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class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modifiable_areal_unit_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Modifiable areal unit problem</span> </div> </a> <ul id="toc-Modifiable_areal_unit_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modifiable_temporal_unit_problem" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modifiable_temporal_unit_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Modifiable temporal unit problem</span> </div> </a> <ul id="toc-Modifiable_temporal_unit_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Neighborhood_effect_averaging_problem" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Neighborhood_effect_averaging_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.4</span> <span>Neighborhood effect averaging problem</span> </div> </a> <ul id="toc-Neighborhood_effect_averaging_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Travelling_salesman_problem" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Travelling_salesman_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.5</span> <span>Travelling salesman problem</span> </div> </a> <ul id="toc-Travelling_salesman_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uncertain_geographic_context_problem" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Uncertain_geographic_context_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.6</span> <span>Uncertain geographic context problem</span> </div> </a> <ul id="toc-Uncertain_geographic_context_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weber_problem" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Weber_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.7</span> <span>Weber problem</span> </div> </a> <ul id="toc-Weber_problem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Spatial_characterization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_characterization"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Spatial characterization</span> </div> </a> <ul id="toc-Spatial_characterization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spatial_dependence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_dependence"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Spatial dependence</span> </div> </a> <ul id="toc-Spatial_dependence-sublist" class="vector-toc-list"> <li id="toc-Spatial_auto-correlation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Spatial_auto-correlation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Spatial auto-correlation</span> </div> </a> <ul id="toc-Spatial_auto-correlation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Spatial_association" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_association"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Spatial association</span> </div> </a> <ul id="toc-Spatial_association-sublist" class="vector-toc-list"> <li id="toc-The_second_dimension_of_spatial_association" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_second_dimension_of_spatial_association"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.1</span> <span>The second dimension of spatial association</span> </div> </a> <ul id="toc-The_second_dimension_of_spatial_association-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Scaling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaling"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Scaling</span> </div> </a> <ul id="toc-Scaling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sampling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sampling"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Sampling</span> </div> </a> <ul id="toc-Sampling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Common_errors_in_spatial_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Common_errors_in_spatial_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Common errors in spatial analysis</span> </div> </a> <ul id="toc-Common_errors_in_spatial_analysis-sublist" class="vector-toc-list"> <li id="toc-Length" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Length"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7.1</span> <span>Length</span> </div> </a> <ul id="toc-Length-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Locational_fallacy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Locational_fallacy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7.2</span> <span>Locational fallacy</span> </div> </a> <ul id="toc-Locational_fallacy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Atomic_fallacy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Atomic_fallacy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7.3</span> <span>Atomic fallacy</span> </div> </a> <ul id="toc-Atomic_fallacy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ecological_fallacy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ecological_fallacy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7.4</span> <span>Ecological fallacy</span> </div> </a> <ul id="toc-Ecological_fallacy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Solutions_to_the_fundamental_issues" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solutions_to_the_fundamental_issues"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Solutions to the fundamental issues</span> </div> </a> <ul id="toc-Solutions_to_the_fundamental_issues-sublist" class="vector-toc-list"> <li id="toc-Geographic_space" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Geographic_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8.1</span> <span>Geographic space</span> </div> </a> <ul id="toc-Geographic_space-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Types" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Types</span> </div> </a> <button aria-controls="toc-Types-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 Types subsection</span> </button> <ul id="toc-Types-sublist" class="vector-toc-list"> <li id="toc-Spatial_data_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_data_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Spatial data analysis</span> </div> </a> <ul id="toc-Spatial_data_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spatial_autocorrelation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_autocorrelation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Spatial autocorrelation</span> </div> </a> <ul id="toc-Spatial_autocorrelation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spatial_heterogeneity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_heterogeneity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Spatial heterogeneity</span> </div> </a> <ul id="toc-Spatial_heterogeneity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spatial_interaction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_interaction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Spatial interaction</span> </div> </a> <ul id="toc-Spatial_interaction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spatial_interpolation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_interpolation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Spatial interpolation</span> </div> </a> <ul id="toc-Spatial_interpolation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spatial_regression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_regression"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Spatial regression</span> </div> </a> <ul id="toc-Spatial_regression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spatial_neural_networks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spatial_neural_networks"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Spatial neural networks</span> </div> </a> <ul id="toc-Spatial_neural_networks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simulation_and_modeling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simulation_and_modeling"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Simulation and modeling</span> </div> </a> <ul id="toc-Simulation_and_modeling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiple-point_geostatistics_(MPS)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple-point_geostatistics_(MPS)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Multiple-point geostatistics (MPS)</span> </div> </a> <ul id="toc-Multiple-point_geostatistics_(MPS)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geospatial_and_hydrospatial_analysis" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Geospatial_and_hydrospatial_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Geospatial and hydrospatial analysis</span> </div> </a> <button aria-controls="toc-Geospatial_and_hydrospatial_analysis-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 Geospatial and hydrospatial analysis subsection</span> </button> <ul id="toc-Geospatial_and_hydrospatial_analysis-sublist" class="vector-toc-list"> <li id="toc-Geographical_information_system_usage" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geographical_information_system_usage"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Geographical information system usage</span> </div> </a> <ul id="toc-Geographical_information_system_usage-sublist" class="vector-toc-list"> <li id="toc-Basic_applications" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Basic_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Basic applications</span> </div> </a> <ul id="toc-Basic_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_operations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Basic_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>Basic operations</span> </div> </a> <ul id="toc-Basic_operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Advanced_operations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Advanced_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.3</span> <span>Advanced operations</span> </div> </a> <ul id="toc-Advanced_operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mobile_geospatial_and_hydrospatial_Computing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mobile_geospatial_and_hydrospatial_Computing"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.4</span> <span>Mobile geospatial and hydrospatial Computing</span> </div> </a> <ul id="toc-Mobile_geospatial_and_hydrospatial_Computing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geographic_information_science_and_spatial_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geographic_information_science_and_spatial_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Geographic information science and spatial analysis</span> </div> </a> <ul id="toc-Geographic_information_science_and_spatial_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label 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">Spatial analysis</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 18 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-18" 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">18 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D9%85%D9%83%D8%A7%D9%86%D9%8A" title="تحليل مكاني – Arabic" lang="ar" hreflang="ar" data-title="تحليل مكاني" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%AA%E0%A5%87%E0%A4%B6%E0%A4%BF%E0%A4%AF%E0%A4%B2_%E0%A4%8F%E0%A4%A8%E0%A4%BE%E0%A4%B2%E0%A4%BF%E0%A4%B8%E0%A4%BF%E0%A4%B8" title="स्पेशियल एनालिसिस – Bhojpuri" lang="bh" hreflang="bh" data-title="स्पेशियल एनालिसिस" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_espacial" title="Anàlisi espacial – Catalan" lang="ca" hreflang="ca" data-title="Anàlisi espacial" 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/Prostorov%C3%A1_anal%C3%BDza" title="Prostorová analýza – Czech" lang="cs" hreflang="cs" data-title="Prostorová analýza" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_espacial" title="Análisis espacial – Spanish" lang="es" hreflang="es" data-title="Análisis espacial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%DB%8C%D9%84_%D9%81%D8%B6%D8%A7%DB%8C%DB%8C" title="تحلیل فضایی – Persian" lang="fa" hreflang="fa" data-title="تحلیل فضایی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_spatiale" title="Analyse spatiale – French" lang="fr" hreflang="fr" data-title="Analyse spatiale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%B5%EA%B0%84_%EB%B6%84%EC%84%9D" title="공간 분석 – Korean" lang="ko" hreflang="ko" data-title="공간 분석" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BE%E0%A4%A8%E0%A4%BF%E0%A4%95_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="स्थानिक विश्लेषण – Hindi" lang="hi" hreflang="hi" data-title="स्थानिक विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Prostorna_analiza" title="Prostorna analiza – Croatian" lang="hr" hreflang="hr" data-title="Prostorna analiza" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A9%BA%E9%96%93%E5%88%86%E6%9E%90" title="空間分析 – Japanese" lang="ja" hreflang="ja" data-title="空間分析" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_przestrzenna" title="Analiza przestrzenna – Polish" lang="pl" hreflang="pl" data-title="Analiza przestrzenna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt 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0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Spatial_analysis" title="Special:EditPage/Spatial analysis">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Spatial+analysis%22">"Spatial analysis"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Spatial+analysis%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Spatial+analysis%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Spatial+analysis%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Spatial+analysis%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Spatial+analysis%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">January 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Snow-cholera-map.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Snow-cholera-map.jpg/300px-Snow-cholera-map.jpg" decoding="async" width="300" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Snow-cholera-map.jpg/450px-Snow-cholera-map.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Snow-cholera-map.jpg/600px-Snow-cholera-map.jpg 2x" data-file-width="889" data-file-height="869" /></a><figcaption>Map by Dr. <a href="/wiki/John_Snow_(physician)" class="mw-redirect" title="John Snow (physician)">John Snow</a> of <a href="/wiki/London" title="London">London</a>, showing <a href="/wiki/Cluster_(epidemiology)" class="mw-redirect" title="Cluster (epidemiology)">clusters</a> of cholera cases in the <a href="/wiki/1854_Broad_Street_cholera_outbreak" title="1854 Broad Street cholera outbreak">1854 Broad Street cholera outbreak</a>. This was one of the first uses of map-based spatial analysis.