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id="toc-Estimate_of_Positive_Correctness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gini_impurity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gini_impurity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Gini impurity</span> </div> </a> <ul id="toc-Gini_impurity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Information_gain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Information_gain"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Information gain</span> </div> </a> <ul id="toc-Information_gain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variance_reduction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variance_reduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Variance reduction</span> </div> </a> <ul id="toc-Variance_reduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measure_of_"goodness"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measure_of_"goodness""> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Measure of "goodness"</span> </div> </a> <ul id="toc-Measure_of_"goodness"-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Uses" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Uses"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Uses</span> </div> </a> <button aria-controls="toc-Uses-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 Uses subsection</span> </button> <ul id="toc-Uses-sublist" class="vector-toc-list"> <li id="toc-Advantages" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Advantages"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Advantages</span> </div> </a> <ul id="toc-Advantages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limitations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Limitations</span> </div> </a> <ul id="toc-Limitations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Implementations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Implementations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Implementations</span> </div> </a> <ul id="toc-Implementations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Extensions</span> </div> </a> <button aria-controls="toc-Extensions-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 Extensions subsection</span> </button> <ul id="toc-Extensions-sublist" class="vector-toc-list"> <li id="toc-Decision_graphs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decision_graphs"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Decision graphs</span> </div> </a> <ul id="toc-Decision_graphs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_search_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_search_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Alternative search methods</span> </div> </a> <ul id="toc-Alternative_search_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Decision tree learning</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" 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Available in 19 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-19" 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">19 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%B9%D9%84%D9%85_%D8%B4%D8%AC%D8%B1%D8%A9_%D8%A7%D9%84%D9%82%D8%B1%D8%A7%D8%B1" title="تعلم شجرة القرار – Arabic" lang="ar" hreflang="ar" data-title="تعلم شجرة القرار" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%BF%E0%A6%B8%E0%A6%BF%E0%A6%B6%E0%A6%A8_%E0%A6%9F%E0%A7%8D%E0%A6%B0%E0%A6%BF_%E0%A6%B2%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A8%E0%A6%BF%E0%A6%82" title="ডিসিশন ট্রি লার্নিং – Bangla" lang="bn" hreflang="bn" data-title="ডিসিশন ট্রি লার্নিং" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Aprenentatge_basat_en_arbres_de_decisi%C3%B3" title="Aprenentatge basat en arbres de decisió – Catalan" lang="ca" hreflang="ca" data-title="Aprenentatge basat en arbres de decisió" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Aprendizaje_basado_en_%C3%A1rboles_de_decisi%C3%B3n" title="Aprendizaje basado en árboles de decisión – Spanish" lang="es" hreflang="es" data-title="Aprendizaje basado en árboles de decisión" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erabaki-zuhaitzen_bidezko_ikaskuntza" title="Erabaki-zuhaitzen bidezko ikaskuntza – Basque" lang="eu" hreflang="eu" data-title="Erabaki-zuhaitzen bidezko ikaskuntza" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DB%8C%D8%A7%D8%AF%DA%AF%DB%8C%D8%B1%DB%8C_%D8%AF%D8%B1%D8%AE%D8%AA_%D8%AA%D8%B5%D9%85%DB%8C%D9%85" title="یادگیری درخت تصمیم – Persian" lang="fa" hreflang="fa" data-title="یادگیری درخت تصمیم" 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/Arbre_de_d%C3%A9cision_(apprentissage)" title="Arbre de décision (apprentissage) – French" lang="fr" hreflang="fr" data-title="Arbre de décision (apprentissage)" 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%B2%B0%EC%A0%95_%ED%8A%B8%EB%A6%AC_%ED%95%99%EC%8A%B5%EB%B2%95" title="결정 트리 학습법 – Korean" lang="ko" hreflang="ko" data-title="결정 트리 학습법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he badge-Q70894304 mw-list-item" title=""><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A5_%D7%94%D7%97%D7%9C%D7%98%D7%94_%D7%9C%D7%95%D7%9E%D7%93" title="עץ החלטה לומד – Hebrew" lang="he" hreflang="he" data-title="עץ החלטה לומד" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/CART_algoritms" title="CART algoritms – Latvian" lang="lv" hreflang="lv" data-title="CART algoritms" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Sprendim%C5%B3_med%C5%BEi%C5%B3_mokymas" title="Sprendimų medžių mokymas – Lithuanian" lang="lt" hreflang="lt" data-title="Sprendimų medžių mokymas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%A1%E0%AC%BF%E0%AC%B8%E0%AC%BF%E0%AC%B8%E0%AC%A8_%E0%AC%9F%E0%AD%8D%E0%AC%B0%E0%AC%BF_%E0%AC%B2%E0%AC%B0%E0%AD%8D%E0%AC%A3%E0%AC%BF%E0%AC%82" title="ଡିସିସନ ଟ୍ରି ଲର୍ଣିଂ – Odia" lang="or" hreflang="or" data-title="ଡିସିସନ ଟ୍ରି ଲର୍ଣିଂ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Drzewa_klasyfikacyjne" title="Drzewa klasyfikacyjne – Polish" lang="pl" hreflang="pl" data-title="Drzewa klasyfikacyjne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Aprendizagem_de_%C3%A1rvore_de_decis%C3%A3o" title="Aprendizagem de árvore de decisão – Portuguese" lang="pt" hreflang="pt" data-title="Aprendizagem de árvore de decisão" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%83%D1%87%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B4%D0%B5%D1%80%D0%B5%D0%B2%D0%B0_%D1%80%D0%B5%D1%88%D0%B5%D0%BD%D0%B8%D0%B9" title="Обучение