlapping_charts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Separated_charts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Separated_charts"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Separated charts</span> </div> </a> <ul id="toc-Separated_charts-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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" > <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 id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Time series</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 45 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-45" 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">45 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/%D9%85%D8%AA%D8%B3%D9%84%D8%B3%D9%84%D8%A9_%D8%B2%D9%85%D9%86%D9%8A%D8%A9" title="متسلسلة زمنية – Arabic" lang="ar" hreflang="ar" data-title="متسلسلة زمنية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Zaman_s%C4%B1ralar%C4%B1" title="Zaman sıraları – Azerbaijani" lang="az" hreflang="az" data-title="Zaman sıraları" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D1%80%D0%B5%D0%BC%D0%B5%D0%B2%D0%B8_%D1%80%D0%B5%D0%B4" title="Времеви ред – Bulgarian" lang="bg" hreflang="bg" data-title="Времеви ред" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/S%C3%A8rie_temporal" title="Sèrie temporal – Catalan" lang="ca" hreflang="ca" data-title="Sèrie temporal" 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/%C4%8Casov%C3%A1_%C5%99ada" title="Časová řada – Czech" lang="cs" hreflang="cs" data-title="Časová řada" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cbk-zam mw-list-item"><a href="https://cbk-zam.wikipedia.org/wiki/Temporal_serie" title="Temporal serie – Chavacano" lang="cbk" hreflang="cbk" data-title="Temporal serie" data-language-autonym="Chavacano de Zamboanga" data-language-local-name="Chavacano" class="interlanguage-link-target"><span>Chavacano de Zamboanga</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Tidsserie" title="Tidsserie – Danish" lang="da" hreflang="da" data-title="Tidsserie" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zeitreihenanalyse" title="Zeitreihenanalyse – German" lang="de" hreflang="de" data-title="Zeitreihenanalyse" 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/Aegrida" title="Aegrida – Estonian" lang="et" hreflang="et" data-title="Aegrida" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A7%CF%81%CE%BF%CE%BD%CE%BF%CE%BB%CE%BF%CE%B3%CE%B9%CE%BA%CE%AD%CF%82_%CE%A3%CE%B5%CE%B9%CF%81%CE%AD%CF%82" title="Χρονολογικές Σειρές – Greek" lang="el" hreflang="el" data-title="Χρονολογικές Σειρές" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Serie_temporal" title="Serie temporal – Spanish" lang="es" hreflang="es" data-title="Serie temporal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Denbora_serie" title="Denbora serie – Basque" lang="eu" hreflang="eu" data-title="Denbora serie" 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%B3%D8%B1%DB%8C_%D8%B2%D9%85%D8%A7%D9%86%DB%8C" title="سری زمانی – Persian" lang="fa" hreflang="fa" data-title="سری زمانی" 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/S%C3%A9rie_temporelle" title="Série temporelle – French" lang="fr" hreflang="fr" data-title="Série temporelle" 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/Serie_temporal" title="Serie temporal – Galician" lang="gl" hreflang="gl" data-title="Serie temporal" 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%8B%9C%EA%B3%84%EC%97%B4" title="시계열 – Korean" lang="ko" hreflang="ko" data-title="시계열" 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/Deret_waktu" title="Deret waktu – Indonesian" lang="id" hreflang="id" data-title="Deret waktu" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Serie_storica" title="Serie storica – Italian" lang="it" hreflang="it" data-title="Serie storica" 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%A1%D7%93%D7%A8%D7%94_%D7%A2%D7%AA%D7%99%D7%AA" title="סדרה עתית – Hebrew" lang="he" hreflang="he" data-title="סדרה עתית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Dh%C3%A8r%C3%A8t_wektu" title="Dhèrèt wektu – Javanese" lang="jv" hreflang="jv" data-title="Dhèrèt wektu" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A3%D0%B0%D2%9B%D1%8B%D1%82_%D2%9B%D0%B0%D1%82%D0%B0%D1%80%D1%8B" title="Уақыт қатары – Kazakh" lang="kk" hreflang="kk" data-title="Уақыт қатары" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dinamin%C4%97_eilut%C4%97" title="Dinaminė eilutė – Lithuanian" lang="lt" hreflang="lt" data-title="Dinaminė eilutė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Id%C5%91sor" title="Idősor – Hungarian" lang="hu" hreflang="hu" data-title="Idősor" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D1%81%D0%BA%D0%B0_%D0%BD%D0%B8%D0%B7%D0%B0" title="Временска низа – Macedonian" lang="mk" hreflang="mk" data-title="Временска низа" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tijdreeksanalyse" title="Tijdreeksanalyse – Dutch" lang="nl" hreflang="nl" data-title="Tijdreeksanalyse" 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/%E6%99%82%E7%B3%BB%E5%88%97" title="時系列 – Japanese" lang="ja" hreflang="ja" data-title="時系列" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tidsrekke" title="Tidsrekke – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tidsrekke" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tidsrekkjeanalyse" title="Tidsrekkjeanalyse – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Tidsrekkjeanalyse" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Szereg_czasowy" title="Szereg czasowy – Polish" lang="pl" hreflang="pl" data-title="Szereg czasowy" 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/S%C3%A9rie_temporal" title="Série temporal – Portuguese" lang="pt" hreflang="pt" data-title="Série temporal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Serie_temporal%C4%83" title="Serie temporală – Romanian" lang="ro" hreflang="ro" data-title="Serie temporală" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D0%BD%D0%BE%D0%B9_%D1%80%D1%8F%D0%B4" title="Временной ряд – Russian" lang="ru" hreflang="ru" data-title="Временной ряд" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Serit%C3%AB_kohore" title="Seritë kohore – Albanian" lang="sq" hreflang="sq" data-title="Seritë kohore" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Time_Series" title="Time Series – Simple English" lang="en-simple" hreflang="en-simple" data-title="Time Series" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/%C4%8Casovne_vrste" title="Časovne vrste – Slovenian" lang="sl" hreflang="sl" data-title="Časovne vrste" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/D%C3%A9r%C3%A9t_waktu" title="Dérét waktu – Sundanese" lang="su" hreflang="su" data-title="Dérét waktu" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aikasarja" title="Aikasarja – Finnish" lang="fi" hreflang="fi" data-title="Aikasarja" 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/Tidsserie" title="Tidsserie – Swedish" lang="sv" hreflang="sv" data-title="Tidsserie" 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/Serye_ng_panahon" title="Serye ng panahon – Tagalog" lang="tl" hreflang="tl" data-title="Serye ng panahon" 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/Zaman_serisi" title="Zaman serisi – Turkish" lang="tr" hreflang="tr" data-title="Zaman serisi" 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%A7%D0%B0%D1%81%D0%BE%D0%B2%D0%B8%D0%B9_%D1%80%D1%8F%D0%B4" title="Часовий ряд – Ukrainian" lang="uk" hreflang="uk" data-title="Часовий ряд" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B3%D9%84%D8%B3%D9%84%DB%81_%D8%B2%D9%85%D8%A7%DA%BA" title="سلسلہ زماں – Urdu" lang="ur" hreflang="ur" data-title="سلسلہ زماں" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Sequence of data points over time</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">Not to be confused with <a href="/wiki/Time_(TV_series)" class="mw-redirect mw-disambig" title="Time (TV series)"><i>Time</i> (TV series)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Random-data-plus-trend-r2.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Random-data-plus-trend-r2.png/250px-Random-data-plus-trend-r2.png" decoding="async" width="250" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Random-data-plus-trend-r2.png/375px-Random-data-plus-trend-r2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Random-data-plus-trend-r2.png/500px-Random-data-plus-trend-r2.png 2x" data-file-width="601" data-file-height="447" /></a><figcaption>Time series: random data plus trend, with best-fit line and different applied filters</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>time series</b> is a series of <a href="/wiki/Data_point" class="mw-redirect" title="Data point">data points</a> indexed (or listed or graphed) in time order. Most commonly, a time series is a <a href="/wiki/Sequence" title="Sequence">sequence</a> taken at successive equally spaced points in time. Thus it is a sequence of <a href="/wiki/Discrete-time" class="mw-redirect" title="Discrete-time">discrete-time</a> data. Examples of time series are heights of ocean <a href="/wiki/Tides" class="mw-redirect" title="Tides">tides</a>, counts of <a href="/wiki/Sunspots" class="mw-redirect" title="Sunspots">sunspots</a>, and the daily closing value of the <a href="/wiki/Dow_Jones_Industrial_Average" title="Dow Jones Industrial Average">Dow Jones Industrial Average</a>. </p><p>A time series is very frequently plotted via a <a href="/wiki/Run_chart" title="Run chart">run chart</a> (which is a temporal <a href="/wiki/Line_chart" title="Line chart">line chart</a>). Time series are used in <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="/wiki/Pattern_recognition" title="Pattern recognition">pattern recognition</a>, <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>, <a href="/wiki/Mathematical_finance" title="Mathematical finance">mathematical finance</a>, <a href="/wiki/Weather_forecasting" title="Weather forecasting">weather forecasting</a>, <a href="/wiki/Earthquake_prediction" title="Earthquake prediction">earthquake prediction</a>, <a href="/wiki/Electroencephalography" title="Electroencephalography">electroencephalography</a>, <a href="/wiki/Control_engineering" title="Control engineering">control engineering</a>, <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, <a