</figcaption></figure> <p><b>Spatial analysis</b> is any of the formal <a href="/wiki/Scientific_technique" title="Scientific technique">techniques</a> which studies entities using their <a href="/wiki/Topological" class="mw-redirect" title="Topological">topological</a>, <a href="/wiki/Geometric" class="mw-redirect" title="Geometric">geometric</a>, or <a href="/wiki/Geographic" class="mw-redirect" title="Geographic">geographic</a> properties, primarily used in <a href="/wiki/Urban_design" title="Urban design">Urban Design</a>. Spatial analysis includes a variety of techniques using different analytic approaches, especially <i><a href="/wiki/Spatial_statistics" title="Spatial statistics">spatial statistics</a></i>. It may be applied in fields as diverse as <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, with its studies of the placement of galaxies in the <a href="/wiki/Cosmos" title="Cosmos">cosmos</a>, or to chip fabrication engineering, with its use of "place and route" <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> to build complex wiring structures. In a more restricted sense, spatial analysis is <b>geospatial analysis</b>, the technique applied to structures at the human scale, most notably in the analysis of <a href="/wiki/Geographic_data" class="mw-redirect" title="Geographic data">geographic data</a>. It may also applied to genomics, as in <a href="/wiki/Spatial_transcriptomics" title="Spatial transcriptomics">transcriptomics data</a>, but is primarily for spatial data. </p><p>Complex issues arise in spatial analysis, many of which are neither clearly defined nor completely resolved, but form the basis for current research. The most fundamental of these is the problem of defining the spatial location of the entities being studied. Classification of the techniques of spatial analysis is difficult because of the large number of different fields of research involved, the different fundamental approaches which can be chosen, and the many forms the data can take. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spatial analysis began with early attempts at <a href="/wiki/Cartography" title="Cartography">cartography</a> and <a href="/wiki/Surveying" title="Surveying">surveying</a>. Land surveying goes back to at least 1,400 B.C in Egypt: the dimensions of taxable land plots were measured with measuring ropes and plumb bobs.<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> Many fields have contributed to its rise in modern form. <a href="/wiki/Biology" title="Biology">Biology</a> contributed through <a href="/wiki/Botany" title="Botany">botanical</a> studies of global plant distributions and local plant locations, <a href="/wiki/Ethology" title="Ethology">ethological</a> studies of animal movement, <a href="/wiki/Landscape_ecology" title="Landscape ecology">landscape ecological</a> studies of vegetation blocks, <a href="/wiki/Ecology" title="Ecology">ecological</a> studies of spatial population dynamics, and the study of <a href="/wiki/Biogeography" title="Biogeography">biogeography</a>. <a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a> contributed with early work on disease mapping, notably <a href="/wiki/John_Snow_(physician)" class="mw-redirect" title="John Snow (physician)">John Snow</a>'s work of mapping an outbreak of cholera, with research on mapping the spread of disease and with location studies for health care delivery. <a href="/wiki/Statistics" title="Statistics">Statistics</a> has contributed greatly through work in spatial statistics. <a href="/wiki/Economics" title="Economics">Economics</a> has contributed notably through <a href="/wiki/Spatial_econometrics" title="Spatial econometrics">spatial econometrics</a>. <a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a> is currently a major contributor due to the importance of geographic software in the modern analytic toolbox. <a href="/wiki/Remote_sensing" title="Remote sensing">Remote sensing</a> has contributed extensively in morphometric and clustering analysis. <a href="/wiki/Computer_science" title="Computer science">Computer science</a> has contributed extensively through the study of algorithms, notably in <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a>. <a href="/wiki/Mathematics" title="Mathematics">Mathematics</a> continues to provide the fundamental tools for analysis and to reveal the complexity of the spatial realm, for example, with recent work on <a href="/wiki/Fractals" class="mw-redirect" title="Fractals">fractals</a> and <a href="/wiki/Scale_invariance" title="Scale invariance">scale invariance</a>. <a href="/wiki/Scientific_modelling" title="Scientific modelling">Scientific modelling</a> provides a useful framework for new approaches.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2021)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Fundamental_issues">Fundamental issues</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=2" title="Edit section: Fundamental issues"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spatial analysis confronts many fundamental issues in the definition of its objects of study, in the construction of the analytic operations to be used, in the use of computers for analysis, in the limitations and particularities of the analyses which are known, and in the presentation of analytic results. Many of these issues are active subjects of modern research.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2021)">citation needed</span></a></i>]</sup> </p><p>Common errors often arise in spatial analysis, some due to the mathematics of space, some due to the particular ways data are presented spatially, some due to the tools which are available. Census data, because it protects individual privacy by aggregating data into local units, raises a number of statistical issues. The fractal nature of coastline makes precise measurements of its length difficult if not impossible. A computer software fitting straight lines to the curve of a coastline, can easily calculate the lengths of the lines which it defines. However these straight lines may have no inherent meaning in the real world, as was shown for the <a href="/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Self-Similarity_and_Fractional_Dimension" class="mw-redirect" title="How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension">coastline of Britain</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2021)">citation needed</span></a></i>]</sup> </p><p>These problems represent a challenge in spatial analysis because of the power of maps as media of presentation. When results are presented as maps, the presentation combines spatial data which are generally accurate with analytic results which may be inaccurate, leading to an impression that analytic results are more accurate than the data would indicate.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Formal_Problems">Formal Problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=3" title="Edit section: Formal Problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Boundary_problem">Boundary problem</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=4" title="Edit section: Boundary problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><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 dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Boundary_problem_(spatial_analysis)" title="Boundary problem (spatial analysis)">Boundary problem (spatial analysis)</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Boundary_problem_(spatial_analysis)&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> A <a href="/wiki/Boundary_problem_(spatial_analysis)" title="Boundary problem (spatial analysis)">boundary problem</a> in analysis is a phenomenon in which geographical patterns are differentiated by the shape and arrangement of boundaries that are drawn for administrative or measurement purposes. The boundary problem occurs because of the loss of neighbors in analyses that depend on the values of the neighbors. While geographic phenomena are measured and analyzed within a specific unit, identical spatial data can appear either dispersed or clustered depending on the boundary placed around the data. In analysis with point data, dispersion is evaluated as dependent of the boundary. In analysis with areal data, statistics should be interpreted based upon the boundary.</div></div> <div class="mw-heading mw-heading4"><h4 id="Modifiable_areal_unit_problem">Modifiable areal unit problem</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=5" title="Edit section: Modifiable areal unit problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Modifiable_areal_unit_problem" title="Modifiable areal unit problem">Modifiable areal unit problem</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Modifiable_areal_unit_problem&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Maup_rate_numbers.png" class="mw-file-description"><img alt="MAUP distortion example" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Maup_rate_numbers.png/298px-Maup_rate_numbers.png" decoding="async" width="298" height="284" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Maup_rate_numbers.png/447px-Maup_rate_numbers.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Maup_rate_numbers.png/596px-Maup_rate_numbers.png 2x" data-file-width="970" data-file-height="925" /></a><figcaption>An example of the modifiable areal unit problem and the distortion of rate calculations</figcaption></figure> <p>The <a href="/wiki/Modifiable_areal_unit_problem" title="Modifiable areal unit problem">modifiable areal unit problem</a> (MAUP) is a source of <a href="/wiki/Statistical_bias" class="mw-redirect" title="Statistical bias">statistical bias</a> that can significantly impact the results of <a href="/wiki/Statistical_hypothesis_test" title="Statistical hypothesis test">statistical hypothesis tests</a>. MAUP affects results when point-based measures of spatial phenomena are <a href="/wiki/Aggregate_data" title="Aggregate data">aggregated</a> into spatial partitions or <i>areal units</i> (such as <a href="/wiki/Regions" class="mw-redirect" title="Regions">regions</a> or <a href="/wiki/District" title="District">districts</a>) as in, for example, <a href="/wiki/Population_density" title="Population density">population density</a> or <a href="/wiki/Illness_rate" class="mw-redirect" title="Illness rate">illness rates</a>.<sup id="cite_ref-Modifiable_areal_unit_problem_Openshaw1_3-0" class="reference"><a href="#cite_note-Modifiable_areal_unit_problem_Openshaw1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Modifiable_areal_unit_problem_Chen1_4-0" class="reference"><a href="#cite_note-Modifiable_areal_unit_problem_Chen1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The resulting summary values (e.g., totals, rates, proportions, densities) are influenced by both the shape and <a href="/wiki/Scale_(geography)" title="Scale (geography)">scale</a> of the aggregation unit.<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> </p> For example, census data may be aggregated into county districts, census tracts, postcode areas, police precincts, or any other arbitrary spatial partition. Thus the results of data aggregation are dependent on the mapmaker's choice of which "modifiable areal unit" to use in their analysis. A census <a href="/wiki/Choropleth_map" title="Choropleth map">choropleth map</a> calculating population density using state boundaries will yield radically different results than a map that calculates density based on county boundaries. Furthermore, census district boundaries are also subject to change over time,<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> meaning the MAUP must be considered when comparing past data to current data.</div></div> <div class="mw-heading mw-heading4"><h4 id="Modifiable_temporal_unit_problem">Modifiable temporal unit problem</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=6" title="Edit section: Modifiable temporal unit problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Modifiable_temporal_unit_problem" title="Modifiable temporal unit problem">Modifiable temporal unit problem</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Modifiable_temporal_unit_problem&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Time_units.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Time_units.svg/220px-Time_units.svg.png" decoding="async" width="220" height="320" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Time_units.svg/330px-Time_units.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Time_units.svg/440px-Time_units.svg.png 2x" data-file-width="624" data-file-height="907" /></a><figcaption>Flowchart illustrating selected units of time. The graphic also shows the three celestial objects that are related to the units of time.</figcaption></figure> The <a href="/wiki/Modifiable_temporal_unit_problem" title="Modifiable temporal unit problem">Modified Temporal Unit Problem</a> (MTUP) is a source of <a href="/wiki/Statistical_bias" class="mw-redirect" title="Statistical bias">statistical bias</a> that occurs in time series and spatial analysis when using temporal data that has been <a href="/wiki/Aggregate_data" title="Aggregate data">aggregated</a> into <a href="/wiki/Unit_of_time" title="Unit of time">temporal units</a>.<sup id="cite_ref-Modifiable_temporal_unit_problem_Cheng1_7-0" class="reference"><a href="#cite_note-Modifiable_temporal_unit_problem_Cheng1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Modifiable_temporal_unit_problem_Jong1_8-0" class="reference"><a href="#cite_note-Modifiable_temporal_unit_problem_Jong1-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In such cases, choosing a temporal unit (e.g., days, months, years) can affect the analysis results and lead to inconsistencies or errors in <a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">statistical hypothesis testing</a>.<sup id="cite_ref-Modifiable_temporal_unit_problem_Deckard1_9-0" class="reference"><a href="#cite_note-Modifiable_temporal_unit_problem_Deckard1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></div></div> <div class="mw-heading mw-heading4"><h4 id="Neighborhood_effect_averaging_problem">Neighborhood effect averaging problem</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=7" title="Edit section: Neighborhood effect averaging problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Neighborhood_effect_averaging_problem" title="Neighborhood effect averaging problem">Neighborhood effect averaging problem</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Neighborhood_effect_averaging_problem&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> The <a href="/wiki/Neighborhood_effect_averaging_problem" title="Neighborhood effect averaging problem">neighborhood effect averaging problem</a> or NEAP delves into the challenges associated with understanding the influence of aggregating neighborhood-level phenomena on individuals when mobility-dependent exposures influence the phenomena.<sup id="cite_ref-Neighborhood_effect_averaging_problem_Kwan1_10-0" class="reference"><a href="#cite_note-Neighborhood_effect_averaging_problem_Kwan1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Neighborhood_effect_averaging_problem_Kwan2_11-0" class="reference"><a href="#cite_note-Neighborhood_effect_averaging_problem_Kwan2-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Neighborhood_effect_averaging_problem_Xu1_12-0" class="reference"><a href="#cite_note-Neighborhood_effect_averaging_problem_Xu1-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The problem confounds the <a href="/wiki/Neighbourhood_effect" title="Neighbourhood effect">neighbourhood effect</a>, which suggests that a person's neighborhood impacts their individual characteristics, such as health.<sup id="cite_ref-Neighborhood_effect_averaging_problem_Ham1_13-0" class="reference"><a href="#cite_note-Neighborhood_effect_averaging_problem_Ham1-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Neighborhood_effect_averaging_problem_Parry1_14-0" class="reference"><a href="#cite_note-Neighborhood_effect_averaging_problem_Parry1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> It relates to the <a href="/wiki/Boundary_problem_(spatial_analysis)" title="Boundary problem (spatial analysis)">boundary problem</a>, in that delineated neighborhoods used for analysis may not fully account for an individual's activity space if the borders are permeable, and individual mobility crosses the boundaries. The term was first coined by <a href="/wiki/Mei-Po_Kwan" title="Mei-Po Kwan">Mei-Po Kwan</a> in the peer-reviewed journal "International Journal of Environmental Research and Public Health" in 2018.