дерева решений – Russian" lang="ru" hreflang="ru" data-title="Обучение дерева решений" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%DB%8E%D8%B1%D8%A8%D9%88%D9%88%D9%86%DB%8C_%D8%AF%D8%B1%DB%95%D8%AE%D8%AA%DB%8C_%D8%A8%DA%95%DB%8C%D8%A7%D8%B1" title="فێربوونی درەختی بڕیار – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فێربوونی درەختی بڕیار" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B5%D1%80%D0%B5%D0%B2%D0%B0_%D1%80%D1%96%D1%88%D0%B5%D0%BD%D1%8C_%D1%83_%D0%BC%D0%B0%D1%88%D0%B8%D0%BD%D0%BD%D0%BE%D0%BC%D1%83_%D0%BD%D0%B0%D0%B2%D1%87%D0%B0%D0%BD%D0%BD%D1%96" title="Дерева рішень у машинному навчанні – Ukrainian" lang="uk" hreflang="uk" data-title="Дерева рішень у машинному навчанні" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B1%BA%E7%AD%96%E6%A8%B9%E5%AD%B8%E7%BF%92" title="決策樹學習 – Cantonese" lang="yue" hreflang="yue" data-title="決策樹學習" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%86%B3%E7%AD%96%E6%A0%91%E5%AD%A6%E4%B9%A0" title="决策树学习 – Chinese" lang="zh" hreflang="zh" data-title="决策树学习" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q16766476#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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text-align:center;;color: var(--color-base)">Paradigms</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">Semi-supervised learning</a></li> <li><a href="/wiki/Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Meta-learning_(computer_science)" title="Meta-learning (computer science)">Meta-learning</a></li> <li><a href="/wiki/Online_machine_learning" title="Online machine learning">Online learning</a></li> <li><a href="/wiki/Batch_learning" class="mw-redirect" title="Batch learning">Batch learning</a></li> <li><a href="/wiki/Curriculum_learning" title="Curriculum learning">Curriculum learning</a></li> <li><a href="/wiki/Rule-based_machine_learning" title="Rule-based machine learning">Rule-based learning</a></li> <li><a href="/wiki/Neuro-symbolic_AI" title="Neuro-symbolic AI">Neuro-symbolic AI</a></li> <li><a href="/wiki/Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">Neuromorphic engineering</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Problems</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Generative_model" title="Generative model">Generative modeling</a></li> <li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></li> <li><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></li> <li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li> <li><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></li> <li><a href="/wiki/Data_cleaning" class="mw-redirect" title="Data cleaning">Data cleaning</a></li> <li><a href="/wiki/Automated_machine_learning" title="Automated machine learning">AutoML</a></li> <li><a href="/wiki/Association_rule_learning" title="Association rule learning">Association rules</a></li> <li><a href="/wiki/Semantic_analysis_(machine_learning)" title="Semantic analysis (machine learning)">Semantic analysis</a></li> <li><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></li> <li><a href="/wiki/Feature_engineering" title="Feature engineering">Feature engineering</a></li> <li><a href="/wiki/Feature_learning" title="Feature learning">Feature learning</a></li> <li><a href="/wiki/Learning_to_rank" title="Learning to rank">Learning to rank</a></li> <li><a href="/wiki/Grammar_induction" title="Grammar induction">Grammar induction</a></li> <li><a href="/wiki/Ontology_learning" title="Ontology learning">Ontology learning</a></li> <li><a href="/wiki/Multimodal_learning" title="Multimodal learning">Multimodal learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a><br /><span class="nobold"><span style="font-size:85%;">(<b><a href="/wiki/Statistical_classification" title="Statistical classification">classification</a></b> • <b><a href="/wiki/Regression_analysis" title="Regression analysis">regression</a></b>)</span></span> </div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Apprenticeship_learning" title="Apprenticeship learning">Apprenticeship learning</a></li> <li><a class="mw-selflink selflink">Decision trees</a></li> <li><a href="/wiki/Ensemble_learning" title="Ensemble learning">Ensembles</a> <ul><li><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bagging</a></li> <li><a href="/wiki/Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a></li> <li><a href="/wiki/Random_forest" title="Random forest">Random forest</a></li></ul></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes</a></li> <li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural networks</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Perceptron" title="Perceptron">Perceptron</a></li> <li><a href="/wiki/Relevance_vector_machine" title="Relevance vector machine">Relevance vector machine (RVM)</a></li> <li><a href="/wiki/Support_vector_machine" title="Support vector machine">Support vector machine (SVM)</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/BIRCH" title="BIRCH">BIRCH</a></li> <li><a href="/wiki/CURE_algorithm" title="CURE algorithm">CURE</a></li> <li><a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">Hierarchical</a></li> <li><a href="/wiki/K-means_clustering" title="K-means clustering"><i>k</i>-means</a></li> <li><a href="/wiki/Fuzzy_clustering" title="Fuzzy clustering">Fuzzy</a></li> <li><a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">Expectation–maximization (EM)</a></li> <li><br /><a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a></li> <li><a href="/wiki/OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a></li> <li><a href="/wiki/Mean_shift" title="Mean shift">Mean shift</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">CCA</a></li> <li><a href="/wiki/Independent_component_analysis" title="Independent component analysis">ICA</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">LDA</a></li> <li><a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">NMF</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">PCA</a></li> <li><a href="/wiki/Proper_generalized_decomposition" title="Proper generalized decomposition">PGD</a></li> <li><a href="/wiki/T-distributed_stochastic_neighbor_embedding" title="T-distributed stochastic neighbor embedding">t-SNE</a></li> <li><a href="/wiki/Sparse_dictionary_learning" title="Sparse dictionary learning">SDL</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Graphical_model" title="Graphical