href="/wiki/Communications_engineering" class="mw-redirect" title="Communications engineering">communications engineering</a>, and largely in any domain of applied <a href="/wiki/Applied_science" title="Applied science">science</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a> which involves <a href="/wiki/Time" title="Time">temporal</a> measurements. </p><p><b>Time series <i>analysis</i></b> comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. <b>Time series <i>forecasting</i></b> is the use of a <a href="/wiki/Model_(abstract)" class="mw-redirect" title="Model (abstract)">model</a> to predict future values based on previously observed values. Generally, time series data is modelled as a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a>. While <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a> is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series. </p><p>Time series data have a natural temporal ordering. This makes time series analysis distinct from <a href="/wiki/Cross-sectional_study" title="Cross-sectional study">cross-sectional studies</a>, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from <a href="/wiki/Spatial_data_analysis" class="mw-redirect" title="Spatial data analysis">spatial data analysis</a> where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A <a href="/wiki/Stochastic" title="Stochastic">stochastic</a> model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see <a href="/wiki/Time_reversibility" title="Time reversibility">time reversibility</a>). </p><p>Time series analysis can be applied to <a href="/wiki/Real_number" title="Real number">real-valued</a>, continuous data, <a href="https://en.wiktionary.org/wiki/discrete" class="extiw" title="wikt:discrete">discrete</a> <a href="/wiki/Data_type#Numeric_types" title="Data type">numeric</a> data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the <a href="/wiki/English_language" title="English language">English language</a><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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Methods_for_analysis">Methods for analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=1" title="Edit section: Methods for analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Methods for time series analysis may be divided into two classes: <a href="/wiki/Frequency-domain" class="mw-redirect" title="Frequency-domain">frequency-domain</a> methods and <a href="/wiki/Time-domain" class="mw-redirect" title="Time-domain">time-domain</a> methods. The former include <a href="/wiki/Frequency_spectrum#Spectrum_analysis" class="mw-redirect" title="Frequency spectrum">spectral analysis</a> and <a href="/wiki/Wavelet_analysis" class="mw-redirect" title="Wavelet analysis">wavelet analysis</a>; the latter include <a href="/wiki/Auto-correlation" class="mw-redirect" title="Auto-correlation">auto-correlation</a> and <a href="/wiki/Cross-correlation" title="Cross-correlation">cross-correlation</a> analysis. In the time domain, correlation and analysis can be made in a filter-like manner using <a href="/wiki/Scaled_correlation" title="Scaled correlation">scaled correlation</a>, thereby mitigating the need to operate in the frequency domain. </p><p>Additionally, time series analysis techniques may be divided into <a href="/wiki/Parametric_estimation" class="mw-redirect" title="Parametric estimation">parametric</a> and <a href="/wiki/Non-parametric_statistics" class="mw-redirect" title="Non-parametric statistics">non-parametric</a> methods. The <a href="/wiki/Parametric_estimation" class="mw-redirect" title="Parametric estimation">parametric approaches</a> assume that the underlying <a href="/wiki/Stationary_process" title="Stationary process">stationary stochastic process</a> has a certain structure which can be described using a small number of parameters (for example, using an <a href="/wiki/Autoregressive" class="mw-redirect" title="Autoregressive">autoregressive</a> or <a href="/wiki/Moving-average_model" title="Moving-average model">moving-average model</a>). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, <a href="/wiki/Non-parametric_statistics" class="mw-redirect" title="Non-parametric statistics">non-parametric approaches</a> explicitly estimate the <a href="/wiki/Covariance" title="Covariance">covariance</a> or the <a href="/wiki/Spectrum" title="Spectrum">spectrum</a> of the process without assuming that the process has any particular structure. </p><p>Methods of time series analysis may also be divided into <a href="/wiki/Linear_regression" title="Linear regression">linear</a> and <a href="/wiki/Nonlinear_regression" title="Nonlinear regression">non-linear</a>, and <a href="/wiki/Univariate_analysis" class="mw-redirect" title="Univariate analysis">univariate</a> and <a href="/wiki/Multivariate_analysis" class="mw-redirect" title="Multivariate analysis">multivariate</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Panel_data">Panel data</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=2" title="Edit section: Panel data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A time series is one type of <a href="/wiki/Panel_data" title="Panel data">panel data</a>. Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel (as is a <a href="/wiki/Cross-sectional_data" title="Cross-sectional data">cross-sectional dataset</a>). A data set may exhibit characteristics of both panel data and time series data. One way to tell is to ask what makes one data record unique from the other records. If the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a time data field and an additional identifier which is unrelated to time (e.g. student ID, stock symbol, country code), then it is panel data candidate. If the differentiation lies on the non-time identifier, then the data set is a cross-sectional data set candidate. </p> <div class="mw-heading mw-heading2"><h2 id="Analysis">Analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=3" title="Edit section: Analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several types of motivation and data analysis available for time series which are appropriate for different purposes. </p> <div class="mw-heading mw-heading3"><h3 id="Motivation">Motivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=4" title="Edit section: Motivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the context of <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>, <a href="/wiki/Quantitative_finance" class="mw-redirect" title="Quantitative finance">quantitative finance</a>, <a href="/wiki/Seismology" title="Seismology">seismology</a>, <a href="/wiki/Meteorology" title="Meteorology">meteorology</a>, and <a href="/wiki/Geophysics" title="Geophysics">geophysics</a> the primary goal of time series analysis is <a href="/wiki/Forecasting" title="Forecasting">forecasting</a>. In the context of <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="/wiki/Control_engineering" title="Control engineering">control engineering</a> and <a href="/wiki/Communication_engineering" class="mw-redirect" title="Communication engineering">communication engineering</a> it is used for signal detection. Other applications are in <a href="/wiki/Data_mining" title="Data mining">data mining</a>, <a href="/wiki/Pattern_recognition" title="Pattern recognition">pattern recognition</a> and <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, where time series analysis can be used for <a href="/wiki/Cluster_analysis" title="Cluster analysis">clustering</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><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> <a href="/wiki/Statistical_classification" title="Statistical classification">classification</a>,<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> query by content,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Anomaly_detection" title="Anomaly detection">anomaly detection</a> as well as <a href="/wiki/Forecasting" title="Forecasting">forecasting</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Exploratory_analysis">Exploratory analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=5" title="Edit section: Exploratory analysis"><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">Further information: <a href="/wiki/Exploratory_analysis" class="mw-redirect" title="Exploratory analysis">Exploratory analysis</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Time_series_TB_US.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Time_series_TB_US.png/220px-Time_series_TB_US.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Time_series_TB_US.png/330px-Time_series_TB_US.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Time_series_TB_US.png/440px-Time_series_TB_US.png 2x" data-file-width="717" data-file-height="720" /></a><figcaption>Time series of tuberculosis deaths in the United States 1954-2021.</figcaption></figure> <p>A simple way to examine a regular time series is manually with a <a href="/wiki/Line_chart" title="Line chart">line chart</a>. The datagraphic shows tuberculosis deaths in the United States,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> along with the yearly change and the percentage change from year to year. The total number of deaths declined in every year until the mid-1980s, after which there were occasional increases, often proportionately - but not absolutely - quite large. </p><p>A study of corporate data analysts found two challenges to exploratory time series analysis: discovering the shape of interesting patterns, and finding an explanation for these patterns.