<sup id="cite_ref-Neighborhood_effect_averaging_problem_Kwan1_10-1" class="reference"><a href="#cite_note-Neighborhood_effect_averaging_problem_Kwan1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Neighborhood_effect_averaging_problem_Kwan2_11-1" class="reference"><a href="#cite_note-Neighborhood_effect_averaging_problem_Kwan2-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></div></div> <div class="mw-heading mw-heading4"><h4 id="Travelling_salesman_problem">Travelling salesman problem</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=8" title="Edit section: Travelling salesman problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Travelling_salesman_problem" title="Travelling salesman problem">Travelling salesman problem</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Travelling_salesman_problem&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:GLPK_solution_of_a_travelling_salesman_problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/GLPK_solution_of_a_travelling_salesman_problem.svg/220px-GLPK_solution_of_a_travelling_salesman_problem.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/GLPK_solution_of_a_travelling_salesman_problem.svg/330px-GLPK_solution_of_a_travelling_salesman_problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/GLPK_solution_of_a_travelling_salesman_problem.svg/440px-GLPK_solution_of_a_travelling_salesman_problem.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot.</figcaption></figure> <p>In the <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">theory of computational complexity</a>, the <a href="/wiki/Travelling_salesman_problem" title="Travelling salesman problem">travelling salesman problem</a> (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an <a href="/wiki/NP-hardness" title="NP-hardness">NP-hard</a> problem in <a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a>, important in <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a> and <a href="/wiki/Operations_research" title="Operations research">operations research</a>. </p><p>The <a href="/wiki/Traveling_purchaser_problem" title="Traveling purchaser problem">travelling purchaser problem</a>, the <a href="/wiki/Vehicle_routing_problem" title="Vehicle routing problem">vehicle routing problem</a> and the <a href="/wiki/Ring_star_problem" title="Ring star problem">ring star problem</a><sup id="cite_ref-Travelling_salesman_problem_Labbe2004_15-0" class="reference"><a href="#cite_note-Travelling_salesman_problem_Labbe2004-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> are three generalizations of TSP. </p><p>The decision version of the TSP (where given a length <i>L</i>, the task is to decide whether the graph has a tour whose length is at most <i>L</i>) belongs to the class of <a href="/wiki/NP-completeness" title="NP-completeness">NP-complete</a> problems. Thus, it is possible that the <a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">worst-case</a> <a href="/wiki/Time_complexity" title="Time complexity">running time</a> for any algorithm for the TSP increases <a href="/wiki/Time_complexity#Superpolynomial_time" title="Time complexity">superpolynomially</a> (but no more than <a href="/wiki/Exponential_time_hypothesis" title="Exponential time hypothesis">exponentially</a>) with the number of cities. </p> The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. It is used as a <a href="/wiki/Benchmark_(computing)" title="Benchmark (computing)">benchmark</a> for many optimization methods. Even though the problem is computationally difficult, many <a href="/wiki/Heuristic" title="Heuristic">heuristics</a> and <a href="/wiki/Exact_algorithm" title="Exact algorithm">exact algorithms</a> are known, so that some instances with tens of thousands of cities can be solved completely, and even problems with millions of cities can be approximated within a small fraction of 1%.<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></div></div> <div class="mw-heading mw-heading4"><h4 id="Uncertain_geographic_context_problem">Uncertain geographic context problem</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=9" title="Edit section: Uncertain geographic context problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Uncertain_geographic_context_problem" title="Uncertain geographic context problem">Uncertain geographic context problem</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Uncertain_geographic_context_problem&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> The <a href="/wiki/Uncertain_geographic_context_problem" title="Uncertain geographic context problem">uncertain geographic context problem</a> or UGCoP is a source of <a href="/wiki/Statistical_bias" class="mw-redirect" title="Statistical bias">statistical bias</a> that can significantly impact the results of spatial analysis when dealing with aggregate data.<sup id="cite_ref-Uncertain_geographic_context_problem_Kwan1_17-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Kwan1-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Kwan2_18-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Kwan2-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Matthews1_19-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Matthews1-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The UGCoP is very closely related to the <a href="/wiki/Modifiable_areal_unit_problem" title="Modifiable areal unit problem">Modifiable areal unit problem</a> (MAUP), and like the MAUP, arises from how we divide the land into areal units.<sup id="cite_ref-Uncertain_geographic_context_problem_Openshaw1_20-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Openshaw1-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Chen2_21-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Chen2-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> It is caused by the difficulty, or impossibility, of understanding how phenomena under investigation (such as people within a census tract) in different enumeration units interact between enumeration units, and outside of a study area over time.<sup id="cite_ref-Uncertain_geographic_context_problem_Kwan1_17-1" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Kwan1-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Gao1_22-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Gao1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> It is particularly important to consider the UGCoP within the discipline of <a href="/wiki/Time_geography" title="Time geography">time geography</a>, where phenomena under investigation can move between spatial enumeration units during the study period.<sup id="cite_ref-Uncertain_geographic_context_problem_Kwan2_18-1" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Kwan2-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Examples of research that needs to consider the UGCoP include food access and human mobility.<sup id="cite_ref-Uncertain_geographic_context_problem_Chen1_23-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Chen1-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Zhou1_24-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Zhou1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Space-time_prism_1500x1125.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Space-time_prism_1500x1125.png/280px-Space-time_prism_1500x1125.png" decoding="async" width="280" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Space-time_prism_1500x1125.png/420px-Space-time_prism_1500x1125.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Space-time_prism_1500x1125.png/560px-Space-time_prism_1500x1125.png 2x" data-file-width="1500" data-file-height="1127" /></a><figcaption>Schematic and example of a <i>space-time prism</i> using transit network data: On the right is a <a href="/wiki/Schematic" title="Schematic">schematic</a> diagram of a space-time prism, and on the left is a map of the potential path area for two different time budgets.<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></figcaption></figure> The uncertain geographic context problem, or UGCoP, was first coined by Dr. <a href="/wiki/Mei-Po_Kwan" title="Mei-Po Kwan">Mei-Po Kwan</a> in 2012.<sup id="cite_ref-Uncertain_geographic_context_problem_Kwan1_17-2" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Kwan1-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Kwan2_18-2" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Kwan2-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The problem is highly related to the <a href="/wiki/Ecological_fallacy" title="Ecological fallacy">ecological fallacy</a>, edge effect, and <a href="/wiki/Modifiable_areal_unit_problem" title="Modifiable areal unit problem">Modifiable areal unit problem</a> (MAUP) in that, it relates to aggregate units as they apply to individuals.<sup id="cite_ref-Uncertain_geographic_context_problem_Chen2_21-1" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Chen2-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> The crux of the problem is that the boundaries we use for aggregation are arbitrary and may not represent the actual neighborhood of the individuals within them.<sup id="cite_ref-Uncertain_geographic_context_problem_Openshaw1_20-1" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Openshaw1-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Chen2_21-2" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Chen2-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> While a particular enumeration unit, such as a <a href="/wiki/Census_tract" title="Census tract">census tract</a>, contains a person's location, they may cross its boundaries to work, go to school, and shop in completely different areas.<sup id="cite_ref-Uncertain_geographic_context_problem_Zhao2_26-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Zhao2-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Zhao3_27-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Zhao3-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Thus, the geographic phenomena under investigation extends beyond the delineated boundary .<sup id="cite_ref-Uncertain_geographic_context_problem_Gao1_22-1" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Gao1-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Tobler1_28-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Tobler1-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Salvo1_29-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Salvo1-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Different individuals, or groups may have completely different <a href="/wiki/Activity_space" title="Activity space">activity spaces</a>, making an enumeration unit that is relevant for one person meaningless to another.<sup id="cite_ref-Uncertain_geographic_context_problem_Chen1_23-1" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Chen1-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Thrift1_30-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Thrift1-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> For example, a map that aggregates people by school districts will be more meaningful when studying a population of students than the general population.<sup id="cite_ref-Uncertain_geographic_context_problem_Smool1_31-0" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Smool1-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Traditional spatial analysis, by necessity, treats each discrete areal unit as a self-contained neighborhood and does not consider the daily activity of crossing the boundaries.<sup id="cite_ref-Uncertain_geographic_context_problem_Kwan1_17-3" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Kwan1-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uncertain_geographic_context_problem_Kwan2_18-3" class="reference"><a href="#cite_note-Uncertain_geographic_context_problem_Kwan2-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></div></div> <div class="mw-heading mw-heading4"><h4 id="Weber_problem">Weber problem</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=10" title="Edit section: Weber problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Weber_problem" title="Weber problem">Weber problem</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Weber_problem&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, the <a href="/wiki/Weber_problem" title="Weber problem">Weber problem</a>, named after <a href="/wiki/Alfred_Weber" title="Alfred Weber">Alfred Weber</a>, is one of the most famous problems in <a href="/wiki/Location_theory" title="Location theory">location theory</a>. It requires finding a point in the plane that minimizes the sum of the transportation costs from this point to <span class="texhtml mvar" style="font-style:italic;">n</span> destination points, where different destination points are associated with different costs per unit distance. </p> The Weber problem generalizes the <a href="/wiki/Geometric_median" title="Geometric median">geometric median</a>, which assumes transportation costs per unit distance are the same for all destination points, and the problem of computing the <a href="/wiki/Fermat_point" title="Fermat point">Fermat point</a>, the geometric median of three points. For this reason it is sometimes called the Fermat–Weber problem, although the same name has also been used for the unweighted geometric median problem. The Weber problem is in turn generalized by the <a href="/w/index.php?title=Attraction%E2%80%93repulsion_problem&action=edit&redlink=1" class="new" title="Attraction–repulsion problem (page does not exist)">attraction–repulsion problem</a>, which allows some of the costs to be negative, so that greater distance from some points is better.</div></div> <div class="mw-heading mw-heading3"><h3 id="Spatial_characterization">Spatial characterization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=11" title="Edit section: Spatial characterization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bubonic_plague-en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Bubonic_plague-en.svg/240px-Bubonic_plague-en.svg.png" decoding="async" width="240" height="272" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Bubonic_plague-en.svg/360px-Bubonic_plague-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Bubonic_plague-en.svg/480px-Bubonic_plague-en.svg.png 2x" data-file-width="696" data-file-height="790" /></a><figcaption>Spread of bubonic plague in medieval Europe.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2013)">citation needed</span></a></i>]</sup> The colors indicate the spatial distribution of plague outbreaks over time.</figcaption></figure><p>The definition of the spatial presence of an entity constrains the possible analysis which can be applied to that entity and influences the final conclusions that can be reached. While this property is fundamentally true of all <a href="/wiki/Analysis" title="Analysis">analysis</a>, it is particularly important in spatial analysis because the tools to define and study entities favor specific characterizations of the entities being studied. Statistical techniques favor the spatial definition of objects as points because there are very few statistical techniques which operate directly on line, area, or volume elements. Computer tools favor the spatial definition of objects as homogeneous and separate elements because of the limited number of <a href="/wiki/Database" title="Database">database</a> elements and computational structures available, and the ease with which these primitive structures can be created.