model">Graphical models</a> <ul><li><a href="/wiki/Bayesian_network" title="Bayesian network">Bayes net</a></li> <li><a href="/wiki/Conditional_random_field" title="Conditional random field">Conditional random field</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Random_sample_consensus" title="Random sample consensus">RANSAC</a></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a href="/wiki/Local_outlier_factor" title="Local outlier factor">Local outlier factor</a></li> <li><a href="/wiki/Isolation_forest" title="Isolation forest">Isolation forest</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural network</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Autoencoder" title="Autoencoder">Autoencoder</a></li> <li><a href="/wiki/Deep_learning" title="Deep learning">Deep learning</a></li> <li><a href="/wiki/Feedforward_neural_network" title="Feedforward neural network">Feedforward neural network</a></li> <li><a href="/wiki/Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network</a> <ul><li><a href="/wiki/Long_short-term_memory" title="Long short-term memory">LSTM</a></li> <li><a href="/wiki/Gated_recurrent_unit" title="Gated recurrent unit">GRU</a></li> <li><a href="/wiki/Echo_state_network" title="Echo state network">ESN</a></li> <li><a href="/wiki/Reservoir_computing" title="Reservoir computing">reservoir computing</a></li></ul></li> <li><a href="/wiki/Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a> <ul><li><a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted</a></li></ul></li> <li><a href="/wiki/Generative_adversarial_network" title="Generative adversarial network">GAN</a></li> <li><a href="/wiki/Diffusion_model" title="Diffusion model">Diffusion model</a></li> <li><a href="/wiki/Self-organizing_map" title="Self-organizing map">SOM</a></li> <li><a href="/wiki/Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network</a> <ul><li><a href="/wiki/U-Net" title="U-Net">U-Net</a></li> <li><a href="/wiki/LeNet" title="LeNet">LeNet</a></li> <li><a href="/wiki/AlexNet" title="AlexNet">AlexNet</a></li> <li><a href="/wiki/DeepDream" title="DeepDream">DeepDream</a></li></ul></li> <li><a href="/wiki/Neural_radiance_field" title="Neural radiance field">Neural radiance field</a></li> <li><a href="/wiki/Transformer_(machine_learning_model)" class="mw-redirect" title="Transformer (machine learning model)">Transformer</a> <ul><li><a href="/wiki/Vision_transformer" title="Vision transformer">Vision</a></li></ul></li> <li><a href="/wiki/Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li> <li><a href="/wiki/Spiking_neural_network" title="Spiking neural network">Spiking neural network</a></li> <li><a href="/wiki/Memtransistor" title="Memtransistor">Memtransistor</a></li> <li><a href="/wiki/Electrochemical_RAM" title="Electrochemical RAM">Electrochemical RAM</a> (ECRAM)</li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Q-learning" title="Q-learning">Q-learning</a></li> <li><a href="/wiki/State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li> <li><a href="/wiki/Temporal_difference_learning" title="Temporal difference learning">Temporal difference (TD)</a></li> <li><a href="/wiki/Multi-agent_reinforcement_learning" title="Multi-agent reinforcement learning">Multi-agent</a> <ul><li><a href="/wiki/Self-play_(reinforcement_learning_technique)" class="mw-redirect" title="Self-play (reinforcement learning technique)">Self-play</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Learning with humans</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Active_learning_(machine_learning)" title="Active learning (machine learning)">Active learning</a></li> <li><a href="/wiki/Crowdsourcing" title="Crowdsourcing">Crowdsourcing</a></li> <li><a href="/wiki/Human-in-the-loop" title="Human-in-the-loop">Human-in-the-loop</a></li> <li><a href="/wiki/Reinforcement_learning_from_human_feedback" title="Reinforcement learning from human feedback">RLHF</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Model diagnostics</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li> <li><a href="/wiki/Learning_curve_(machine_learning)" title="Learning curve (machine learning)">Learning curve</a></li> <li><a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">ROC curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Mathematical foundations</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Kernel_machines" class="mw-redirect" title="Kernel machines">Kernel machines</a></li> <li><a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li> <li><a href="/wiki/Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li> <li><a href="/wiki/Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></li> <li><a href="/wiki/Occam_learning" title="Occam learning">Occam learning</a></li> <li><a href="/wiki/Probably_approximately_correct_learning" title="Probably approximately correct learning">PAC learning</a></li> <li><a href="/wiki/Statistical_learning_theory" title="Statistical learning theory">Statistical learning</a></li> <li><a href="/wiki/Vapnik%E2%80%93Chervonenkis_theory" title="Vapnik–Chervonenkis theory">VC theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Journals and conferences</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/ECML_PKDD" title="ECML PKDD">ECML PKDD</a></li> <li><a href="/wiki/Conference_on_Neural_Information_Processing_Systems" title="Conference on Neural Information Processing Systems">NeurIPS</a></li> <li><a href="/wiki/International_Conference_on_Machine_Learning" title="International Conference on Machine Learning">ICML</a></li> <li><a href="/wiki/International_Conference_on_Learning_Representations" title="International Conference on Learning Representations">ICLR</a></li> <li><a href="/wiki/International_Joint_Conference_on_Artificial_Intelligence" title="International Joint Conference on Artificial Intelligence">IJCAI</a></li> <li><a href="/wiki/Machine_Learning_(journal)" title="Machine Learning (journal)">ML</a></li> <li><a href="/wiki/Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">JMLR</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Related articles</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Glossary of artificial intelligence</a></li> <li><a href="/wiki/List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a> <ul><li><a href="/wiki/List_of_datasets_in_computer_vision_and_image_processing" title="List of datasets in computer vision and image processing">List of datasets in computer