<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> Visual tools that represent time series data as <a href="/wiki/Heat_map" title="Heat map">heat map matrices</a> can help overcome these challenges. </p> <div class="mw-heading mw-heading3"><h3 id="Estimation,_filtering,_and_smoothing"><span id="Estimation.2C_filtering.2C_and_smoothing"></span>Estimation, filtering, and smoothing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=6" title="Edit section: Estimation, filtering, and smoothing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This approach may be based on <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> and filtering of signals in the <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a> using the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, and <a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">spectral density estimation</a>. Its development was significantly accelerated during <a href="/wiki/World_War_II" title="World War II">World War II</a> by mathematician <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a>, electrical engineers <a href="/wiki/Rudolf_E._K%C3%A1lm%C3%A1n" title="Rudolf E. Kálmán">Rudolf E. Kálmán</a>, <a href="/wiki/Dennis_Gabor" title="Dennis Gabor">Dennis Gabor</a> and others for filtering signals from noise and predicting signal values at a certain point in time. </p><p>An equivalent effect may be achieved in the time domain, as in a <a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a>; see <a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">filtering</a> and <a href="/wiki/Smoothing" title="Smoothing">smoothing</a> for more techniques. </p><p>Other related techniques include: </p> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation</a> analysis to examine <a href="/wiki/Serial_dependence" class="mw-redirect" title="Serial dependence">serial dependence</a></li> <li><a href="/wiki/Frequency_spectrum#Spectrum_analysis" class="mw-redirect" title="Frequency spectrum">Spectral analysis</a> to examine cyclic behavior which need not be related to <a href="/wiki/Seasonality" title="Seasonality">seasonality</a>. For example, sunspot activity varies over 11 year cycles.<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-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity.</li> <li>Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see <a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">trend estimation</a> and <a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">decomposition of time series</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Curve_fitting">Curve fitting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=7" title="Edit section: Curve fitting"><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/Curve_fitting" title="Curve fitting">Curve fitting</a></div> <p>Curve fitting<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><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> is the process of constructing a <a href="/wiki/Curve" title="Curve">curve</a>, or <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">mathematical function</a>, that has the best fit to a series of <a href="/wiki/Data" title="Data">data</a> points,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> possibly subject to constraints.<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><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> Curve fitting can involve either <a href="/wiki/Interpolation" title="Interpolation">interpolation</a>,<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><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> where an exact fit to the data is required, or <a href="/wiki/Smoothing" title="Smoothing">smoothing</a>,<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><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> in which a "smooth" function is constructed that approximately fits the data. A related topic is <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><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> which focuses more on questions of <a href="/wiki/Statistical_inference" title="Statistical inference">statistical inference</a> such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> to infer values of a function where no data are available,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> and to summarize the relationships among two or more variables.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Extrapolation" title="Extrapolation">Extrapolation</a> refers to the use of a fitted curve beyond the <a href="/wiki/Range_(statistics)" title="Range (statistics)">range</a> of the observed data,<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and is subject to a <a href="/wiki/Uncertainty" title="Uncertainty">degree of uncertainty</a><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> since it may reflect the method used to construct the curve as much as it reflects the observed data. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Growth_equations.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Growth_equations.png/220px-Growth_equations.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Growth_equations.png/330px-Growth_equations.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Growth_equations.png/440px-Growth_equations.png 2x" data-file-width="1248" data-file-height="1248" /></a><figcaption>Growth equations</figcaption></figure> <p>For processes that are expected to generally grow in magnitude one of the curves in the graphic (and many others) can be fitted by estimating their parameters. </p><p>The construction of economic time series involves the estimation of some components for some dates by <a href="/wiki/Interpolation" title="Interpolation">interpolation</a> between values ("benchmarks") for earlier and later dates. Interpolation is estimation of an unknown quantity between two known quantities (historical data), or drawing conclusions about missing information from the available information ("reading between the lines").<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Interpolation is useful where the data surrounding the missing data is available and its trend, seasonality, and longer-term cycles are known. This is often done by using a related series known for all relevant dates.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Alternatively <a href="/wiki/Polynomial_interpolation" title="Polynomial interpolation">polynomial interpolation</a> or <a href="/wiki/Spline_interpolation" title="Spline interpolation">spline interpolation</a> is used where piecewise <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> functions are fitted in time intervals such that they fit smoothly together. A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function (also called <a href="/wiki/Polynomial_regression" title="Polynomial regression">regression</a>). The main difference between regression and interpolation is that polynomial regression gives a single polynomial that models the entire data set. Spline interpolation, however, yield a piecewise continuous function composed of many polynomials to model the data set. </p><p><a href="/wiki/Extrapolation" title="Extrapolation">Extrapolation</a> is the process of estimating, beyond the original observation range, the value of a variable on the basis of its relationship with another variable. It is similar to <a href="/wiki/Interpolation" title="Interpolation">interpolation</a>, which produces estimates between known observations, but extrapolation is subject to greater <a href="/wiki/Uncertainty" title="Uncertainty">uncertainty</a> and a higher risk of producing meaningless results. </p> <div class="mw-heading mw-heading3"><h3 id="Function_approximation">Function approximation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=8" title="Edit section: Function approximation"><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/Function_approximation" title="Function approximation">Function approximation</a></div> <p>In general, a function approximation problem asks us to select a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> among a well-defined class that closely matches ("approximates") a target function in a task-specific way. One can distinguish two major classes of function approximation problems: First, for known target functions, <a href="/wiki/Approximation_theory" title="Approximation theory">approximation theory</a> is the branch of <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a> that investigates how certain known functions (for example, <a href="/wiki/Special_function" class="mw-redirect" title="Special function">special functions</a>) can be approximated by a specific class of functions (for example, <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> or <a href="/wiki/Rational_function" title="Rational function">rational functions</a>) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.). </p><p>Second, the target function, call it <i>g</i>, may be unknown; instead of an explicit formula, only a set of points (a time series) of the form (<i>x</i>, <i>g</i>(<i>x</i>)) is provided. Depending on the structure of the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> and <a href="/wiki/Codomain" title="Codomain">codomain</a> of <i>g</i>, several techniques for approximating <i>g</i> may be applicable. For example, if <i>g</i> is an operation on the <a href="/wiki/Real_number" title="Real number">real numbers</a>, techniques of <a href="/wiki/Interpolation" title="Interpolation">interpolation</a>, <a href="/wiki/Extrapolation" title="Extrapolation">extrapolation</a>, <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>, and <a href="/wiki/Curve_fitting" title="Curve fitting">curve fitting</a> can be used. If the <a href="/wiki/Codomain" title="Codomain">codomain</a> (range or target set) of <i>g</i> is a finite set, one is dealing with a <a href="/wiki/Statistical_classification" title="Statistical classification">classification</a> problem instead. A related problem of <i>online</i> time series approximation<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> is to summarize the data in one-pass and construct an approximate representation that can support a variety of time series queries with bounds on worst-case error. </p><p>To some extent, the different problems (<a href="/wiki/Regression_analysis" title="Regression analysis">regression</a>, <a href="/wiki/Statistical_classification" title="Statistical classification">classification</a>, <a href="/wiki/Fitness_approximation" title="Fitness approximation">fitness approximation</a>) have received a unified treatment in <a href="/wiki/Statistical_learning_theory" title="Statistical learning theory">statistical learning theory</a>, where they are viewed as <a href="/wiki/Supervised_learning" title="Supervised learning">supervised learning</a> problems. </p> <div class="mw-heading mw-heading3"><h3 id="Prediction_and_forecasting">Prediction and forecasting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=9" title="Edit section: Prediction and forecasting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Prediction" title="Prediction">prediction</a> is a part of <a href="/wiki/Statistical_inference" title="Statistical inference">statistical inference</a>. One particular approach to such inference is known as <a href="/wiki/Predictive_inference" class="mw-redirect" title="Predictive inference">predictive inference</a>, but the prediction can be undertaken within any of the several approaches to statistical inference. Indeed, one description of statistics is that it provides a means of transferring knowledge about a sample of a population to the whole population, and to other related populations, which is not necessarily the same as prediction over time. When information is transferred across time, often to specific points in time, the process is known as <a href="/wiki/Forecasting" title="Forecasting">forecasting</a>. </p> <ul><li>Fully formed statistical models for <a href="/wiki/Stochastic_simulation" title="Stochastic simulation">stochastic simulation</a> purposes, so as to generate alternative versions of the time series, representing what might happen over non-specific time-periods in the future</li> <li>Simple or fully formed statistical models to describe the likely outcome of the time series in the immediate future, given knowledge of the most recent outcomes (forecasting).