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2021)">citation needed</span></a></i>]</sup> </p><div class="mw-heading mw-heading3"><h3 id="Spatial_dependence">Spatial dependence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=12" title="Edit section: Spatial dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Spatial dependence</b> is the spatial relationship of variable values (for themes defined over space, such as <a href="/wiki/Rainfall" class="mw-redirect" title="Rainfall">rainfall</a>) or locations (for themes defined as objects, such as cities). Spatial dependence is measured as the existence of <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">statistical dependence</a> in a collection of <a href="/wiki/Random_variables" class="mw-redirect" title="Random variables">random variables</a>, each of which is associated with a different <a href="/wiki/Geographical_location" class="mw-redirect" title="Geographical location">geographical location</a>. Spatial dependence is of importance in applications where it is reasonable to postulate the existence of corresponding set of random variables at locations that have not been included in a sample. Thus <a href="/wiki/Rainfall" class="mw-redirect" title="Rainfall">rainfall</a> may be measured at a set of rain gauge locations, and such measurements can be considered as outcomes of random variables, but rainfall clearly occurs at other locations and would again be random. Because <a href="/wiki/Rainfall" class="mw-redirect" title="Rainfall">rainfall</a> exhibits properties of <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a>, spatial interpolation techniques can be used to estimate <a href="/wiki/Rainfall" class="mw-redirect" title="Rainfall">rainfall</a> amounts at locations near measured locations.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>As with other types of statistical dependence, the presence of spatial dependence generally leads to estimates of an average value from a sample being less accurate than had the samples been independent, although if negative dependence exists a sample average can be better than in the independent case. A different problem than that of estimating an overall average is that of <a href="/wiki/Spatial_interpolation" class="mw-redirect" title="Spatial interpolation">spatial interpolation</a>: here the problem is to estimate the unobserved random outcomes of variables at locations intermediate to places where measurements are made, on that there is spatial dependence between the observed and unobserved random variables.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2021)">citation needed</span></a></i>]</sup> </p><p>Tools for exploring spatial dependence include: <a href="/wiki/Spatial_correlation" class="mw-redirect" title="Spatial correlation">spatial correlation</a>, <a href="/wiki/Spatial_covariance_function" class="mw-redirect" title="Spatial covariance function">spatial covariance functions</a> and <a href="/wiki/Semivariogram" class="mw-redirect" title="Semivariogram">semivariograms</a>. Methods for spatial interpolation include <a href="/wiki/Kriging" title="Kriging">Kriging</a>, which is a type of <a href="/wiki/Best_linear_unbiased_prediction" title="Best linear unbiased prediction">best linear unbiased prediction</a>. The topic of spatial dependence is of importance to <a href="/wiki/Geostatistics" title="Geostatistics">geostatistics</a> and spatial analysis.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2021)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading4"><h4 id="Spatial_auto-correlation">Spatial auto-correlation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=13" title="Edit section: Spatial auto-correlation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spatial dependency is the co-variation of properties within geographic space: characteristics at proximal locations appear to be correlated, either positively or negatively.<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> Spatial dependency leads to the <b>spatial <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a></b> problem in statistics since, like temporal autocorrelation, this violates standard statistical techniques that assume independence among observations. For example, <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> analyses that do not compensate for spatial dependency can have unstable parameter estimates and yield unreliable significance tests. Spatial regression models (see below) capture these relationships and do not suffer from these weaknesses. It is also appropriate to view spatial dependency as a source of information rather than something to be corrected.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>Locational effects also manifest as spatial <a href="/wiki/Heterogeneity" class="mw-redirect" title="Heterogeneity">heterogeneity</a>, or the apparent variation in a process with respect to location in geographic space. Unless a space is uniform and boundless, every location will have some degree of uniqueness relative to the other locations. This affects the spatial dependency relations and therefore the spatial process. Spatial heterogeneity means that overall parameters estimated for the entire system may not adequately describe the process at any given location.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2021)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Spatial_association">Spatial association</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=14" title="Edit section: Spatial association"><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/Indicators_of_spatial_association" title="Indicators of spatial association">Indicators of spatial association</a></div> <p><b>Spatial association</b> is the degree to which things are similarly arranged in space. Analysis of the distribution patterns of two phenomena is done by map overlay. If the distributions are similar, then the spatial association is strong, and vice versa.<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> In a <a href="/wiki/Geographic_Information_System" class="mw-redirect" title="Geographic Information System">Geographic Information System</a>, the analysis can be done quantitatively. For example, a set of observations (as points or extracted from raster cells) at matching locations can be intersected and examined by <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>. </p><p>Like <a href="/wiki/Spatial_autocorrelation" class="mw-redirect" title="Spatial autocorrelation">spatial autocorrelation</a>, this can be a useful tool for spatial prediction. In spatial modeling, the concept of spatial association allows the use of covariates in a regression equation to predict the geographic field and thus produce a map. </p> <div class="mw-heading mw-heading4"><h4 id="The_second_dimension_of_spatial_association">The second dimension of spatial association</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=15" title="Edit section: The second dimension of spatial association"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The second dimension of spatial association (SDA) reveals the association between spatial variables through extracting geographical information at locations outside samples. SDA effectively uses the missing geographical information outside sample locations in methods of the first dimension of spatial association (FDA), which explore spatial association using observations at sample locations.<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> <div class="mw-heading mw-heading3"><h3 id="Scaling">Scaling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=16" title="Edit section: Scaling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spatial <a href="/wiki/Measurement" title="Measurement">measurement</a> scale is a persistent issue in spatial analysis; more detail is available at the <a href="/wiki/Modifiable_areal_unit_problem" title="Modifiable areal unit problem">modifiable areal unit problem</a> (MAUP) topic entry. Landscape ecologists developed a series of <a href="/wiki/Scale_invariant" class="mw-redirect" title="Scale invariant">scale invariant</a> metrics for aspects of ecology that are <a href="/wiki/Fractal" title="Fractal">fractal</a> in nature.<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> In more general terms, no scale independent method of <a href="/wiki/Analysis" title="Analysis">analysis</a> is widely agreed upon for spatial statistics.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2021)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Sampling">Sampling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=17" title="Edit section: Sampling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spatial <a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">sampling</a> involves determining a limited number of locations in geographic space for faithfully measuring phenomena that are subject to dependency and heterogeneity. <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2010)">citation needed</span></a></i>]</sup> Dependency suggests that since one location can predict the value of another location, we do not need observations in both places. But heterogeneity suggests that this relation can change across space, and therefore we cannot trust an observed degree of dependency beyond a region that may be small. Basic spatial sampling schemes include random, clustered and systematic. These basic schemes can be applied at multiple levels in a designated spatial hierarchy (e.g., urban area, city, neighborhood). It is also possible to exploit ancillary data, for example, using property values as a guide in a spatial sampling scheme to measure educational attainment and income. Spatial models such as autocorrelation statistics, regression and interpolation (see below) can also dictate sample design.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2010)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Common_errors_in_spatial_analysis">Common errors in spatial analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=18" title="Edit section: Common errors in spatial analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fundamental issues in spatial analysis lead to numerous problems in analysis including bias, distortion and outright errors in the conclusions reached. These issues are often interlinked but various attempts have been made to separate out particular issues from each other.<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-heading4"><h4 id="Length">Length</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=19" title="Edit section: Length"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In discussing the <a href="/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Self-Similarity_and_Fractional_Dimension" class="mw-redirect" title="How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension">coastline of Britain</a>, <a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a> showed that certain spatial concepts are inherently nonsensical despite presumption of their validity. Lengths in ecology depend directly on the scale at which they are measured and experienced. So while surveyors commonly measure the length of a river, this length only has meaning in the context of the relevance of the measuring technique to the question under study.<sup id="cite_ref-SGAM-Penn-State-U_39-0" class="reference"><a href="#cite_note-SGAM-Penn-State-U-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <ul class="gallery mw-gallery-traditional" style="max-width: 489px;"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Britain-fractal-coastline-200km.png" class="mw-file-description" title="Britain measured using a 200 km linear measurement"><img alt="Britain measured using a 200 km linear measurement" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Britain-fractal-coastline-200km.png/62px-Britain-fractal-coastline-200km.png" decoding="async" width="62" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Britain-fractal-coastline-200km.png/93px-Britain-fractal-coastline-200km.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Britain-fractal-coastline-200km.png/124px-Britain-fractal-coastline-200km.png 2x" data-file-width="311" data-file-height="600" /></a></span></div> <div class="gallerytext">Britain measured using a 200 km linear measurement</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Britain-fractal-coastline-100km.png" class="mw-file-description" title="Britain measured using a 100 km linear measurement"><img alt="Britain measured using a 100 km linear measurement" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Britain-fractal-coastline-100km.png/62px-Britain-fractal-coastline-100km.png" decoding="async" width="62" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Britain-fractal-coastline-100km.png/93px-Britain-fractal-coastline-100km.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Britain-fractal-coastline-100km.png/124px-Britain-fractal-coastline-100km.png 2x" data-file-width="311" data-file-height="600" /></a></span></div> <div class="gallerytext">Britain measured using a 100 km linear measurement</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Britain-fractal-coastline-50km.png" class="mw-file-description" title="Britain measured using a 50 km linear measurement"><img alt="Britain measured using a 50 km linear measurement" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Britain-fractal-coastline-50km.png/62px-Britain-fractal-coastline-50km.png" decoding="async" width="62" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Britain-fractal-coastline-50km.png/93px-Britain-fractal-coastline-50km.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Britain-fractal-coastline-50km.png/124px-Britain-fractal-coastline-50km.png 2x" data-file-width="311" data-file-height="600" /></a></span></div> <div class="gallerytext">Britain measured using a 50 km linear measurement</div> </li> </ul> <div class="mw-heading mw-heading4"><h4 id="Locational_fallacy">Locational fallacy</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=20" title="Edit section: Locational fallacy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The locational fallacy refers to error due to the particular spatial characterization chosen for the elements of study, in particular choice of placement for the spatial presence of the element.<sup id="cite_ref-SGAM-Penn-State-U_39-1" class="reference"><a href="#cite_note-SGAM-Penn-State-U-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p>Spatial characterizations may be simplistic or even wrong. Studies of humans often reduce the spatial existence of humans to a single point, for instance their home address. This can easily lead to poor analysis, for example, when considering disease transmission which can happen at work or at school and therefore far from the home.<sup id="cite_ref-SGAM-Penn-State-U_39-2" class="reference"><a href="#cite_note-SGAM-Penn-State-U-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p>The spatial characterization may implicitly limit the subject of study. For example, the spatial analysis of crime data has recently become popular but these studies can only describe the particular kinds of crime which can be described spatially. This leads to many maps of assault but not to any maps of embezzlement with political consequences in the conceptualization of crime and the design of policies to address the issue.