vision and image processing</a></li></ul></li> <li><a href="/wiki/Outline_of_machine_learning" title="Outline of machine learning">Outline of machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Machine_learning" title="Template:Machine learning"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Machine_learning" title="Template talk:Machine learning"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Machine_learning" title="Special:EditPage/Template:Machine learning"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Decision tree learning</b> is a <a href="/wiki/Supervised_learning" title="Supervised learning">supervised learning</a> approach used in <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Data_mining" title="Data mining">data mining</a> and <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>. In this formalism, a classification or regression <a href="/wiki/Decision_tree" title="Decision tree">decision tree</a> is used as a <a href="/wiki/Predictive_model" class="mw-redirect" title="Predictive model">predictive model</a> to draw conclusions about a set of observations. </p><p>Tree models where the target variable can take a discrete set of values are called <b><a href="/wiki/Statistical_classification" title="Statistical classification">classification</a> <a href="/wiki/Decision_tree" title="Decision tree">trees</a></b>; in these tree structures, <a href="/wiki/Leaf_node" class="mw-redirect" title="Leaf node">leaves</a> represent class labels and branches represent <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunctions</a> of features that lead to those class labels. Decision trees where the target variable can take continuous values (typically <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>) are called <b><a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> <a href="/wiki/Decision_tree" title="Decision tree">trees</a></b>. More generally, the concept of regression tree can be extended to any kind of object equipped with pairwise dissimilarities such as categorical sequences.<sup id="cite_ref-:1_1-0" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Decision trees are among the most popular machine learning algorithms given their intelligibility and simplicity.<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><p>In decision analysis, a decision tree can be used to visually and explicitly represent decisions and <a href="/wiki/Decision_making" class="mw-redirect" title="Decision making">decision making</a>. In <a href="/wiki/Data_mining" title="Data mining">data mining</a>, a decision tree describes data (but the resulting classification tree can be an input for decision making). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="General">General</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=1" title="Edit section: General"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Decision_Tree.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Decision_Tree.jpg/220px-Decision_Tree.jpg" decoding="async" width="220" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Decision_Tree.jpg/330px-Decision_Tree.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Decision_Tree.jpg/440px-Decision_Tree.jpg 2x" data-file-width="457" data-file-height="473" /></a><figcaption>A tree showing survival of passengers on the <a href="/wiki/Titanic" title="Titanic">Titanic</a> ("sibsp" is the number of spouses or siblings aboard). The figures under the leaves show the probability of survival and the percentage of observations in the leaf. Summarizing: Your chances of survival were good if you were (i) a female or (ii) a male at most 9.5 years old with strictly fewer than 3 siblings.</figcaption></figure> <p>Decision tree learning is a method commonly used in data mining.<sup id="cite_ref-tdidt_3-0" class="reference"><a href="#cite_note-tdidt-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The goal is to create a model that predicts the value of a target variable based on several input variables. </p><p>A decision tree is a simple representation for classifying examples. For this section, assume that all of the input <a href="/wiki/Feature_(machine_learning)" title="Feature (machine learning)">features</a> have finite discrete domains, and there is a single target feature called the "classification". Each element of the domain of the classification is called a <i>class</i>. A decision tree or a classification tree is a tree in which each internal (non-leaf) node is labeled with an input feature. The arcs coming from a node labeled with an input feature are labeled with each of the possible values of the target feature or the arc leads to a subordinate decision node on a different input feature. Each leaf of the tree is labeled with a class or a probability distribution over the classes, signifying that the data set has been classified by the tree into either a specific class, or into a particular probability distribution (which, if the decision tree is well-constructed, is skewed towards certain subsets of classes). </p><p>A tree is built by splitting the source <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, constituting the root node of the tree, into subsets—which constitute the successor children. The splitting is based on a set of splitting rules based on classification features.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This process is repeated on each derived subset in a recursive manner called <a href="/wiki/Recursive_partitioning" title="Recursive partitioning">recursive partitioning</a>. The <a href="/wiki/Recursion" title="Recursion">recursion</a> is completed when the subset at a node has all the same values of the target variable, or when splitting no longer adds value to the predictions. This process of <i>top-down induction of decision trees</i> (TDIDT)<sup id="cite_ref-Quinlan86_5-0" class="reference"><a href="#cite_note-Quinlan86-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> is an example of a <a href="/wiki/Greedy_algorithm" title="Greedy algorithm">greedy algorithm</a>, and it is by far the most common strategy for learning decision trees from data.