</li> <li>Forecasting on time series is usually done using automated statistical software packages and programming languages, such as <a href="/wiki/Julia_(programming_language)" title="Julia (programming language)">Julia</a>, <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>, <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>, <a href="/wiki/SAS_(software)" title="SAS (software)">SAS</a>, <a href="/wiki/SPSS" title="SPSS">SPSS</a> and many others.</li> <li>Forecasting on large scale data can be done with <a href="/wiki/Apache_Spark" title="Apache Spark">Apache Spark</a> using the Spark-TS library, a third-party package.<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></li></ul> <div class="mw-heading mw-heading3"><h3 id="Classification">Classification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=10" title="Edit section: Classification"><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/Statistical_classification" title="Statistical classification">Statistical classification</a></div> <p>Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in <a href="/wiki/Sign_language" title="Sign language">sign language</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Segmentation">Segmentation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=11" title="Edit section: Segmentation"><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/Time-series_segmentation" title="Time-series segmentation">Time-series segmentation</a></div> <p>Splitting a time-series into a sequence of segments. It is often the case that a time-series can be represented as a sequence of individual segments, each with its own characteristic properties. For example, the audio signal from a conference call can be partitioned into pieces corresponding to the times during which each person was speaking. In time-series segmentation, the goal is to identify the segment boundary points in the time-series, and to characterize the dynamical properties associated with each segment. One can approach this problem using <a href="/wiki/Change_detection" title="Change detection">change-point detection</a>, or by modeling the time-series as a more sophisticated system, such as a Markov jump linear system. </p> <div class="mw-heading mw-heading3"><h3 id="Clustering">Clustering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=12" title="Edit section: Clustering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Time series data may be clustered, however special care has to be taken when considering subsequence clustering.<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> Time series clustering may be split into </p> <ul><li>whole time series clustering (multiple time series for which to find a cluster)</li> <li>subsequence time series clustering (single timeseries, split into chunks using sliding windows)</li> <li>time point clustering</li></ul> <div class="mw-heading mw-heading4"><h4 id="Subsequence_time_series_clustering">Subsequence time series clustering</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=13" title="Edit section: Subsequence time series clustering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Subsequence time series clustering resulted in unstable (random) clusters <i>induced by the feature extraction</i> using chunking with sliding windows.<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> It was found that the cluster centers (the average of the time series in a cluster - also a time series) follow an arbitrarily shifted sine pattern (regardless of the dataset, even on realizations of a <a href="/wiki/Random_walk" title="Random walk">random walk</a>). This means that the found cluster centers are non-descriptive for the dataset because the cluster centers are always nonrepresentative sine waves. </p> <div class="mw-heading mw-heading2"><h2 id="Models">Models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=14" title="Edit section: Models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Models for time series data can have many forms and represent different <a href="/wiki/Stochastic_processes" class="mw-redirect" title="Stochastic processes">stochastic processes</a>. When modeling variations in the level of a process, three broad classes of practical importance are the <i><a href="/wiki/Autoregressive" class="mw-redirect" title="Autoregressive">autoregressive</a></i> (AR) models, the <i>integrated</i> (I) models, and the <i><a href="/wiki/Moving-average_model" title="Moving-average model">moving-average</a></i> (MA) models. These three classes depend <i>linearly</i> on previous data points.<sup id="cite_ref-linear_time_series_34-0" class="reference"><a href="#cite_note-linear_time_series-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Combinations of these ideas produce <a href="/wiki/Autoregressive_moving-average_model" title="Autoregressive moving-average model">autoregressive moving-average</a> (ARMA) and <a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">autoregressive integrated moving-average</a> (ARIMA) models. The <a href="/wiki/Autoregressive_fractionally_integrated_moving_average" title="Autoregressive fractionally integrated moving average">autoregressive fractionally integrated moving-average</a> (ARFIMA) model generalizes the former three. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series models and sometimes the preceding acronyms are extended by including an initial "V" for "vector", as in VAR for <a href="/wiki/Vector_autoregression" title="Vector autoregression">vector autoregression</a>. An additional set of extensions of these models is available for use where the observed time-series is driven by some "forcing" time-series (which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's control. For these models, the acronyms are extended with a final "X" for "exogenous". </p><p>Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a <a href="/wiki/Chaos_theory" title="Chaos theory">chaotic</a> time series. However, more importantly, empirical investigations can indicate the advantage of using predictions derived from non-linear models, over those from linear models, as for example in <a href="/wiki/Nonlinear_autoregressive_exogenous_model" title="Nonlinear autoregressive exogenous model">nonlinear autoregressive exogenous models</a>. Further references on nonlinear time series analysis: (Kantz and Schreiber),<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> and (Abarbanel)<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> </p><p>Among other types of non-linear time series models, there are models to represent the changes of variance over time (<a href="/wiki/Heteroskedasticity" class="mw-redirect" title="Heteroskedasticity">heteroskedasticity</a>). These models represent <a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">autoregressive conditional heteroskedasticity</a> (ARCH) and the collection comprises a wide variety of representation (<a href="/wiki/GARCH" class="mw-redirect" title="GARCH">GARCH</a>, TARCH, EGARCH, FIGARCH, CGARCH, etc.). Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in contrast to other possible representations of locally varying variability, where the variability might be modelled as being driven by a separate time-varying process, as in a <a href="/wiki/Doubly_stochastic_model" title="Doubly stochastic model">doubly stochastic model</a>. </p><p>In recent work on model-free analyses, wavelet transform based methods (for example locally stationary wavelets and wavelet decomposed neural networks) have gained favor.<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> Multiscale (often referred to as multiresolution) techniques decompose a given time series, attempting to illustrate time dependence at multiple scales. See also <a href="/wiki/Markov_switching_multifractal" title="Markov switching multifractal">Markov switching multifractal</a> (MSMF) techniques for modeling volatility evolution. </p><p>A <a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">hidden Markov model</a> (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (hidden) states. An HMM can be considered as the simplest <a href="/wiki/Dynamic_Bayesian_network" title="Dynamic Bayesian network">dynamic Bayesian network</a>. HMM models are widely used in <a href="/wiki/Speech_recognition" title="Speech recognition">speech recognition</a>, for translating a time series of spoken words into text. </p><p>Many of these models are collected in the python package <a href="/w/index.php?title=Sktime&action=edit&redlink=1" class="new" title="Sktime (page does not exist)">sktime</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=15" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A number of different notations are in use for time-series analysis. A common notation specifying a time series <i>X</i> that is indexed by the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> is written </p> <dl><dd><span class="texhtml"><i>X</i> = (<i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>, ...)</span>.