<sup id="cite_ref-SGAM-Penn-State-U_39-3" class="reference"><a href="#cite_note-SGAM-Penn-State-U-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Atomic_fallacy">Atomic fallacy</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=21" title="Edit section: Atomic fallacy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This describes errors due to treating elements as separate 'atoms' outside of their spatial context.<sup id="cite_ref-SGAM-Penn-State-U_39-4" class="reference"><a href="#cite_note-SGAM-Penn-State-U-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> The fallacy is about transferring individual conclusions to spatial units.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Ecological_fallacy">Ecological fallacy</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=22" title="Edit section: Ecological fallacy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Ecological_fallacy" title="Ecological fallacy">ecological fallacy</a> describes errors due to performing analyses on aggregate data when trying to reach conclusions on the individual units.<sup id="cite_ref-SGAM-Penn-State-U_39-5" class="reference"><a href="#cite_note-SGAM-Penn-State-U-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><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> Errors occur in part from spatial aggregation. For example, a <a href="/wiki/Pixel" title="Pixel">pixel</a> represents the average surface temperatures within an area. Ecological fallacy would be to assume that all points within the area have the same temperature. </p> <div class="mw-heading mw-heading3"><h3 id="Solutions_to_the_fundamental_issues">Solutions to the fundamental issues</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=23" title="Edit section: Solutions to the fundamental issues"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Geographic_space">Geographic space</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=24" title="Edit section: Geographic space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Manhattan_distance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/200px-Manhattan_distance.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/300px-Manhattan_distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/400px-Manhattan_distance.svg.png 2x" data-file-width="283" data-file-height="283" /></a><figcaption>Manhattan distance versus Euclidean distance: The red, blue, and yellow lines have the same length (12) in both Euclidean and taxicab geometry. In Euclidean geometry, the green line has length 6×<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> ≈ 8.48, and is the unique shortest path. In taxicab geometry, the green line's length is still 12, making it no shorter than any other path shown.</figcaption></figure><p>A mathematical space exists whenever we have a set of observations and quantitative measures of their attributes. For example, we can represent individuals' incomes or years of education within a coordinate system where the location of each individual can be specified with respect to both dimensions. The distance between individuals within this space is a quantitative measure of their differences with respect to income and education. However, in spatial analysis, we are concerned with specific types of mathematical spaces, namely, geographic space. In geographic space, the observations correspond to locations in a spatial measurement framework that capture their proximity in the real world. The locations in a spatial measurement framework often represent locations on the surface of the Earth, but this is not strictly necessary. A spatial measurement framework can also capture proximity with respect to, say, interstellar space or within a biological entity such as a liver. The fundamental tenet is <a href="/wiki/Tobler%27s_first_law_of_geography" title="Tobler's first law of geography">Tobler's First Law of Geography</a>: if the interrelation between entities increases with proximity in the real world, then representation in geographic space and assessment using spatial analysis techniques are appropriate. </p><p>The <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> between locations often represents their proximity, although this is only one possibility. There are an infinite number of distances in addition to Euclidean that can support quantitative analysis. For example, "Manhattan" (or "<a href="/wiki/Taxicab_geometry" title="Taxicab geometry">Taxicab</a>") distances where movement is restricted to paths parallel to the axes can be more meaningful than Euclidean distances in urban settings. In addition to distances, other geographic relationships such as connectivity<b> (e.g., the existence or degree of shared borders) and direction</b> can also influence the relationships among entities. It is also possible to compute minimal cost paths across a cost surface; for example, this can represent proximity among locations when travel must occur across rugged terrain. </p> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=25" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <blockquote> <p> Spatial data comes in many varieties and it is not easy to arrive at a system of classification that is simultaneously exclusive, exhaustive, imaginative, and satisfying. -- G. Upton & B. Fingelton<sup id="cite_ref-Upton+:1985_42-0" class="reference"><a href="#cite_note-Upton+:1985-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Spatial_data_analysis">Spatial data analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=26" title="Edit section: Spatial data analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Urban and Regional Studies deal with large tables of spatial data obtained from censuses and surveys. It is necessary to simplify the huge amount of detailed information in order to extract the main trends. Multivariable analysis (or <a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a>, FA) allows a change of variables, transforming the many variables of the census, usually correlated between themselves, into fewer independent "Factors" or "Principal Components" which are, actually, the <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvectors</a> of the data correlation matrix weighted by the inverse of their eigenvalues. This change of variables has two main advantages: </p> <ol><li>Since information is concentrated on the first new factors, it is possible to keep only a few of them while losing only a small amount of information; mapping them produces fewer and more significant maps</li> <li>The factors, actually the eigenvectors, are orthogonal by construction, i.e. not correlated. In most cases, the dominant factor (with the largest eigenvalue) is the Social Component, separating rich and poor in the city. Since factors are not-correlated, other smaller processes than social status, which would have remained hidden otherwise, appear on the second, third, ... factors.</li></ol> <p>Factor analysis depends on measuring distances between observations : the choice of a significant metric is crucial. The Euclidean metric (Principal Component Analysis), the Chi-Square distance (Correspondence Analysis) or the Generalized Mahalanobis distance (Discriminant Analysis) are among the more widely used.<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> More complicated models, using communalities or rotations have been proposed.<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> </p><p>Using multivariate methods in spatial analysis began really in the 1950s (although some examples go back to the beginning of the century) and culminated in the 1970s, with the increasing power and accessibility of computers. Already in 1948, in a seminal publication, two sociologists, <a href="/wiki/Wendell_Bell" title="Wendell Bell">Wendell Bell</a> and Eshref Shevky,<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> had shown that most city populations in the US and in the world could be represented with three independent factors : 1- the « socio-economic status » opposing rich and poor districts and distributed in sectors running along highways from the city center, 2- the « life cycle », i.e. the age structure of households, distributed in concentric circles, and 3- « race and ethnicity », identifying patches of migrants located within the city. In 1961, in a groundbreaking study, British geographers used FA to classify British towns.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Brian J Berry, at the University of Chicago, and his students made a wide use of the method,<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> applying it to most important cities in the world and exhibiting common social structures.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> The use of Factor Analysis in Geography, made so easy by modern computers, has been very wide but not always very wise.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p><p>Since the vectors extracted are determined by the data matrix, it is not possible to compare factors obtained from different censuses. A solution consists in fusing together several census matrices in a unique table which, then, may be analyzed. This, however, assumes that the definition of the variables has not changed over time and produces very large tables, difficult to manage. A better solution, proposed by psychometricians,<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> groups the data in a « cubic matrix », with three entries (for instance, locations, variables, time periods). A Three-Way Factor Analysis produces then three groups of factors related by a small cubic « core matrix ».<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> This method, which exhibits data evolution over time, has not been widely used in geography.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> In Los Angeles,<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> however, it has exhibited the role, traditionally ignored, of Downtown as an organizing center for the whole city during several decades. </p> <div class="mw-heading mw-heading3"><h3 id="Spatial_autocorrelation">Spatial autocorrelation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=27" title="Edit section: Spatial autocorrelation"><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/Tobler%27s_first_law_of_geography" title="Tobler's first law of geography">Tobler's first law of geography</a></div> <p>Spatial <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a> statistics measure and analyze the degree of dependency among observations in a geographic space. Classic spatial autocorrelation statistics include <a href="/wiki/Moran%27s_I" title="Moran's I">Moran's <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span></a>, <a href="/wiki/Geary%27s_C" title="Geary's C">Geary's <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span></a>, <a href="/w/index.php?title=Getis%27s_G&action=edit&redlink=1" class="new" title="Getis's G (page does not exist)">Getis's</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> and the <a href="/w/index.php?title=Standard_deviational_ellipse&action=edit&redlink=1" class="new" title="Standard deviational ellipse (page does not exist)">standard deviational ellipse</a>. These statistics require measuring a <a href="/wiki/Spatial_weights_matrix" class="mw-redirect" title="Spatial weights matrix">spatial weights matrix</a> that reflects the intensity of the geographic relationship between observations in a neighborhood, e.g., the distances between neighbors, the lengths of shared border, or whether they fall into a specified directional class such as "west". Classic spatial autocorrelation statistics compare the spatial weights to the covariance relationship at pairs of locations. Spatial autocorrelation that is more positive than expected from random indicate the clustering of similar values across geographic space, while significant negative spatial autocorrelation indicates that neighboring values are more dissimilar than expected by chance, suggesting a spatial pattern similar to a chess board. </p><p>Spatial autocorrelation statistics such as Moran's <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> and Geary's <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> are global in the sense that they estimate the overall degree of spatial autocorrelation for a dataset. The possibility of spatial heterogeneity suggests that the estimated degree of autocorrelation may vary significantly across geographic space. <b>Local spatial autocorrelation statistics</b> provide estimates disaggregated to the level of the spatial analysis units, allowing assessment of the dependency relationships across space. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> statistics compare neighborhoods to a global average and identify local regions of strong autocorrelation. Local versions of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> statistics are also available. </p> <div class="mw-heading mw-heading3"><h3 id="Spatial_heterogeneity">Spatial heterogeneity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=28" title="Edit section: Spatial heterogeneity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Spatial_heterogeneity" title="Spatial heterogeneity">Spatial heterogeneity</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Spatial_heterogeneity&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:NLCD_landcover_MSN_area.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/NLCD_landcover_MSN_area.png/220px-NLCD_landcover_MSN_area.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/NLCD_landcover_MSN_area.png/330px-NLCD_landcover_MSN_area.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/NLCD_landcover_MSN_area.png/440px-NLCD_landcover_MSN_area.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Land cover surrounding Madison, WI. Fields are colored yellow and brown, water is colored blue, and urban surfaces are colored red.</figcaption></figure> <a href="/wiki/Spatial_heterogeneity" title="Spatial heterogeneity">Spatial heterogeneity</a> is a property generally ascribed to a <a href="/wiki/Landscape" title="Landscape">landscape</a> or to a <a href="/wiki/Population" title="Population">population</a>. It refers to the uneven distribution of various concentrations of each <a href="/wiki/Species" title="Species">species</a> within an area. A landscape with spatial heterogeneity has a mix of concentrations of multiple species of plants or animals (biological), or of <a href="/wiki/Terrain" title="Terrain">terrain</a> formations (geological), or environmental characteristics (e.g. rainfall, temperature, wind) filling its area. A population showing spatial heterogeneity is one where various concentrations of individuals of this species are unevenly distributed across an area; nearly synonymous with "patchily distributed."