<sup id="cite_ref-top-downDT_6-0" class="reference"><a href="#cite_note-top-downDT-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Data_mining" title="Data mining">data mining</a>, decision trees can be described also as the combination of mathematical and computational techniques to aid the description, categorization and generalization of a given set of data. </p><p>Data comes in records of the form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\textbf {x}},Y)=(x_{1},x_{2},x_{3},...,x_{k},Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\textbf {x}},Y)=(x_{1},x_{2},x_{3},...,x_{k},Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9354e3bfc0c65eb88a0bf7b6b625dcdbc9e74248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.55ex; height:2.843ex;" alt="{\displaystyle ({\textbf {x}},Y)=(x_{1},x_{2},x_{3},...,x_{k},Y)}"></span></dd></dl> <p>The dependent variable, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, is the target variable that we are trying to understand, classify or generalize. The vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85631435f001c884eca834164392982c621f40e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle {\textbf {x}}}"></span> is composed of the features, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},x_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},x_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4102ba3aa87d8bd353467896b23eae57f4fb06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.22ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},x_{3}}"></span> etc., that are used for that task. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cart_tree_kyphosis.png" class="mw-file-description"><img alt="Three different representations of a regression tree of kyphosis data" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Cart_tree_kyphosis.png/800px-Cart_tree_kyphosis.png" decoding="async" width="800" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/25/Cart_tree_kyphosis.png 1.5x" data-file-width="1200" data-file-height="300" /></a><figcaption> An example tree which estimates the probability of <a href="/wiki/Kyphosis" title="Kyphosis">kyphosis</a> after spinal surgery, given the age of the patient and the vertebra at which surgery was started. The same tree is shown in three different ways. <b>Left</b> The colored leaves show the probability of kyphosis after spinal surgery, and percentage of patients in the leaf. <b>Middle</b> The tree as a perspective plot. <b>Right</b> Aerial view of the middle plot. The probability of kyphosis after surgery is higher in the darker areas. (Note: The treatment of <a href="/wiki/Kyphosis" title="Kyphosis">kyphosis</a> has advanced considerably since this rather small set of data was collected.<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 2019)">citation needed</span></a></i>]</sup>) </figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Decision_tree_types">Decision tree types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=2" title="Edit section: Decision tree types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Decision trees used in <a href="/wiki/Data_mining" title="Data mining">data mining</a> are of two main types: </p> <ul><li><b><a href="/wiki/Classification_tree" class="mw-redirect" title="Classification tree">Classification tree</a></b> analysis is when the predicted outcome is the class (discrete) to which the data belongs.</li> <li><b>Regression tree</b> analysis is when the predicted outcome can be considered a real number (e.g. the price of a house, or a patient's length of stay in a hospital).</li></ul> <p>The term <b>classification and regression tree (CART)</b> analysis is an <a href="/wiki/Umbrella_term" class="mw-redirect" title="Umbrella term">umbrella term</a> used to refer to either of the above procedures, first introduced by <a href="/wiki/Leo_Breiman" title="Leo Breiman">Breiman</a> et al. in 1984.<sup id="cite_ref-bfos_7-0" class="reference"><a href="#cite_note-bfos-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Trees used for regression and trees used for classification have some similarities – but also some differences, such as the procedure used to determine where to split.<sup id="cite_ref-bfos_7-1" class="reference"><a href="#cite_note-bfos-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Some techniques, often called <i>ensemble</i> methods, construct more than one decision tree: </p> <ul><li><b><a href="/wiki/Gradient_boosted_trees" class="mw-redirect" title="Gradient boosted trees">Boosted trees</a></b> Incrementally building an ensemble by training each new instance to emphasize the training instances previously mis-modeled. A typical example is <a href="/wiki/AdaBoost" title="AdaBoost">AdaBoost</a>. These can be used for regression-type and classification-type problems.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li><b>Committees of decision trees</b> (also called k-DT<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>), an early method that used randomized decision tree algorithms to generate multiple different trees from the training data, and then combine them using majority voting to generate output.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li> <li><b><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bootstrap aggregated</a></b> (or bagged) decision trees, an early ensemble method, builds multiple decision trees by repeatedly <a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">resampling training data with replacement</a>, and voting the trees for a consensus prediction.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <ul><li>A <b><a href="/wiki/Random_forest" title="Random forest">random forest</a></b> classifier is a specific type of <a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">bootstrap aggregating</a></li></ul></li> <li><b>Rotation forest</b> – in which every decision tree is trained by first applying <a href="/wiki/Principal_component_analysis" title="Principal component analysis">principal component analysis</a> (PCA) on a random subset of the input features.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul> <p>A special case of a decision tree is a <a href="/wiki/Decision_list" title="Decision list">decision list</a>,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> which is a one-sided decision tree, so that every internal node has exactly 1 leaf node and exactly 1 internal node as a child (except for the bottommost node, whose only child is a single leaf node). While less expressive, decision lists are arguably easier to understand than general decision trees due to their added sparsity<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 2021)">citation needed</span></a></i>]</sup>, permit non-greedy learning methods<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and monotonic constraints to be imposed.