</dd></dl> <p>Another common notation is </p> <dl><dd><span class="texhtml"><i>Y</i> = (<i>Y<sub>t</sub></i>: <i>t</i> ∈ <i>T</i>)</span>,</dd></dl> <p>where <i>T</i> is the <a href="/wiki/Index_set" title="Index set">index set</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Conditions">Conditions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=16" title="Edit section: Conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are two sets of conditions under which much of the theory is built: </p> <ul><li><a href="/wiki/Stationary_process" title="Stationary process">Stationary process</a></li> <li><a href="/wiki/Ergodic_process" title="Ergodic process">Ergodic process</a></li></ul> <p>Ergodicity implies stationarity, but the converse is not necessarily the case. Stationarity is usually classified into <a href="/wiki/Strict_stationarity" class="mw-redirect" title="Strict stationarity">strict stationarity</a> and wide-sense or <a href="/wiki/Stationary_process#Weaker_forms_of_stationarity" title="Stationary process">second-order stationarity</a>. Both models and applications can be developed under each of these conditions, although the models in the latter case might be considered as only partly specified. </p><p>In addition, time-series analysis can be applied where the series are <a href="/wiki/Cyclostationary_process" title="Cyclostationary process">seasonally stationary</a> or non-stationary. Situations where the amplitudes of frequency components change with time can be dealt with in <a href="/wiki/Time-frequency_analysis" class="mw-redirect" title="Time-frequency analysis">time-frequency analysis</a> which makes use of a <a href="/wiki/Time%E2%80%93frequency_representation" title="Time–frequency representation">time–frequency representation</a> of a time-series or signal.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Tools">Tools</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=17" title="Edit section: Tools"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tools for investigating time-series data include: </p> <ul><li>Consideration of the <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation function</a> and the <a href="/wiki/Spectral_density" title="Spectral density">spectral density function</a> (also <a href="/wiki/Cross-correlation_function" class="mw-redirect" title="Cross-correlation function">cross-correlation functions</a> and cross-spectral density functions)</li> <li><a href="/wiki/Scaled_correlation" title="Scaled correlation">Scaled</a> cross- and auto-correlation functions to remove contributions of slow components<sup id="cite_ref-Nikolicetal_39-0" class="reference"><a href="#cite_note-Nikolicetal-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup></li> <li>Performing a <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> to investigate the series in the <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a></li> <li>Discrete, continuous or mixed spectra of time series, depending on whether the time series contains a (generalized) harmonic signal or not</li> <li>Use of a <a href="/wiki/Digital_filter" title="Digital filter">filter</a> to remove unwanted <a href="/wiki/Noise_(physics)" class="mw-redirect" title="Noise (physics)">noise</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal component analysis</a> (or <a href="/wiki/Empirical_orthogonal_function" class="mw-redirect" title="Empirical orthogonal function">empirical orthogonal function</a> analysis)</li> <li><a href="/wiki/Singular_spectrum_analysis" title="Singular spectrum analysis">Singular spectrum analysis</a></li> <li>"Structural" models: <ul><li>General <a href="/wiki/State_space_model" class="mw-redirect" title="State space model">state space models</a></li> <li>Unobserved components models</li></ul></li> <li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a> <ul><li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural networks</a></li> <li><a href="/wiki/Support_vector_machine" title="Support vector machine">Support vector machine</a></li> <li><a href="/wiki/Fuzzy_logic" title="Fuzzy logic">Fuzzy logic</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Genetic_programming" title="Genetic programming">Genetic programming</a></li> <li><a href="/wiki/Gene_expression_programming" title="Gene expression programming">Gene expression programming</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model</a></li> <li><a href="/wiki/Multi_expression_programming" title="Multi expression programming">Multi expression programming</a></li></ul></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a> analysis</li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a> <ul><li><a href="/wiki/Shewhart_individuals_control_chart" title="Shewhart individuals control chart">Shewhart individuals control chart</a></li> <li><a href="/wiki/CUSUM" title="CUSUM">CUSUM</a> chart</li> <li><a href="/wiki/EWMA_chart" title="EWMA chart">EWMA chart</a></li></ul></li> <li><a href="/wiki/Detrended_fluctuation_analysis" title="Detrended fluctuation analysis">Detrended fluctuation analysis</a></li> <li><a href="/wiki/Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">Nonlinear mixed-effects modeling</a></li> <li><a href="/wiki/Dynamic_time_warping" title="Dynamic time warping">Dynamic time warping</a><sup id="cite_ref-Sakoe_1978_40-0" class="reference"><a href="#cite_note-Sakoe_1978-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Dynamic_Bayesian_network" title="Dynamic Bayesian network">Dynamic Bayesian network</a></li> <li><a href="/wiki/Time-frequency_representation" class="mw-redirect" title="Time-frequency representation">Time-frequency analysis techniques:</a> <ul><li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">Fast Fourier transform</a></li> <li><a href="/wiki/Continuous_wavelet_transform" title="Continuous wavelet transform">Continuous wavelet transform</a></li> <li><a href="/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">Short-time Fourier transform</a></li> <li><a href="/wiki/Chirplet_transform" title="Chirplet transform">Chirplet transform</a></li> <li><a href="/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">Fractional Fourier transform</a></li></ul></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaotic analysis</a> <ul><li><a href="/wiki/Correlation_dimension" title="Correlation dimension">Correlation dimension</a></li> <li><a href="/wiki/Recurrence_plot" title="Recurrence plot">Recurrence plots</a></li> <li><a href="/wiki/Recurrence_quantification_analysis" title="Recurrence quantification analysis">Recurrence quantification analysis</a></li> <li><a href="/wiki/Lyapunov_exponent" title="Lyapunov exponent">Lyapunov exponents</a></li> <li><a href="/wiki/Entropy_encoding" class="mw-redirect" title="Entropy encoding">Entropy encoding</a></li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Measures">Measures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=18" title="Edit section: Measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Time-series metrics or <a href="/wiki/Features_(pattern_recognition)" class="mw-redirect" title="Features (pattern recognition)">features</a> that can be used for time series <a href="/wiki/Classification_(machine_learning)" class="mw-redirect" title="Classification (machine learning)">classification</a> or <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</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> </p> <ul><li><b>Univariate linear measures</b> <ul><li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moment (mathematics)</a></li> <li><a href="/w/index.php?title=Spectral_band_power&action=edit&redlink=1" class="new" title="Spectral band power (page does not exist)">Spectral band power</a></li> <li><a href="/wiki/Spectral_edge_frequency" class="mw-redirect" title="Spectral edge frequency">Spectral edge frequency</a></li> <li>Accumulated <a href="/wiki/Energy_(signal_processing)" title="Energy (signal processing)">energy (signal processing)</a></li> <li>Characteristics of the <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a> function</li> <li><a href="/wiki/Hjorth_parameters" title="Hjorth parameters">Hjorth parameters</a></li> <li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a> parameters</li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive model</a> parameters</li> <li><a href="/w/index.php?title=Mann%E2%80%93Kendall_test&action=edit&redlink=1" class="new" title="Mann–Kendall test (page does not exist)">Mann–Kendall test</a></li></ul></li> <li><b>Univariate non-linear measures</b> <ul><li>Measures based on the <a href="/wiki/Correlation" title="Correlation">correlation</a> sum</li> <li><a href="/wiki/Correlation_dimension" title="Correlation dimension">Correlation dimension</a></li> <li><a href="/wiki/Correlation_integral" title="Correlation integral">Correlation integral</a></li> <li><a href="/w/index.php?title=Correlation_density&action=edit&redlink=1" class="new" title="Correlation density (page does not exist)">Correlation density</a></li> <li><a href="/w/index.php?title=Correlation_entropy&action=edit&redlink=1" class="new" title="Correlation entropy (page does not exist)">Correlation entropy</a></li> <li><a href="/wiki/Approximate_entropy" title="Approximate entropy">Approximate entropy</a><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Sample_entropy" title="Sample entropy">Sample entropy</a></li> <li><a href="/w/index.php?title=Fourier_entropy&action=edit&redlink=1" class="new" title="Fourier entropy (page does not exist)">Fourier entropy</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D1%96%D1%8F_%D0%A4%D1%83%D1%80%27%D1%94" class="extiw" title="uk:Ентропія Фур'є">uk</a>]</span></li> <li>Wavelet entropy</li> <li>Dispersion entropy</li> <li>Fluctuation dispersion entropy</li> <li><a href="/wiki/R%C3%A9nyi_entropy" title="Rényi entropy">Rényi entropy</a></li> <li>Higher-order methods</li> <li><a href="/w/index.php?title=Marginal_predictability&action=edit&redlink=1" class="new" title="Marginal predictability (page does not exist)">Marginal predictability</a></li> <li><a href="/w/index.php?title=Dynamical_similarity&action=edit&redlink=1" class="new" title="Dynamical similarity (page does not exist)">Dynamical similarity</a> index</li> <li><a href="/wiki/State_space" class="mw-redirect" title="State space">State space</a> dissimilarity measures</li> <li><a href="/wiki/Lyapunov_exponent" title="Lyapunov exponent">Lyapunov exponent</a></li> <li>Permutation methods</li> <li><a