</div></div> <div class="mw-heading mw-heading3"><h3 id="Spatial_interaction">Spatial interaction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=29" title="Edit section: Spatial interaction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spatial interaction or "<a href="/wiki/Gravity_model" title="Gravity model">gravity models</a>" estimate the flow of people, material or information between locations in geographic space. Factors can include origin propulsive variables such as the number of commuters in residential areas, destination attractiveness variables such as the amount of office space in employment areas, and proximity relationships between the locations measured in terms such as driving distance or travel time. In addition, the topological, or <a href="/wiki/Geospatial_topology" title="Geospatial topology">connective</a>, relationships between areas must be identified, particularly considering the often conflicting relationship between distance and topology; for example, two spatially close neighborhoods may not display any significant interaction if they are separated by a highway. After specifying the functional forms of these relationships, the analyst can estimate model parameters using observed flow data and standard estimation techniques such as ordinary least squares or maximum likelihood. Competing destinations versions of spatial interaction models include the proximity among the destinations (or origins) in addition to the origin-destination proximity; this captures the effects of destination (origin) clustering on flows. </p> <div class="mw-heading mw-heading3"><h3 id="Spatial_interpolation">Spatial interpolation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=30" title="Edit section: Spatial interpolation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Spatial_interpolation" class="mw-redirect" title="Spatial interpolation">Spatial interpolation</a> methods estimate the variables at unobserved locations in geographic space based on the values at observed locations. Basic methods include <a href="/wiki/Inverse_distance_weighting" title="Inverse distance weighting">inverse distance weighting</a>: this attenuates the variable with decreasing proximity from the observed location. <a href="/wiki/Kriging" title="Kriging">Kriging</a> is a more sophisticated method that interpolates across space according to a spatial lag relationship that has both systematic and random components. This can accommodate a wide range of spatial relationships for the hidden values between observed locations. Kriging provides optimal estimates given the hypothesized lag relationship, and error estimates can be mapped to determine if spatial patterns exist. </p> <div class="mw-heading mw-heading3"><h3 id="Spatial_regression">Spatial regression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=31" title="Edit section: Spatial regression"><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">See also: <a href="/wiki/Local_regression" title="Local regression">Local regression</a> and <a href="/wiki/Regression-Kriging" class="mw-redirect" title="Regression-Kriging">Regression-Kriging</a></div> <p>Spatial regression methods capture spatial dependency in <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>, avoiding statistical problems such as unstable parameters and unreliable significance tests, as well as providing information on spatial relationships among the variables involved. Depending on the specific technique, spatial dependency can enter the regression model as relationships between the independent variables and the dependent, between the dependent variables and a spatial lag of itself, or in the error terms. <b>Geographically weighted regression</b> (GWR) is a local version of spatial regression that generates parameters disaggregated by the spatial units of analysis.<sup id="cite_ref-Fotheringham1998_54-0" class="reference"><a href="#cite_note-Fotheringham1998-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> This allows assessment of the spatial heterogeneity in the estimated relationships between the independent and dependent variables. The use of <a href="/wiki/Bayesian_hierarchical_modeling" title="Bayesian hierarchical modeling">Bayesian hierarchical modeling</a><sup id="cite_ref-Banerjee2014_55-0" class="reference"><a href="#cite_note-Banerjee2014-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> in conjunction with <a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> (MCMC) methods have recently shown to be effective in modeling complex relationships using Poisson-Gamma-CAR, Poisson-lognormal-SAR, or Overdispersed logit models. Statistical packages for implementing such Bayesian models using MCMC include <a href="/wiki/WinBugs" class="mw-redirect" title="WinBugs">WinBugs</a>, <a href="/wiki/CrimeStat" title="CrimeStat">CrimeStat</a> and many packages available via <a href="/wiki/R_programming_language" class="mw-redirect" title="R programming language">R programming language</a>.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p><p>Spatial stochastic processes, such as <a href="/wiki/Gaussian_processes" class="mw-redirect" title="Gaussian processes">Gaussian processes</a> are also increasingly being deployed in spatial regression analysis. Model-based versions of GWR, known as spatially varying coefficient models have been applied to conduct Bayesian inference.<sup id="cite_ref-Banerjee2014_55-1" class="reference"><a href="#cite_note-Banerjee2014-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> Spatial stochastic process can become computationally effective and scalable Gaussian process models, such as Gaussian Predictive Processes<sup id="cite_ref-Banerjee2008_57-0" class="reference"><a href="#cite_note-Banerjee2008-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> and Nearest Neighbor Gaussian Processes (NNGP).<sup id="cite_ref-Banerjee2016_58-0" class="reference"><a href="#cite_note-Banerjee2016-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Spatial_neural_networks">Spatial neural networks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=32" title="Edit section: Spatial neural networks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Spatial_neural_network" title="Spatial neural network">Spatial neural network</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Spatial_neural_network&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> Spatial neural networks (SNNs) constitute a supercategory of tailored <a href="/wiki/Artificial_neural_networks" class="mw-redirect" title="Artificial neural networks">neural networks (NNs)</a> for representing and predicting geographic phenomena. They generally improve both the statistical <a href="/wiki/Accuracy_and_precision" title="Accuracy and precision">accuracy</a> and <a href="/wiki/Statistical_reliability" class="mw-redirect" title="Statistical reliability">reliability</a> of the a-spatial/classic NNs whenever they handle <a href="/wiki/Geographic_data_and_information" title="Geographic data and information"> geo-spatial datasets</a>, and also of the other spatial <a href="/wiki/Statistical_model" title="Statistical model">(statistical) models</a> (e.g. spatial regression models) whenever the geo-spatial <a href="/wiki/Data_set" title="Data set">datasets</a>' variables depict <a href="/wiki/Nonlinear_system" title="Nonlinear system">non-linear relations</a>.<sup id="cite_ref-Spatial_neural_network_Morer_et_al._(2020)_59-0" class="reference"><a href="#cite_note-Spatial_neural_network_Morer_et_al._(2020)-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Spatial_neural_network_Gupta_et_al._(2021)_60-0" class="reference"><a href="#cite_note-Spatial_neural_network_Gupta_et_al._(2021)-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Spatial_neural_network_Hagenauer_et_al._(2022)_61-0" class="reference"><a href="#cite_note-Spatial_neural_network_Hagenauer_et_al._(2022)-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> Examples of SNNs are the OSFA spatial neural networks, SVANNs and GWNNs.</div></div> <div class="mw-heading mw-heading3"><h3 id="Simulation_and_modeling">Simulation and modeling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=33" title="Edit section: Simulation and modeling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spatial interaction models are aggregate and top-down: they specify an overall governing relationship for flow between locations. This characteristic is also shared by urban models such as those based on mathematical programming, flows among economic sectors, or bid-rent theory. An alternative modeling perspective is to represent the system at the highest possible level of disaggregation and study the bottom-up emergence of complex patterns and relationships from behavior and interactions at the individual level. <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2010)">citation needed</span></a></i>]</sup> </p><p><a href="/wiki/Complex_adaptive_systems" class="mw-redirect" title="Complex adaptive systems">Complex adaptive systems</a> theory as applied to spatial analysis suggests that simple interactions among proximal entities can lead to intricate, persistent and functional spatial entities at aggregate levels. Two fundamentally spatial simulation methods are cellular automata and agent-based modeling. <a href="/wiki/Cellular_automata" class="mw-redirect" title="Cellular automata">Cellular automata</a> modeling imposes a fixed spatial framework such as grid cells and specifies rules that dictate the state of a cell based on the states of its neighboring cells. As time progresses, spatial patterns emerge as cells change states based on their neighbors; this alters the conditions for future time periods. For example, cells can represent locations in an urban area and their states can be different types of land use. Patterns that can emerge from the simple interactions of local land uses include office districts and <a href="/wiki/Urban_sprawl" title="Urban sprawl">urban sprawl</a>. <a href="/wiki/Agent-based_model" title="Agent-based model">Agent-based modeling</a> uses software entities (agents) that have purposeful behavior (goals) and can react, interact and modify their environment while seeking their objectives. Unlike the cells in cellular automata, simulysts can allow agents to be mobile with respect to space. For example, one could model traffic flow and dynamics using agents representing individual vehicles that try to minimize travel time between specified origins and destinations. While pursuing minimal travel times, the agents must avoid collisions with other vehicles also seeking to minimize their travel times. Cellular automata and agent-based modeling are complementary modeling strategies. They can be integrated into a common geographic automata system where some agents are fixed while others are mobile. </p><p>Calibration plays a pivotal role in both CA and ABM simulation and modelling approaches. Initial approaches to CA proposed robust calibration approaches based on stochastic, Monte Carlo methods.<sup id="cite_ref-Silva_62-0" class="reference"><a href="#cite_note-Silva-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Silva_EPS_63-0" class="reference"><a href="#cite_note-Silva_EPS-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> ABM approaches rely on agents' decision rules (in many cases extracted from qualitative research base methods such as questionnaires).<sup id="cite_ref-silva3_64-0" class="reference"><a href="#cite_note-silva3-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> Recent Machine Learning Algorithms calibrate using training sets, for instance in order to understand the qualities of the built environment.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Multiple-point_geostatistics_(MPS)"><span id="Multiple-point_geostatistics_.28MPS.29"></span>Multiple-point geostatistics (MPS)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=34" title="Edit section: Multiple-point geostatistics (MPS)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spatial analysis of a conceptual geological model is the main purpose of any MPS algorithm. The method analyzes the spatial statistics of the geological model, called the training image, and generates realizations of the phenomena that honor those input multiple-point statistics. </p><p>A recent MPS algorithm used to accomplish this task is the pattern-based method by Honarkhah.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> In this method, a distance-based approach is employed to analyze the patterns in the training image. This allows the reproduction of the multiple-point statistics, and the complex geometrical features of the training image. Each output of the MPS algorithm is a realization that represents a random field. Together, several realizations may be used to quantify spatial uncertainty. </p><p>One of the recent methods is presented by Tahmasebi et al.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> uses a cross-correlation function to improve the spatial pattern reproduction. They call their MPS simulation method as the CCSIM algorithm. This method is able to quantify the spatial connectivity, variability and uncertainty. Furthermore, the method is not sensitive to any type of data and is able to simulate both categorical and continuous scenarios. CCSIM algorithm is able to be used for any stationary, non-stationary and multivariate systems and it can provide high quality visual appeal model.,<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Geospatial_and_hydrospatial_analysis">Geospatial and hydrospatial analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=35" title="Edit section: Geospatial and hydrospatial analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Cleanup_merge plainlinks metadata ambox ambox-move" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/50px-Split-arrows.svg.png" decoding="async" width="50" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/75px-Split-arrows.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/100px-Split-arrows.svg.png 2x" data-file-width="60" data-file-height="20" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><b>This section may need to be cleaned up.</b> It has been merged from <i><a href="/wiki/Geospatial_analysis" class="mw-redirect" title="Geospatial analysis">Geospatial analysis</a></i>.</div></td></tr></tbody></table> <p><b>Geospatial and hydrospatial analysis</b>, or just <b>spatial analysis</b>,<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> is an approach to applying <a href="/wiki/Statistical_analysis" class="mw-redirect" title="Statistical analysis">statistical analysis</a> and other analytic techniques to data which has a geographical or spatial aspect. Such analysis would typically employ software capable of rendering maps processing spatial data, and applying <a href="/wiki/Scientific_modelling" title="Scientific modelling">analytical</a> methods to terrestrial or <a href="/wiki/Geographic" class="mw-redirect" title="Geographic">geographic</a> datasets, including the use of <a href="/wiki/Geographic_information_system" title="Geographic information system">geographic information systems</a> and <a href="/wiki/Geomatics" title="Geomatics">geomatics</a>.