<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> </p><p>Notable decision tree algorithms include: </p> <ul><li><a href="/wiki/ID3_algorithm" title="ID3 algorithm">ID3</a> (Iterative Dichotomiser 3)</li> <li><a href="/wiki/C4.5_algorithm" title="C4.5 algorithm">C4.5</a> (successor of ID3)</li> <li>CART (Classification And Regression Tree)<sup id="cite_ref-bfos_7-2" class="reference"><a href="#cite_note-bfos-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>OC1 (Oblique classifier 1). First method that created multivariate splits at each node.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Chi-square_automatic_interaction_detection" title="Chi-square automatic interaction detection">Chi-square automatic interaction detection</a> (CHAID). Performs multi-level splits when computing classification trees.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">MARS</a>: extends decision trees to handle numerical data better.</li> <li>Conditional Inference Trees. Statistics-based approach that uses non-parametric tests as splitting criteria, corrected for multiple testing to avoid overfitting. This approach results in unbiased predictor selection and does not require pruning.<sup id="cite_ref-Hothorn2006_21-0" class="reference"><a href="#cite_note-Hothorn2006-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Strobl2009_22-0" class="reference"><a href="#cite_note-Strobl2009-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li></ul> <p>ID3 and CART were invented independently at around the same time (between 1970 and 1980)<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. (August 2014)">citation needed</span></a></i>]</sup>, yet follow a similar approach for learning a decision tree from training tuples. </p><p>It has also been proposed to leverage concepts of <a href="/wiki/Fuzzy_set_theory" class="mw-redirect" title="Fuzzy set theory">fuzzy set theory</a> for the definition of a special version of decision tree, known as Fuzzy Decision Tree (FDT).<sup id="cite_ref-Janikow1998_23-0" class="reference"><a href="#cite_note-Janikow1998-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> In this type of fuzzy classification, generally, an input vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85631435f001c884eca834164392982c621f40e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle {\textbf {x}}}"></span> is associated with multiple classes, each with a different confidence value. Boosted ensembles of FDTs have been recently investigated as well, and they have shown performances comparable to those of other very efficient fuzzy classifiers.<sup id="cite_ref-Barsacchi2020_24-0" class="reference"><a href="#cite_note-Barsacchi2020-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Metrics">Metrics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=3" title="Edit section: Metrics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algorithms for constructing decision trees usually work top-down, by choosing a variable at each step that best splits the set of items.<sup id="cite_ref-top-downDT_6-1" class="reference"><a href="#cite_note-top-downDT-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Different algorithms use different metrics for measuring "best". These generally measure the homogeneity of the target variable within the subsets. Some examples are given below. These metrics are applied to each candidate subset, and the resulting values are combined (e.g., averaged) to provide a measure of the quality of the split. Depending on the underlying metric, the performance of various heuristic algorithms for decision tree learning may vary significantly.<sup id="cite_ref-thask_25-0" class="reference"><a href="#cite_note-thask-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Estimate_of_Positive_Correctness">Estimate of Positive Correctness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=4" title="Edit section: Estimate of Positive Correctness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple and effective metric can be used to identify the degree to which true positives outweigh false positives (see <a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a>). This metric, "Estimate of Positive Correctness" is defined below: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{P}=TP-FP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> <mi>P</mi> <mo>−</mo> <mi>F</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{P}=TP-FP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f229fb6c97298f859d3a3fb8ef03eec150d0e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.988ex; height:2.509ex;" alt="{\displaystyle E_{P}=TP-FP}"></span> </p><p>In this equation, the total false positives (FP) are subtracted from the total true positives (TP). The resulting number gives an estimate on how many positive examples the feature could correctly identify within the data, with higher numbers meaning that the feature could correctly classify more positive samples. Below is an example of how to use the metric when the full confusion matrix of a certain feature is given: </p><p><b>Feature A Confusion Matrix</b> </p> <table class="wikitable"> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">Predicted<br />Class</div><div style="margin-right:2em;text-align:left">Actual Class</div> </th> <th>Cancer </th> <th>Non-cancer </th></tr> <tr> <th>Cancer </th> <td><u>8</u> </td> <td>3 </td></tr> <tr> <th>Non-cancer </th> <td><u>2</u> </td> <td>5 </td></tr></tbody></table> <p>Here we can see that the TP value would be 8 and the FP value would be 2 (the underlined numbers in the table). When we plug these numbers in the equation we are able to calculate the estimate: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{p}=TP-FP=8-2=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> <mi>P</mi> <mo>−</mo> <mi>F</mi> <mi>P</mi> <mo>=</mo> <mn>8</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{p}=TP-FP=8-2=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2346b8ae3d68959e7c32be66ae0ca31de6aac2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.106ex; height:2.843ex;" alt="{\displaystyle E_{p}=TP-FP=8-2=6}"></span>. This means that using the estimate on this feature would have it receive a score of 6. </p><p>However, it should be worth noting that this number is only an estimate. For example, if two features both had a FP value of 2 while one of the