href="/wiki/Local_flow" class="mw-redirect" title="Local flow">Local flow</a></li></ul></li> <li><b>Other univariate measures</b> <ul><li><a href="/wiki/Algorithmic_information_theory" title="Algorithmic information theory">Algorithmic complexity</a></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a> estimates</li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model</a> states</li> <li><a href="/wiki/Rough_path#Signature" title="Rough path">Rough path signature</a><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></li> <li>Surrogate time series and surrogate correction</li> <li>Loss of recurrence (degree of non-stationarity)</li></ul></li> <li><b>Bivariate linear measures</b> <ul><li>Maximum linear <a href="/wiki/Cross-correlation" title="Cross-correlation">cross-correlation</a></li> <li>Linear <a href="/wiki/Coherence_(signal_processing)" title="Coherence (signal processing)">Coherence (signal processing)</a></li></ul></li> <li><b>Bivariate non-linear measures</b> <ul><li>Non-linear interdependence</li> <li>Dynamical Entrainment (physics)</li> <li>Measures for <a href="/wiki/Phase_synchronization" title="Phase synchronization">phase synchronization</a></li> <li>Measures for <a href="/wiki/Phase_locking" class="mw-redirect" title="Phase locking">phase locking</a></li></ul></li> <li><b>Similarity measures</b>:<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> <ul><li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation</a></li> <li><a href="/wiki/Dynamic_time_warping" title="Dynamic time warping">Dynamic time warping</a><sup id="cite_ref-Sakoe_1978_40-1" class="reference"><a href="#cite_note-Sakoe_1978-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model</a></li> <li><a href="/wiki/Edit_distance" title="Edit distance">Edit distance</a></li> <li><a href="/wiki/Total_correlation" title="Total correlation">Total correlation</a></li> <li><a href="/wiki/Newey%E2%80%93West_estimator" title="Newey–West estimator">Newey–West estimator</a></li> <li><a href="/wiki/Prais%E2%80%93Winsten_estimation" title="Prais–Winsten estimation">Prais–Winsten transformation</a></li> <li>Data as vectors in a metrizable space <ul><li><a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a></li> <li><a href="/wiki/Mahalanobis_distance" title="Mahalanobis distance">Mahalanobis distance</a></li></ul></li> <li>Data as time series with envelopes <ul><li>Global <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a></li> <li>Local <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a></li> <li>Windowed <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a></li></ul></li> <li>Data interpreted as stochastic series <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment correlation coefficient</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's rank correlation coefficient</a></li></ul></li> <li>Data interpreted as a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> function <ul><li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov test</a></li> <li><a href="/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion" title="Cramér–von Mises criterion">Cramér–von Mises criterion</a></li></ul></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Visualization">Visualization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=19" title="Edit section: Visualization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Time series can be visualized with two categories of chart: Overlapping Charts and Separated Charts. Overlapping Charts display all-time series on the same layout while Separated Charts presents them on different layouts (but aligned for comparison purpose)<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> </p> <div class="mw-heading mw-heading3"><h3 id="Overlapping_charts">Overlapping charts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=20" title="Edit section: Overlapping charts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Braided_graph&action=edit&redlink=1" class="new" title="Braided graph (page does not exist)">Braided graphs</a></li> <li>Line charts</li> <li>Slope graphs</li> <li><a href="/w/index.php?title=GapChart&action=edit&redlink=1" class="new" title="GapChart (page does not exist)">GapChart</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://fr.wikipedia.org/wiki/GapChart" class="extiw" title="fr:GapChart">fr</a>]</span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Separated_charts">Separated charts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_series&action=edit&section=21" title="Edit section: Separated charts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Horizon_chart" title="Horizon chart">Horizon graphs</a></li> <li>Reduced line chart (small multiples)</li> <li>Silhouette graph</li> <li>Circular silhouette graph</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=Time_series&action=edit&section=22" 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: 30em;"> <ul><li><a href="/wiki/Anomaly_time_series" class="mw-redirect" title="Anomaly time series">Anomaly time series</a></li> <li><a href="/wiki/Chirp" title="Chirp">Chirp</a></li> <li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition of time series</a></li> <li><a href="/wiki/Detrended_fluctuation_analysis" title="Detrended fluctuation analysis">Detrended fluctuation analysis</a></li> <li><a href="/wiki/Digital_signal_processing" title="Digital signal processing">Digital signal processing</a></li> <li><a href="/wiki/Distributed_lag" title="Distributed lag">Distributed lag</a></li> <li><a href="/wiki/Estimation_theory" title="Estimation theory">Estimation theory</a></li> <li><a href="/wiki/Forecasting" title="Forecasting">Forecasting</a></li> <li><a href="/wiki/Frequency_spectrum" class="mw-redirect" title="Frequency spectrum">Frequency spectrum</a></li> <li><a href="/wiki/Hurst_exponent" title="Hurst exponent">Hurst exponent</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/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a></li> <li><a href="/wiki/Panel_analysis" title="Panel analysis">Panel analysis</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a></li> <li><a href="/wiki/Scaled_correlation" title="Scaled correlation">Scaled correlation</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Sequence_analysis" title="Sequence analysis">Sequence analysis</a></li> <li><a href="/wiki/Signal_processing" title="Signal processing">Signal processing</a></li> <li><a href="/wiki/Time_series_database" title="Time series database">Time series database</a> (TSDB)</li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend estimation</a></li> <li><a href="/wiki/Unevenly_spaced_time_series" title="Unevenly spaced time series">Unevenly spaced time series</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=Time_series&action=edit&section=23" 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 reflist-columns-2"> <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 a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLinKeoghLonardiChiu2003" class="citation book cs1">Lin, Jessica; Keogh, Eamonn; Lonardi, Stefano; Chiu, Bill (2003). "A symbolic representation of time series, with implications for streaming algorithms". <i>Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery</i>. New York: ACM Press. pp. 2–11. <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=10.1.1.14.5597">10.1.1.14.5597</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/10.1145%2F882082.882086">10.1145/882082.882086</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781450374224" title="Special:BookSources/9781450374224"><bdi>9781450374224</bdi></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:6084733">6084733</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+symbolic+representation+of+time+series%2C+with+implications+for+streaming+algorithms&rft.btitle=Proceedings+of+the+8th+ACM+SIGMOD+workshop+on+Research+issues+in+data+mining+and+knowledge+discovery&rft.place=New+York&rft.pages=2-11&rft.pub=ACM+Press&rft.date=2003&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.14.5597%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6084733%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F882082.882086&rft.isbn=9781450374224&rft.aulast=Lin&rft.aufirst=Jessica&rft.au=Keogh%2C+Eamonn&rft.au=Lonardi%2C+Stefano&rft.au=Chiu%2C+Bill&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarren_Liao2005" class="citation journal cs1">Warren Liao, T. 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"Time-series clustering – A decade review". <i>Information Systems</i>. <b>53</b>: 16–38. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.is.2015.04.007">10.1016/j.is.2015.04.007</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:158707">158707</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+Systems&rft.atitle=Time-series+clustering+%E2%80%93+A+decade+review&rft.volume=53&rft.pages=16-38&rft.date=2015-10&rft_id=info%3Adoi%2F10.1016%2Fj.is.2015.04.007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A158707%23id-name%3DS2CID&rft.aulast=Aghabozorgi&rft.aufirst=Saeed&rft.au=Seyed+Shirkhorshidi%2C+Ali&rft.au=Ying+Wah%2C+Teh&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeoghKasetty2002" class="citation book cs1">Keogh, Eamonn; Kasetty, Shruti (2002). 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Vol. 730. pp. 69–84. <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%2F3-540-57301-1_5">10.1007/3-540-57301-1_5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-57301-2" title="Special:BookSources/978-3-540-57301-2"><bdi>978-3-540-57301-2</bdi></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:16748451">16748451</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Efficient+similarity+search+in+sequence+databases&rft.btitle=Foundations+of+Data+Organization+and+Algorithms&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=69-84&rft.date=1993&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16748451%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F3-540-57301-1_5&rft.isbn=978-3-540-57301-2&rft.aulast=Agrawal&rft.aufirst=Rakesh&rft.au=Faloutsos%2C+Christos&rft.au=Swami%2C+Arun&rft_id=https%3A%2F%2Ffigshare.com%2Farticles%2Fjournal_contribution%2F6605123&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChenChiu2021" class="citation journal cs1">Chen, Cathy W. S.; Chiu, L. M. (4 September 2021). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8469594">"Ordinal Time Series Forecasting of the Air Quality Index"</a>. <i>Entropy</i>. <b>23</b> (9): 1167. <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/2021Entrp..23.1167C">2021Entrp..23.1167C</a>. <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/10.3390%2Fe23091167">10.3390/e23091167</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8469594">8469594</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34573792">34573792</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Entropy&rft.atitle=Ordinal+Time+Series+Forecasting+of+the+Air+Quality+Index&rft.volume=23&rft.issue=9&rft.pages=1167&rft.date=2021-09-04&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8469594%23id-name%3DPMC&rft_id=info%3Apmid%2F34573792&rft_id=info%3Adoi%2F10.3390%2Fe23091167&rft_id=info%3Abibcode%2F2021Entrp..23.1167C&rft.aulast=Chen&rft.aufirst=Cathy+W.+S.&rft.au=Chiu%2C+L.+M.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8469594&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://www.cdc.gov/tb/statistics/reports/2022/table1.htm">http://www.cdc.gov/tb/statistics/reports/2022/table1.htm</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSarkarSpottBlackwellJamnik2016" class="citation book cs1">Sarkar, Advait; Spott, Martin; Blackwell, Alan F.; Jamnik, Mateja (2016). <a rel="nofollow" class="external text" href="https://www.repository.cam.ac.uk/handle/1810/260285">"Visual discovery and model-driven explanation of time series patterns"</a>. <i>2016 IEEE Symposium on Visual Languages and Human-Centric Computing (VL/HCC)</i>. pp. 78–86. <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%2Fvlhcc.2016.7739668">10.1109/vlhcc.2016.7739668</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-5090-0252-8" title="Special:BookSources/978-1-5090-0252-8"><bdi>978-1-5090-0252-8</bdi></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:9787931">9787931</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Visual+discovery+and+model-driven+explanation+of+time+series+patterns&rft.btitle=2016+IEEE+Symposium+on+Visual+Languages+and+Human-Centric+Computing+%28VL%2FHCC%29&rft.pages=78-86&rft.date=2016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9787931%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2Fvlhcc.2016.7739668&rft.isbn=978-1-5090-0252-8&rft.aulast=Sarkar&rft.aufirst=Advait&rft.au=Spott%2C+Martin&rft.au=Blackwell%2C+Alan+F.&rft.au=Jamnik%2C+Mateja&rft_id=https%3A%2F%2Fwww.repository.cam.ac.uk%2Fhandle%2F1810%2F260285&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBloomfield1976" class="citation book cs1">Bloomfield, Peter (1976). <i>Fourier Analysis of Time Series: An Introduction</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-08256-9" title="Special:BookSources/978-0-471-08256-9"><bdi>978-0-471-08256-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Analysis+of+Time+Series%3A+An+Introduction&rft.pub=Wiley&rft.date=1976&rft.isbn=978-0-471-08256-9&rft.aulast=Bloomfield&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span><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. (January 2024)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShumway1988" class="citation book cs1">Shumway, Robert H. (1988). <i>Applied Statistical Time Series Analysis</i>. Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-041500-4" title="Special:BookSources/978-0-13-041500-4"><bdi>978-0-13-041500-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Statistical+Time+Series+Analysis&rft.pub=Prentice-Hall&rft.date=1988&rft.isbn=978-0-13-041500-4&rft.aulast=Shumway&rft.aufirst=Robert+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span><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. (January 2024)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArlinghaus1994" class="citation book cs1">Arlinghaus, Sandra (1994). <i>Practical Handbook of Curve Fitting</i>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-0143-8" title="Special:BookSources/978-0-8493-0143-8"><bdi>978-0-8493-0143-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+Handbook+of+Curve+Fitting&rft.pub=CRC+Press&rft.date=1994&rft.isbn=978-0-8493-0143-8&rft.aulast=Arlinghaus&rft.aufirst=Sandra&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span><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. (January 2024)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKolb1984" class="citation book cs1">Kolb, William M. (1984). <i>Curve Fitting for Programmable Calculators</i>. SYNTEC. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-943494-02-9" title="Special:BookSources/978-0-943494-02-9"><bdi>978-0-943494-02-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Curve+Fitting+for+Programmable+Calculators&rft.pub=SYNTEC&rft.date=1984&rft.isbn=978-0-943494-02-9&rft.aulast=Kolb&rft.aufirst=William+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span><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. (January 2024)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalliRao1992" class="citation book cs1">Halli, S. S.; Rao, K. V. (1992). <i>Advanced Techniques of Population Analysis</i>. Springer Science & Business Media. p. 165. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-306-43997-1" title="Special:BookSources/978-0-306-43997-1"><bdi>978-0-306-43997-1</bdi></a>. <q>functions are fulfilled if we have a good to moderate fit for the observed data.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Techniques+of+Population+Analysis&rft.pages=165&rft.pub=Springer+Science+%26+Business+Media&rft.date=1992&rft.isbn=978-0-306-43997-1&rft.aulast=Halli&rft.aufirst=S.+S.&rft.au=Rao%2C+K.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><i><a rel="nofollow" class="external text" href="https://archive.org/details/signalnoisewhymo00silv">The Signal and the Noise</a>: Why So Many Predictions Fail-but Some Don't.</i> By Nate Silver</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPyle1999" class="citation book cs1">Pyle, Dorian (1999). <i>Data Preparation for Data Mining</i>. Morgan Kaufmann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-55860-529-9" title="Special:BookSources/978-1-55860-529-9"><bdi>978-1-55860-529-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Data+Preparation+for+Data+Mining&rft.pub=Morgan+Kaufmann&rft.date=1999&rft.isbn=978-1-55860-529-9&rft.aulast=Pyle&rft.aufirst=Dorian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span><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. (January 2024)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Numerical Methods in Engineering with MATLAB®. By <a href="/w/index.php?title=Jaan_Kiusalaas&action=edit&redlink=1" class="new" title="Jaan Kiusalaas (page does not exist)">Jaan Kiusalaas</a>. Page 24.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKiusalaas2013" class="citation book cs1">Kiusalaas, Jaan (2013). <i>Numerical Methods in Engineering with Python 3</i>. Cambridge University Press. p. 21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-62058-1" title="Special:BookSources/978-1-139-62058-1"><bdi>978-1-139-62058-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Methods+in+Engineering+with+Python+3&rft.pages=21&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-1-139-62058-1&rft.aulast=Kiusalaas&rft.aufirst=Jaan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuest2012" class="citation book cs1">Guest, Philip George (2012). <i>Numerical Methods of Curve Fitting</i>. Cambridge University Press. p. 349. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-64695-7" title="Special:BookSources/978-1-107-64695-7"><bdi>978-1-107-64695-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Methods+of+Curve+Fitting&rft.pages=349&rft.pub=Cambridge+University+Press&rft.date=2012&rft.isbn=978-1-107-64695-7&rft.aulast=Guest&rft.aufirst=Philip+George&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">See also: <a href="/wiki/Mollifier" title="Mollifier">Mollifier</a></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMotulskyChristopoulos2004" class="citation book cs1">Motulsky, Harvey; Christopoulos, Arthur (2004). <i>Fitting Models to Biological Data Using Linear and Nonlinear Regression: A Practical Guide to Curve Fitting</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-803834-4" title="Special:BookSources/978-0-19-803834-4"><bdi>978-0-19-803834-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fitting+Models+to+Biological+Data+Using+Linear+and+Nonlinear+Regression%3A+A+Practical+Guide+to+Curve+Fitting&rft.pub=Oxford+University+Press&rft.date=2004&rft.isbn=978-0-19-803834-4&rft.aulast=Motulsky&rft.aufirst=Harvey&rft.au=Christopoulos%2C+Arthur&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span><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. (January 2024)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Regression Analysis By Rudolf J. Freund, William J. Wilson, Ping Sa. Page 269.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="Please supply a full or month-and-year "|date=" of publication, or use "|year=year not specified" for intentional omission.">date missing</span></a></i>]</sup></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaudSagayanYahyaNajwati2009" class="citation book cs1">Daud, Hanita; Sagayan, Vijanth; Yahya, Noorhana; Najwati, Wan (2009). "Modeling of Electromagnetic Waves Using Statistical and Numerical Techniques". <i>Visual Informatics: Bridging Research and Practice</i>. Lecture Notes in Computer Science. Vol. 5857. pp. 686–695. <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%2F978-3-642-05036-7_65">10.1007/978-3-642-05036-7_65</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-05035-0" title="Special:BookSources/978-3-642-05035-0"><bdi>978-3-642-05035-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Modeling+of+Electromagnetic+Waves+Using+Statistical+and+Numerical+Techniques&rft.btitle=Visual+Informatics%3A+Bridging+Research+and+Practice&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=686-695&rft.date=2009&rft_id=info%3Adoi%2F10.1007%2F978-3-642-05036-7_65&rft.isbn=978-3-642-05035-0&rft.aulast=Daud&rft.aufirst=Hanita&rft.au=Sagayan%2C+Vijanth&rft.au=Yahya%2C+Noorhana&rft.au=Najwati%2C+Wan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHauser2009" class="citation book cs1">Hauser, John R. (2009). <i>Numerical Methods for Nonlinear Engineering Models</i>. Springer Science & Business Media. p. 227. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-9920-5" title="Special:BookSources/978-1-4020-9920-5"><bdi>978-1-4020-9920-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Methods+for+Nonlinear+Engineering+Models&rft.pages=227&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009&rft.isbn=978-1-4020-9920-5&rft.aulast=Hauser&rft.aufirst=John+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam1976" class="citation book cs1">William, Dudley, ed. (1976). 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"A Primer on the Signature Method in Machine Learning". <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/1603.03788">1603.03788</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+Primer+on+the+Signature+Method+in+Machine+Learning&rft.date=2016&rft_id=info%3Aarxiv%2F1603.03788&rft.aulast=Chevyrev&rft.aufirst=Ilya&rft.au=Kormilitzin%2C+Andrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRopellaNagHunt2003" class="citation book cs1">Ropella, G.E.P.; Nag, D.A.; Hunt, C.A. (2003). "Similarity measures for automated comparison of in silico and in vitro experimental results". <i>Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE Cat. No.03CH37439)</i>. pp. 2933–2936. <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%2FIEMBS.2003.1280532">10.1109/IEMBS.2003.1280532</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7803-7789-9" title="Special:BookSources/978-0-7803-7789-9"><bdi>978-0-7803-7789-9</bdi></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:17798157">17798157</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Similarity+measures+for+automated+comparison+of+in+silico+and+in+vitro+experimental+results&rft.btitle=Proceedings+of+the+25th+Annual+International+Conference+of+the+IEEE+Engineering+in+Medicine+and+Biology+Society+%28IEEE+Cat.+No.03CH37439%29&rft.pages=2933-2936&rft.date=2003&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17798157%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FIEMBS.2003.1280532&rft.isbn=978-0-7803-7789-9&rft.aulast=Ropella&rft.aufirst=G.E.P.&rft.au=Nag%2C+D.A.&rft.au=Hunt%2C+C.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTominskiAigner" class="citation web cs1">Tominski, Christian; Aigner, Wolfgang. <a rel="nofollow" class="external text" href="http://survey.timeviz.net/">"The TimeViz Browser:A Visual Survey of Visualization Techniques for Time-Oriented Data"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">1 June</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+TimeViz+Browser%3AA+Visual+Survey+of+Visualization+Techniques+for+Time-Oriented+Data&rft.aulast=Tominski&rft.aufirst=Christian&rft.au=Aigner%2C+Wolfgang&rft_id=http%3A%2F%2Fsurvey.timeviz.net%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" 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=Time_series&action=edit&section=24" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe_GooijerHyndman2006" class="citation journal cs1">De Gooijer, Jan G.; <a href="/wiki/Rob_J._Hyndman" title="Rob J. Hyndman">Hyndman, Rob J.</a> (2006). "25 Tears of Time Series Forecasting". <i>International Journal of Forecasting</i>. Twenty Five Years of Forecasting. <b>22</b> (3): 443–473. <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=10.1.1.154.9227">10.1.1.154.9227</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/10.1016%2Fj.ijforecast.2006.01.001">10.1016/j.ijforecast.2006.01.001</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:14996235">14996235</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Forecasting&rft.atitle=25+Tears+of+Time+Series+Forecasting&rft.volume=22&rft.issue=3&rft.pages=443-473&rft.date=2006&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.154.9227%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14996235%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.ijforecast.2006.01.001&rft.aulast=De+Gooijer&rft.aufirst=Jan+G.&rft.au=Hyndman%2C+Rob+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoxJenkins1976" class="citation cs2"><a href="/wiki/George_E._P._Box" title="George E. P. Box">Box, George</a>; Jenkins, Gwilym (1976), <i>Time Series Analysis: forecasting and control, rev. ed.</i>, Oakland, California: Holden-Day</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time+Series+Analysis%3A+forecasting+and+control%2C+rev.+ed.&rft.place=Oakland%2C+California&rft.pub=Holden-Day&rft.date=1976&rft.aulast=Box&rft.aufirst=George&rft.au=Jenkins%2C+Gwilym&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></li> <li><a href="/wiki/James_Durbin" title="James Durbin">Durbin J.</a>, Koopman S.J. (2001), <i>Time Series Analysis by State Space Methods</i>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGershenfeld2000" class="citation cs2">Gershenfeld, Neil (2000), <i>The Nature of Mathematical Modeling</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57095-4" title="Special:BookSources/978-0-521-57095-4"><bdi>978-0-521-57095-4</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/174825352">174825352</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nature+of+Mathematical+Modeling&rft.pub=Cambridge+University+Press&rft.date=2000&rft_id=info%3Aoclcnum%2F174825352&rft.isbn=978-0-521-57095-4&rft.aulast=Gershenfeld&rft.aufirst=Neil&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton1994" class="citation cs2"><a href="/wiki/James_D._Hamilton" title="James D. Hamilton">Hamilton, James</a> (1994), <i>Time Series Analysis</i>, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-04289-3" title="Special:BookSources/978-0-691-04289-3"><bdi>978-0-691-04289-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time+Series+Analysis&rft.pub=Princeton+University+Press&rft.date=1994&rft.isbn=978-0-691-04289-3&rft.aulast=Hamilton&rft.aufirst=James&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></li> <li><a href="/wiki/Maurice_Priestley" title="Maurice Priestley">Priestley, M. B.</a> (1981), <i>Spectral Analysis and Time Series</i>, <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <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/978-0-12-564901-8" title="Special:BookSources/978-0-12-564901-8">978-0-12-564901-8</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShasha2004" class="citation cs2">Shasha, D. (2004), <i>High Performance Discovery in Time Series</i>, <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-00857-8" title="Special:BookSources/978-0-387-00857-8"><bdi>978-0-387-00857-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=High+Performance+Discovery+in+Time+Series&rft.pub=Springer&rft.date=2004&rft.isbn=978-0-387-00857-8&rft.aulast=Shasha&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></li> <li>Shumway R. H., Stoffer D. S. (2017), <i>Time Series Analysis and its Applications: With R Examples (ed. 4)</i>, Springer, <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/978-3-319-52451-1" title="Special:BookSources/978-3-319-52451-1">978-3-319-52451-1</a></li> <li>Weigend A. S., Gershenfeld N. A. (Eds.) (1994), <i>Time Series Prediction: Forecasting the Future and Understanding the Past</i>. Proceedings of the NATO Advanced Research Workshop on Comparative Time Series Analysis (Santa Fe, May 1992), <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>.</li> <li><a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Wiener, N.</a> (1949), <i>Extrapolation, Interpolation, and Smoothing of Stationary Time Series</i>, <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>.</li> <li>Woodward, W. A., Gray, H. L. & Elliott, A. C. (2012), <i>Applied Time Series Analysis</i>, <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAuffarth2021" class="citation book cs1">Auffarth, Ben (2021). <a rel="nofollow" class="external text" href="https://www.packtpub.com/product/machine-learning-for-time-series-with-python/9781801819626"><i>Machine Learning for Time-Series with Python: Forecast, predict, and detect anomalies with state-of-the-art machine learning methods</i></a> (1st ed.). Packt Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1801819626" title="Special:BookSources/978-1801819626"><bdi>978-1801819626</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">5 November</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Machine+Learning+for+Time-Series+with+Python%3A+Forecast%2C+predict%2C+and+detect+anomalies+with+state-of-the-art+machine+learning+methods&rft.edition=1st&rft.pub=Packt+Publishing&rft.date=2021&rft.isbn=978-1801819626&rft.aulast=Auffarth&rft.aufirst=Ben&rft_id=https%3A%2F%2Fwww.packtpub.com%2Fproduct%2Fmachine-learning-for-time-series-with-python%2F9781801819626&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+series" class="Z3988"></span></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=Time_series&action=edit&section=25" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Time_series" class="extiw" title="commons:Category:Time series">Time series</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4.htm">Introduction to Time series Analysis (Engineering Statistics Handbook)</a> — A practical guide to Time series analysis.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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.navbar{display:none!important}}</style><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 template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Statistics" title="Special:EditPage/Template:Statistics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul> </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="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" 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_collection" 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_inference" 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_analysis" 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/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</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 mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" 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 class="mw-selflink selflink">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 href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a 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</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a class="mw-selflink selflink">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 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