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Geographical_information_system_usage">Geographical information system usage</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=36" title="Edit section: Geographical information system usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information systems</a> (GIS) — a large domain that provides a variety of capabilities designed to capture, store, manipulate, analyze, manage, and present all types of geographical data — utilizes geospatial and hydrospatial analysis in a variety of contexts, operations and applications. </p> <div class="mw-heading mw-heading4"><h4 id="Basic_applications">Basic applications</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=37" title="Edit section: Basic applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Geospatial and Hydrospatial analysis, using <a href="/wiki/GIS" class="mw-redirect" title="GIS">GIS</a>, was developed for problems in the environmental and life sciences, in particular <a href="/wiki/Ecology" title="Ecology">ecology</a>, <a href="/wiki/Geology" title="Geology">geology</a> and <a href="/wiki/Epidemiology" title="Epidemiology">epidemiology</a>. It has extended to almost all industries including defense, intelligence, utilities, Natural Resources (i.e. Oil and Gas, Forestry ... etc.), social sciences, medicine and <a href="/wiki/Public_Safety" class="mw-redirect" title="Public Safety">Public Safety</a> (i.e. emergency management and criminology), disaster risk reduction and management (DRRM), and <a href="/wiki/Climate_change_adaptation" title="Climate change adaptation">climate change adaptation</a> (CCA). Spatial statistics typically result primarily from observation rather than experimentation. Hydrospatial is particularly used for the aquatic side and the members related to the water surface, column, bottom, sub-bottom and the coastal zones. </p> <div class="mw-heading mw-heading4"><h4 id="Basic_operations">Basic operations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=38" title="Edit section: Basic operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vector-based <a href="/wiki/GIS" class="mw-redirect" title="GIS">GIS</a> is typically related to operations such as map overlay (combining two or more maps or map layers according to predefined rules), simple buffering (identifying regions of a map within a specified distance of one or more features, such as towns, roads or rivers) and similar basic operations. This reflects (and is reflected in) the use of the term spatial analysis within the Open Geospatial Consortium (<a href="/wiki/Open_Geospatial_Consortium" title="Open Geospatial Consortium">OGC</a>) “simple feature specifications”. For raster-based GIS, widely used in the environmental sciences and remote sensing, this typically means a range of actions applied to the grid cells of one or more maps (or images) often involving filtering and/or algebraic operations (map algebra). These techniques involve processing one or more raster layers according to simple rules resulting in a new map layer, for example replacing each cell value with some combination of its neighbours’ values, or computing the sum or difference of specific attribute values for each grid cell in two matching raster datasets. Descriptive statistics, such as cell counts, means, variances, maxima, minima, cumulative values, frequencies and a number of other measures and distance computations are also often included in this generic term spatial analysis. Spatial analysis includes a large variety of statistical techniques (descriptive, <a href="/wiki/Exploratory_data_analysis" title="Exploratory data analysis">exploratory</a>, and explanatory <a href="/wiki/Statistics" title="Statistics">statistics</a>) that apply to data that vary spatially and which can vary over time. Some more advanced statistical techniques include Getis-ord Gi* or Anselin Local Moran's I which are used to determine clustering patterns of spatially referenced data. </p> <div class="mw-heading mw-heading4"><h4 id="Advanced_operations">Advanced operations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=39" title="Edit section: Advanced operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Geospatial and Hydrospatial analysis goes beyond 2D and 3D mapping operations and spatial statistics. It is multi-dimensional and also temporal and includes: </p> <ul><li>Surface analysis — in particular analysing the properties of physical surfaces, such as <a href="/wiki/Gradient" title="Gradient">gradient</a>, <a href="/wiki/Aspect_(geography)" title="Aspect (geography)">aspect</a> and <a href="/wiki/Visibility" title="Visibility">visibility</a>, and analysing surface-like data “fields”;</li> <li>Network analysis — examining the properties of natural and man-made networks in order to understand the behaviour of flows within and around such networks; and locational analysis. GIS-based network analysis may be used to address a wide range of practical problems such as route selection and facility location (core topics in the field of <a href="/wiki/Operations_research" title="Operations research">operations research</a>), and problems involving flows such as those found in Hydrospatial and <a href="/wiki/Hydrology" title="Hydrology">hydrology</a> and transportation research. In many instances location problems relate to networks and as such are addressed with tools designed for this purpose, but in others existing networks may have little or no relevance or may be impractical to incorporate within the modeling process. Problems that are not specifically network constrained, such as new road or pipeline routing, regional warehouse location, mobile phone mast positioning or the selection of rural community health care sites, may be effectively analysed (at least initially) without reference to existing physical networks. Locational analysis "in the plane" is also applicable where suitable network datasets are not available, or are too large or expensive to be utilised, or where the location algorithm is very complex or involves the examination or simulation of a very large number of alternative configurations.</li> <li><a href="/wiki/Geovisualization" title="Geovisualization">Geovisualization</a> — the creation and manipulation of images, maps, diagrams, charts, 3D views and their associated tabular datasets. GIS packages increasingly provide a range of such tools, providing static or rotating views, draping images over 2.5D surface representations, providing animations and fly-throughs, dynamic linking and brushing and spatio-temporal visualisations. This latter class of tools is the least developed, reflecting in part the limited range of suitable compatible datasets and the limited set of analytical methods available, although this picture is changing rapidly. All these facilities augment the core tools utilised in spatial analysis throughout the analytical process (exploration of data, identification of patterns and relationships, construction of models, and communication of results)</li></ul> <div class="mw-heading mw-heading4"><h4 id="Mobile_geospatial_and_hydrospatial_Computing">Mobile geospatial and hydrospatial Computing</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=40" title="Edit section: Mobile geospatial and hydrospatial Computing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Traditionally geospatial and hydrospatial computing has been performed primarily on personal computers (PCs) or servers. Due to the increasing capabilities of mobile devices, however, geospatial computing in mobile devices is a fast-growing trend.<sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> The portable nature of these devices, as well as the presence of useful sensors, such as Global Navigation Satellite System (GNSS) receivers and barometric pressure sensors, make them useful for capturing and processing geospatial and hydrospatial information in the field. In addition to the local processing of geospatial information on mobile devices, another growing trend is cloud-based geospatial computing. In this architecture, data can be collected in the field using mobile devices and then transmitted to cloud-based servers for further processing and ultimate storage. In a similar manner, geospatial and hydrospatial information can be made available to connected mobile devices via the cloud, allowing access to vast databases of geospatial and hydrospatial information anywhere where a wireless data connection is available. </p> <div class="mw-heading mw-heading3"><h3 id="Geographic_information_science_and_spatial_analysis">Geographic information science and spatial analysis <span class="anchor" id="GIS"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_analysis&action=edit&section=41" title="Edit section: Geographic information science and spatial 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/Geographic_information_systems#Spatial_analysis" class="mw-redirect" title="Geographic information systems">Geographic information systems § Spatial analysis</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Minard.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Minard.png/300px-Minard.png" decoding="async" width="300" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Minard.png/450px-Minard.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Minard.png/600px-Minard.png 2x" data-file-width="2003" data-file-height="955" /></a><figcaption>This flow map of Napoleon's ill-fated march on Moscow is an early and celebrated example of geovisualization. It shows the army's direction as it traveled, the places the troops passed through, the size of the army as troops died from hunger and wounds, and the freezing temperatures they experienced.</figcaption></figure> <p><a href="/wiki/Geographic_information_systems" class="mw-redirect" title="Geographic information systems">Geographic information systems</a> (GIS) and the underlying <a href="/wiki/Geographic_information_science" title="Geographic information science">geographic information science</a> that advances these technologies have a strong influence on spatial analysis. The increasing ability to capture and handle geographic data means that spatial analysis is occurring within increasingly data-rich environments. Geographic data capture systems include remotely sensed imagery, <a href="/wiki/Environmental_monitoring" title="Environmental monitoring">environmental monitoring</a> systems such as intelligent transportation systems, and location-aware technologies such as mobile devices that can report location in near-real time. GIS provide platforms for managing these data, computing spatial relationships such as distance, connectivity and directional relationships between spatial units, and visualizing both the raw data and spatial analytic results within a cartographic context. Subtypes include: </p> <ul><li><a href="/wiki/Geovisualization" title="Geovisualization">Geovisualization</a> (GVis) combines scientific visualization with <a href="/wiki/Digital_cartography" class="mw-redirect" title="Digital cartography">digital cartography</a> to support the exploration and analysis of geographic data and information, including the results of spatial analysis or simulation. GVis leverages the human orientation towards visual information processing in the exploration, analysis and communication of geographic data and information. In contrast with traditional cartography, GVis is typically three- or four-dimensional (the latter including time) and user-interactive.</li> <li><a href="/w/index.php?title=Geographic_knowledge_discovery&action=edit&redlink=1" class="new" title="Geographic knowledge discovery (page does not exist)">Geographic knowledge discovery</a> (GKD) is the human-centered process of applying efficient computational tools for exploring massive <a href="/wiki/Spatial_database" title="Spatial database">spatial databases</a>. GKD includes geographic <a href="/wiki/Data_mining" title="Data mining">data mining</a>, but also encompasses related activities such as data selection, data cleaning and pre-processing, and interpretation of results. GVis can also serve a central role in the GKD process. GKD is based on the premise that massive databases contain interesting (valid, novel, useful and understandable) patterns that standard analytical techniques cannot find. GKD can serve as a hypothesis-generating process for spatial analysis, producing tentative patterns and relationships that should be confirmed using spatial analytical techniques.</li> <li><a href="/wiki/Spatial_decision_support_system" title="Spatial decision support system">Spatial decision support systems</a> (SDSS) take existing spatial data and use a variety of mathematical models to make projections into the future. This allows urban and regional planners to test intervention decisions prior to implementation.<sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup></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=Spatial_analysis&action=edit&section=42" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>General topics</dt></dl> <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: 25em;"> <ul><li><a href="/wiki/Buffer_analysis" title="Buffer analysis">Buffer analysis</a></li> <li><a href="/wiki/Cartography" title="Cartography">Cartography</a></li> <li><a href="/wiki/Complete_spatial_randomness" title="Complete spatial randomness">Complete spatial randomness</a></li> <li><a href="/wiki/Concepts_and_Techniques_in_Modern_Geography" title="Concepts and Techniques in Modern Geography">Concepts and Techniques in Modern Geography</a></li> <li><a href="/wiki/Cost_distance_analysis" title="Cost distance analysis">Cost distance analysis</a></li> <li><a href="/wiki/Four_traditions_of_geography" title="Four traditions of geography">Four traditions of geography</a></li> <li><a href="/wiki/GeoComputation" class="mw-redirect" title="GeoComputation">GeoComputation</a></li> <li><a href="/wiki/Geospatial_intelligence" title="Geospatial intelligence">Geospatial intelligence</a></li> <li><a href="/wiki/Geospatial_predictive_modeling" class="mw-redirect" title="Geospatial predictive modeling">Geospatial predictive modeling</a></li> <li><a href="/wiki/DE-9IM" title="DE-9IM">Dimensionally Extended nine-Intersection Model</a> (DE-9IM)</li> <li><a href="/wiki/Geographic_information_science" title="Geographic information science">Geographic information science</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Modifiable_areal_unit_problem" title="Modifiable areal unit problem">Modifiable areal unit problem</a></li> <li><a href="/wiki/Modifiable_temporal_unit_problem" title="Modifiable temporal unit problem">Modifiable temporal unit problem</a></li> <li><a href="/wiki/Neighborhood_effect_averaging_problem" title="Neighborhood effect averaging problem">Neighborhood effect averaging problem</a></li> <li><a href="/wiki/Point_process" title="Point process">Point process</a></li> <li><a href="/wiki/Proximity_analysis" title="Proximity analysis">Proximity analysis</a></li> <li><a href="/wiki/Spatial_descriptive_statistics" title="Spatial descriptive statistics">Spatial descriptive statistics</a></li> <li><a href="/wiki/Spatial_relation" title="Spatial relation">Spatial relation</a></li> <li><a href="/wiki/Technical_geography" title="Technical geography">Technical geography</a></li> <li><a href="/wiki/Terrain_analysis" class="mw-redirect" title="Terrain analysis">Terrain analysis</a></li> <li><a href="/wiki/Tobler%27s_first_law_of_geography" title="Tobler's first law of geography">Tobler's first law of geography</a></li> <li><a href="/wiki/Tobler%27s_second_law_of_geography" title="Tobler's second law of geography">Tobler's second law of geography</a></li> <li><a href="/wiki/List_of_spatial_analysis_software" title="List of spatial analysis software">List of spatial analysis software</a></li></ul> </div> <dl><dt>Specific applications</dt></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 25em;"> <ul><li><a href="/wiki/Boundary_problem_(in_spatial_analysis)" class="mw-redirect" title="Boundary problem (in spatial analysis)">Boundary problem (in spatial analysis)</a></li> <li><a href="/wiki/Extrapolation_domain_analysis" title="Extrapolation domain analysis">Extrapolation domain analysis</a></li> <li><a href="/wiki/Fuzzy_architectural_spatial_analysis" title="Fuzzy architectural spatial analysis">Fuzzy architectural spatial analysis</a></li> <li><a href="/wiki/Geodemographic_segmentation" title="Geodemographic segmentation">Geodemographic segmentation</a></li> <li><a href="/wiki/Geographic_information_systems" class="mw-redirect" title="Geographic information systems">Geographic information systems</a></li> <li><a href="/wiki/Geoinformatics" title="Geoinformatics">Geoinformatics</a></li> <li><a href="/wiki/Geostatistics" title="Geostatistics">Geostatistics</a></li> <li><a href="/wiki/Permeability_(spatial_and_transport_planning)" title="Permeability (spatial and transport planning)">Permeability (spatial and transport planning)</a></li> <li><a href="/wiki/Spatial_econometrics" title="Spatial econometrics">Spatial econometrics</a></li> <li><a href="/wiki/Spatial_epidemiology" title="Spatial epidemiology">Spatial epidemiology</a></li> <li><a href="/wiki/Suitability_analysis" title="Suitability analysis">Suitability analysis</a></li> <li><a href="/wiki/Viewshed_analysis" title="Viewshed analysis">Viewshed analysis</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=Spatial_analysis&action=edit&section=43" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The History of Land Surveying. Accessed Dec 17 2020. <a rel="nofollow" class="external free" href="https://info.courthousedirect.com/blog/history-of-land-surveying">https://info.courthousedirect.com/blog/history-of-land-surveying</a></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">Mark Monmonier <i>How to Lie with Maps</i> University of Chicago Press, 1996.</span> </li> <li id="cite_note-Modifiable_areal_unit_problem_Openshaw1-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Modifiable_areal_unit_problem_Openshaw1_3-0">^</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="CITEREFOpenshaw1983" class="citation book cs1">Openshaw, Stan (1983). <a rel="nofollow" class="external text" href="https://alexsingleton.files.wordpress.com/2014/09/38-maup-openshaw.pdf"><i>The Modifiable Areal Unit Problem</i></a> <span class="cs1-format">(PDF)</span>. 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THE CHRONICLE REVIEW. The Chronicle of Higher Education<span class="reference-accessdate">. Retrieved <span class="nowrap">7 October</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Neighborhood+Effect&rft.date=2012-11-05&rft.aulast=Parry&rft.aufirst=Marc&rft_id=https%3A%2F%2Fwww.chronicle.com%2Farticle%2Fthe-neighborhood-effect%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+analysis" class="Z3988"></span></span> </li> <li id="cite_note-Travelling_salesman_problem_Labbe2004-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Travelling_salesman_problem_Labbe2004_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLabbéLaporteMartínGonzález2004" class="citation journal cs1">Labbé, Martine; Laporte, Gilbert; Martín, Inmaculada Rodríguez; González, Juan José Salazar (May 2004). 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Retrieved 5tth August 2012 from CollinsDictionary.com website: <a rel="nofollow" class="external free" href="https://www.collinsdictionary.com/dictionary/english/geospatial">http://www.collinsdictionary.com/dictionary/english/geospatial</a></span> </li> <li id="cite_note-72"><span class="mw-cite-backlink"><b><a href="#cite_ref-72">^</a></b></span> <span class="reference-text">Dictionary.com's 21st Century Lexicon Copyright © 2003-2010 Dictionary.com, LLC <a rel="nofollow" class="external free" href="http://dictionary.reference.com/browse/geospatial">http://dictionary.reference.com/browse/geospatial</a></span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.receiver.vodafone.com/the-geospatial-web">The geospatial web – blending physical and virtual spaces.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111002151826/http://www.receiver.vodafone.com/the-geospatial-web">Archived</a> 2011-10-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Arno Scharl in receiver magazine, Autumn 2008</span> </li> <li id="cite_note-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-74">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChenGuinness2014" class="citation book cs1">Chen, Ruizhi; Guinness, Robert E. 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Adams, and P. Gould (1971) <i>Spatial Organization–The Geographer's View of the World</i>, Englewood Cliffs, NJ: Prentice-Hall.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110919052807/http://www.drs.wisc.edu/documents/articles/curtis/cesoc977/Anselin1995.pdf">Anselin, L. (1995) "Local indicators of spatial association – LISA". <i>Geographical Analysis</i>, 27, 93–115</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAwange,_JosephPaláncz,_Béla2016" class="citation book cs1">Awange, Joseph; Paláncz, Béla (2016). <i>Geospatial Algebraic Computations, Theory and Applications, Third Edition</i>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3319254630" title="Special:BookSources/978-3319254630"><bdi>978-3319254630</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geospatial+Algebraic+Computations%2C+Theory+and+Applications%2C+Third+Edition&rft.place=New+York&rft.pub=Springer&rft.date=2016&rft.isbn=978-3319254630&rft.au=Awange%2C+Joseph&rft.au=Pal%C3%A1ncz%2C+B%C3%A9la&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBanerjeeCarlinGelfand2014" class="citation cs2">Banerjee, Sudipto; Carlin, Bradley P.; Gelfand, Alan E. (2014), <i>Hierarchical Modeling and Analysis for Spatial Data, Second Edition</i>, Monographs on Statistics and Applied Probability (2nd ed.), Chapman and Hall/CRC, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781439819173" title="Special:BookSources/9781439819173"><bdi>9781439819173</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hierarchical+Modeling+and+Analysis+for+Spatial+Data%2C+Second+Edition&rft.series=Monographs+on+Statistics+and+Applied+Probability&rft.edition=2nd&rft.pub=Chapman+and+Hall%2FCRC&rft.date=2014&rft.isbn=9781439819173&rft.aulast=Banerjee&rft.aufirst=Sudipto&rft.au=Carlin%2C+Bradley+P.&rft.au=Gelfand%2C+Alan+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+analysis" class="Z3988"></span></li> <li>Benenson, I. and P. M. Torrens. (2004). <i>Geosimulation: Automata-Based Modeling of Urban Phenomena.</i> Wiley.</li> <li>Fotheringham, A. S., C. Brunsdon and M. Charlton (2000) <i>Quantitative Geography: Perspectives on Spatial Data Analysis</i>, Sage.</li> <li>Fotheringham, A. S. and M. E. O'Kelly (1989) <i>Spatial Interaction Models: Formulations and Applications</i>, Kluwer Academic</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFotheringhamRogerson1993" class="citation journal cs1">Fotheringham, A. S.; Rogerson, P. A. (1993). 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LAP LAMBERT Academic Publishing. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.13140%2F2.1.1560.2247">10.13140/2.1.1560.2247</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=European+Spatial+Structure&rft.pub=LAP+LAMBERT+Academic+Publishing&rft.date=2014&rft_id=info%3Adoi%2F10.13140%2F2.1.1560.2247&rft.aulast=Geza&rft.aufirst=T%C3%B3th&rft.au=%C3%81ron%2C+Kincses&rft.au=Zolt%C3%A1n%2C+Nagy&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+analysis" 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=Spatial_analysis&action=edit&section=45" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box metadata side-box-right"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-abovebelow"> <a href="/wiki/Wikipedia:The_Wikipedia_Library" title="Wikipedia:The Wikipedia Library">Library resources</a> about <br /> <b>Spatial analysis</b> <hr /></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Spatial+analysis">Resources in your library</a></li> <li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Spatial+analysis&library=0CHOOSE0">Resources in other libraries</a></li> </ul></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media 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rel="nofollow" class="external text" href="http://www.ncgia.ucsb.edu/">National Center for Geographic Information and Analysis (NCGIA)</a></li> <li><a rel="nofollow" class="external text" href="http://www.icaci.org">International Cartographic Association (ICA)</a> – the world body for mapping and GIScience professionals</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 ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output 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(graphics)">Educational visualization</a></li> <li><a href="/wiki/Flow_visualization" title="Flow visualization">Flow visualization</a></li> <li><a href="/wiki/Geovisualization" title="Geovisualization">Geovisualization</a></li> <li><a href="/wiki/Information_visualization" class="mw-redirect" title="Information visualization">Information visualization</a></li> <li><a href="/wiki/Mathematical_diagram" title="Mathematical diagram">Mathematical visualization</a></li> <li><a href="/wiki/Medical_imaging" title="Medical imaging">Medical imaging</a></li> <li><a href="/wiki/Molecular_graphics" title="Molecular graphics">Molecular graphics</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Product visualization</a></li> <li><a href="/wiki/Scientific_visualization" title="Scientific visualization">Scientific visualization</a></li> <li><a href="/wiki/Social_visualization" title="Social visualization">Social visualization</a></li> <li><a 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function</a></li> <li><a href="/wiki/Ideogram" title="Ideogram">Ideogram</a></li> <li><a href="/wiki/Map" title="Map">Map</a></li> <li><a href="/wiki/Photograph" title="Photograph">Photograph</a></li> <li><a href="/wiki/Pictogram" title="Pictogram">Pictogram</a></li> <li><a href="/wiki/Plot_(graphics)" title="Plot (graphics)">Plot</a></li> <li><a href="/wiki/Sankey_diagram" title="Sankey diagram">Sankey diagram</a></li> <li><a href="/wiki/Schematic" title="Schematic">Schematic</a></li> <li><a href="/wiki/Skeletal_formula" title="Skeletal formula">Skeletal formula</a></li> <li><a href="/wiki/Statistical_graphics" title="Statistical graphics">Statistical graphics</a></li> <li><a href="/wiki/Table_(information)" title="Table (information)">Table</a></li> <li><a href="/wiki/Technical_drawing" title="Technical drawing">Technical drawings</a></li> <li><a href="/wiki/Technical_illustration" title="Technical illustration">Technical illustration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" 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:1%">Pre-19th century</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/Edmond_Halley" title="Edmond Halley">Edmond Halley</a></li> <li><a href="/wiki/Charles-Ren%C3%A9_de_Fourcroy" title="Charles-René de Fourcroy">Charles-René de Fourcroy</a></li> <li><a href="/wiki/Joseph_Priestley" title="Joseph Priestley">Joseph Priestley</a></li> <li><a href="/wiki/Gaspard_Monge" title="Gaspard Monge">Gaspard Monge</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">19th century</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/Charles_Dupin" title="Charles Dupin">Charles Dupin</a></li> <li><a href="/wiki/Adolphe_Quetelet" title="Adolphe Quetelet">Adolphe Quetelet</a></li> <li><a href="/wiki/Andr%C3%A9-Michel_Guerry" title="André-Michel Guerry">André-Michel Guerry</a></li> <li><a href="/wiki/William_Playfair" title="William Playfair">William Playfair</a></li> <li><a href="/wiki/August_Kekul%C3%A9" title="August Kekulé">August Kekulé</a></li> <li><a href="/wiki/Charles_Joseph_Minard" title="Charles Joseph Minard">Charles Joseph Minard</a></li> <li><a href="/wiki/Francis_Amasa_Walker" title="Francis Amasa Walker">Francis Amasa Walker</a></li> <li><a href="/wiki/John_Venn" title="John Venn">John Venn</a></li> <li><a href="/wiki/Oliver_Byrne_(mathematician)" title="Oliver Byrne (mathematician)">Oliver Byrne</a></li> <li><a href="/wiki/Matthew_Henry_Phineas_Riall_Sankey" title="Matthew Henry Phineas Riall Sankey">Matthew Sankey</a></li> <li><a href="/wiki/Charles_Booth_(social_reformer)" title="Charles Booth (social reformer)">Charles Booth</a></li> <li><a href="/wiki/John_Snow" title="John Snow">John Snow</a></li> <li><a href="/wiki/Florence_Nightingale" title="Florence Nightingale">Florence Nightingale</a></li> <li><a href="/wiki/Karl_Wilhelm_Pohlke" title="Karl Wilhelm Pohlke">Karl Wilhelm Pohlke</a></li> <li><a href="/wiki/Toussaint_Loua" title="Toussaint Loua">Toussaint Loua</a></li> <li><a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Early 20th century</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/Edward_Walter_Maunder" title="Edward Walter Maunder">Edward Walter Maunder</a></li> <li><a href="/wiki/Otto_Neurath" title="Otto Neurath">Otto Neurath</a></li> <li><a href="/wiki/W._E._B._Du_Bois" title="W. E. B. Du Bois">W. E. B. Du Bois</a></li> <li><a href="/wiki/Henry_Gantt" title="Henry Gantt">Henry Gantt</a></li> <li><a href="/wiki/Arthur_Lyon_Bowley" title="Arthur Lyon Bowley">Arthur Lyon Bowley</a></li> <li><a href="/wiki/Howard_G._Funkhouser" title="Howard G. Funkhouser">Howard G. Funkhouser</a></li> <li><a href="/wiki/John_B._Peddle" title="John B. Peddle">John B. Peddle</a></li> <li><a href="/wiki/Ejnar_Hertzsprung" title="Ejnar Hertzsprung">Ejnar Hertzsprung</a></li> <li><a href="/wiki/Henry_Norris_Russell" title="Henry Norris Russell">Henry Norris Russell</a></li> <li><a href="/wiki/Max_O._Lorenz" title="Max O. Lorenz">Max O. Lorenz</a></li> <li><a href="/wiki/Fritz_Kahn" title="Fritz Kahn">Fritz Kahn</a></li> <li><a href="/wiki/Harry_Beck" title="Harry Beck">Harry Beck</a></li> <li><a href="/wiki/Erwin_Raisz" title="Erwin Raisz">Erwin Raisz</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mid 20th century</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/Jacques_Bertin" title="Jacques Bertin">Jacques Bertin</a></li> <li><a href="/wiki/Rudolf_Modley" title="Rudolf Modley">Rudolf Modley</a></li> <li><a href="/wiki/Arthur_H._Robinson" title="Arthur H. Robinson">Arthur H. Robinson</a></li> <li><a href="/wiki/John_Tukey" title="John Tukey">John Tukey</a></li> <li><a href="/wiki/Mary_Eleanor_Spear" title="Mary Eleanor Spear">Mary Eleanor Spear</a></li> <li><a href="/wiki/Edgar_Anderson" title="Edgar Anderson">Edgar Anderson</a></li> <li><a href="/wiki/Howard_T._Fisher" title="Howard T. Fisher">Howard T. Fisher</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Late 20th century</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/Borden_Dent" title="Borden Dent">Borden Dent</a></li> <li><a href="/wiki/Nigel_Holmes" title="Nigel Holmes">Nigel Holmes</a></li> <li><a href="/wiki/William_S._Cleveland" title="William S. Cleveland">William S. Cleveland</a></li> <li><a href="/wiki/George_G._Robertson" title="George G. Robertson">George G. Robertson</a></li> <li><a href="/wiki/Bruce_H._McCormick" title="Bruce H. McCormick">Bruce H. McCormick</a></li> <li><a href="/wiki/Catherine_Plaisant" title="Catherine Plaisant">Catherine Plaisant</a></li> <li><a href="/wiki/Stuart_Card" title="Stuart Card">Stuart Card</a></li> <li><a href="/wiki/Pat_Hanrahan" title="Pat Hanrahan">Pat Hanrahan</a></li> <li><a href="/wiki/Edward_Tufte" title="Edward Tufte">Edward Tufte</a></li> <li><a href="/wiki/Ben_Shneiderman" title="Ben Shneiderman">Ben Shneiderman</a></li> <li><a href="/wiki/Michael_Friendly" title="Michael Friendly">Michael Friendly</a></li> <li><a href="/wiki/Howard_Wainer" title="Howard Wainer">Howard Wainer</a></li> <li><a href="/wiki/Clifford_A._Pickover" title="Clifford A. Pickover">Clifford A. Pickover</a></li> <li><a href="/wiki/Lawrence_J._Rosenblum" title="Lawrence J. Rosenblum">Lawrence J. Rosenblum</a></li> <li><a href="/wiki/Thomas_A._DeFanti" title="Thomas A. DeFanti">Thomas A. DeFanti</a></li> <li><a href="/wiki/George_Furnas" title="George Furnas">George Furnas</a></li> <li><a href="/wiki/Sheelagh_Carpendale" title="Sheelagh Carpendale">Sheelagh Carpendale</a></li> <li><a href="/wiki/Cynthia_Brewer" title="Cynthia Brewer">Cynthia Brewer</a></li> <li><a href="/wiki/Jock_D._Mackinlay" title="Jock D. Mackinlay">Jock D. 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Johnson">Christopher R. 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