features had a higher TP value, that feature would be ranked higher than the other because the resulting estimate when using the equation would give a higher value. This could lead to some inaccuracies when using the metric if some features have more positive samples than others. To combat this, one could use a more powerful metric known as <a href="/wiki/Sensitivity_and_specificity" title="Sensitivity and specificity">Sensitivity</a> that takes into account the proportions of the values from the confusion matrix to give the actual <a href="/wiki/Sensitivity_and_specificity" title="Sensitivity and specificity">true positive rate</a> (TPR). The difference between these metrics is shown in the example below: </p> <table> <caption> </caption> <tbody><tr> <td><b>Feature A Confusion Matrix</b> <table class="wikitable"> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">Predicted<br />Class</div><div style="margin-right:2em;text-align:left">Actual Class</div> </th> <th>Cancer </th> <th>Non-cancer </th></tr> <tr> <th>Cancer </th> <td>8 </td> <td>3 </td></tr> <tr> <th>Non-cancer </th> <td>2 </td> <td>5 </td></tr></tbody></table> </td> <td style="padding-left: 4em;"><b>Feature B Confusion Matrix</b> <table class="wikitable"> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">Predicted<br />Class</div><div style="margin-right:2em;text-align:left">Actual Class</div> </th> <th>Cancer </th> <th>Non-cancer </th></tr> <tr> <th>Cancer </th> <td>6 </td> <td>2 </td></tr> <tr> <th>Non-cancer </th> <td>2 </td> <td>8 </td></tr></tbody></table> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{p}=TP-FP=8-2=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> <mi>P</mi> <mo>−</mo> <mi>F</mi> <mi>P</mi> <mo>=</mo> <mn>8</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{p}=TP-FP=8-2=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2346b8ae3d68959e7c32be66ae0ca31de6aac2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.106ex; height:2.843ex;" alt="{\displaystyle E_{p}=TP-FP=8-2=6}"></span> <p><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 TPR=TP/(TP+FN)=8/(8+3)\approx 0.73}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>P</mi> <mi>R</mi> <mo>=</mo> <mi>T</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>8</mn> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>0.73</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TPR=TP/(TP+FN)=8/(8+3)\approx 0.73}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a742a70750407dc6f035a8d9f12c9c88c54a8582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.255ex; height:2.843ex;" alt="{\displaystyle TPR=TP/(TP+FN)=8/(8+3)\approx 0.73}"></span> </p> </td> <td style="padding-left: 4em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{p}=TP-FP=6-2=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> <mi>P</mi> <mo>−</mo> <mi>F</mi> <mi>P</mi> <mo>=</mo> <mn>6</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{p}=TP-FP=6-2=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68c9bf8838cf6c97c82e1a08336ec93c217aaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.106ex; height:2.843ex;" alt="{\displaystyle E_{p}=TP-FP=6-2=4}"></span> <p><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 TPR=TP/(TP+FN)=6/(6+2)=0.75}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>P</mi> <mi>R</mi> <mo>=</mo> <mi>T</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.75</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TPR=TP/(TP+FN)=6/(6+2)=0.75}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b64d62841ea30e7ffad6cccff7c3f2aa51070cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.255ex; height:2.843ex;" alt="{\displaystyle TPR=TP/(TP+FN)=6/(6+2)=0.75}"></span> </p> </td></tr></tbody></table> <p>In this example, Feature A had an estimate of 6 and a TPR of approximately 0.73 while Feature B had an estimate of 4 and a TPR of 0.75. This shows that although the positive estimate for some feature may be higher, the more accurate TPR value for that feature may be lower when compared to other features that have a lower positive estimate. Depending on the situation and knowledge of the data and decision trees, one may opt to use the positive estimate for a quick and easy solution to their problem. On the other hand, a more experienced user would most likely prefer to use the TPR value to rank the features because it takes into account the proportions of the data and all the samples that should have been classified as positive. </p> <div class="mw-heading mw-heading3"><h3 id="Gini_impurity">Gini impurity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=5" title="Edit section: Gini impurity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Gini impurity</b>, <b>Gini's diversity index</b>,<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> or <b><a href="/wiki/Diversity_index#Gini–Simpson_index" title="Diversity index">Gini-Simpson Index</a></b> in biodiversity research, is named after Italian mathematician <a href="/wiki/Corrado_Gini" title="Corrado Gini">Corrado Gini</a> and used by the CART (classification and regression tree) algorithm for classification trees. Gini impurity measures how often a randomly chosen element of a set would be incorrectly labeled if it were labeled randomly and independently according to the distribution of labels in the set. It reaches its minimum (zero) when all cases in the node fall into a single target category. </p><p>For a set of items with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> classes and relative frequencies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,2,...,J\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>J</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,2,...,J\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7479859c1eacff9b75b2687e150b4f212e3446f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.968ex; height:2.843ex;" alt="{\displaystyle i\in \{1,2,...,J\}}"></span>, the probability of choosing an item with label <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span>, and the probability of miscategorizing that item is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k\neq i}p_{k}=1-p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≠</mo> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k\neq i}p_{k}=1-p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd9588bfbd427cc269d54e7cafcd74758289075b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.071ex; height:6.009ex;" alt="{\displaystyle \sum _{k\neq i}p_{k}=1-p_{i}}"></span>. The Gini impurity is computed by summing pairwise products of these probabilities for each class label: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {I} _{G}(p)=\sum _{i=1}^{J}\left(p_{i}\sum _{k\neq i}p_{k}\right)=\sum _{i=1}^{J}p_{i}(1-p_{i})=\sum _{i=1}^{J}(p_{i}-p_{i}^{2})=\sum _{i=1}^{J}p_{i}-\sum _{i=1}^{J}p_{i}^{2}=1-\sum _{i=1}^{J}p_{i}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≠</mo> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {I} _{G}(p)=\sum _{i=1}^{J}\left(p_{i}\sum _{k\neq i}p_{k}\right)=\sum _{i=1}^{J}p_{i}(1-p_{i})=\sum _{i=1}^{J}(p_{i}-p_{i}^{2})=\sum _{i=1}^{J}p_{i}-\sum _{i=1}^{J}p_{i}^{2}=1-\sum _{i=1}^{J}p_{i}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe8179fbf8ca4addf0e8f5f404fdbf153a04214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:87.437ex; height:7.843ex;" alt="{\displaystyle \operatorname {I} _{G}(p)=\sum _{i=1}^{J}\left(p_{i}\sum _{k\neq i}p_{k}\right)=\sum _{i=1}^{J}p_{i}(1-p_{i})=\sum _{i=1}^{J}(p_{i}-p_{i}^{2})=\sum _{i=1}^{J}p_{i}-\sum _{i=1}^{J}p_{i}^{2}=1-\sum _{i=1}^{J}p_{i}^{2}.}"></span></dd></dl> <p>The Gini impurity is also an information theoretic measure and corresponds to <a href="/wiki/Tsallis_Entropy" class="mw-redirect" title="Tsallis Entropy">Tsallis Entropy</a> with deformation coefficient <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 q=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26622af6012fb982cab4e9584f57dd4f364233b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=2}"></span>, which in physics is associated with the lack of information in out-of-equilibrium, non-extensive, dissipative and quantum systems. For the limit <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 q\to 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\to 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec80cb4e51140ce942dc15b9b857f6a5fb203c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.846ex; height:2.509ex;" alt="{\displaystyle q\to 1}"></span> one recovers the usual Boltzmann-Gibbs or Shannon entropy. In this sense, the Gini impurity is nothing but a variation of the usual entropy measure for decision trees. </p> <div class="mw-heading mw-heading3"><h3 id="Information_gain">Information gain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=6" title="Edit section: Information gain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Information_gain_in_decision_trees" class="mw-redirect" title="Information gain in decision trees">Information gain in decision trees</a></div> <p>Used by the <a href="/wiki/ID3_algorithm" title="ID3 algorithm">ID3</a>, <a href="/wiki/C4.5_algorithm" title="C4.5 algorithm">C4.5</a> and C5.0 tree-generation algorithms. <a href="/wiki/Information_gain" class="mw-redirect" title="Information gain">Information gain</a> is based on the concept of <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">entropy</a> and <a href="/wiki/Information_content" title="Information content">information content</a> from <a href="/wiki/Information_theory" title="Information theory">information theory</a>. </p><p>Entropy is defined as below </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {H} (T)=\operatorname {I} _{E}\left(p_{1},p_{2},\ldots ,p_{J}\right)=-\sum _{i=1}^{J}p_{i}\log _{2}p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H} (T)=\operatorname {I} _{E}\left(p_{1},p_{2},\ldots ,p_{J}\right)=-\sum _{i=1}^{J}p_{i}\log _{2}p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f792af48b1a164791d2c5eeb2ba10d460d82d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.3ex; height:7.343ex;" alt="{\displaystyle \mathrm {H} (T)=\operatorname {I} _{E}\left(p_{1},p_{2},\ldots ,p_{J}\right)=-\sum _{i=1}^{J}p_{i}\log _{2}p_{i}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},p_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2a5f0df47b52b649e76beb97452539b7c058b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.328ex; height:2.009ex;" alt="{\displaystyle p_{1},p_{2},\ldots }"></span> are fractions that add up to 1 and represent the percentage of each class present in the child node that results from a split in the tree.<sup id="cite_ref-Witten_2011_102–103_27-0" class="reference"><a href="#cite_note-Witten_2011_102–103-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \overbrace {IG(T,a)} ^{\text{information gain}}=\overbrace {\mathrm {H} (T)} ^{\text{entropy (parent)}}-\overbrace {\mathrm {H} (T\mid a)} ^{\text{sum of entropies (children)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mi>I</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>information gain</mtext> </mrow> </mover> <mo>=</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>entropy (parent)</mtext> </mrow> </mover> <mo>−</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>∣</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>sum of entropies (children)</mtext> </mrow> </mover> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \overbrace {IG(T,a)} ^{\text{information gain}}=\overbrace {\mathrm {H} (T)} ^{\text{entropy (parent)}}-\overbrace {\mathrm {H} (T\mid a)} ^{\text{sum of entropies (children)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf340151304495b79cf544f2bfd477e084afdbcd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.494ex; height:6.509ex;" alt="{\displaystyle \overbrace {IG(T,a)} ^{\text{information gain}}=\overbrace {\mathrm {H} (T)} ^{\text{entropy (parent)}}-\overbrace {\mathrm {H} (T\mid a)} ^{\text{sum of entropies (children)}}}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =-\sum _{i=1}^{J}p_{i}\log _{2}p_{i}-\sum _{i=1}^{J}-\Pr(i\mid a)\log _{2}\Pr(i\mid a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <mo>−</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>∣</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>∣</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =-\sum _{i=1}^{J}p_{i}\log _{2}p_{i}-\sum _{i=1}^{J}-\Pr(i\mid a)\log _{2}\Pr(i\mid a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b37948bd9afa5c440922f52a00959327ede1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.253ex; height:7.343ex;" alt="{\displaystyle =-\sum _{i=1}^{J}p_{i}\log _{2}p_{i}-\sum _{i=1}^{J}-\Pr(i\mid a)\log _{2}\Pr(i\mid a)}"></span></dd></dl> <p>Averaging over the possible values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \overbrace {E_{A}(\operatorname {IG} (T,a))} ^{\text{expected information gain}}=\overbrace {I(T;A)} ^{{\text{mutual information between }}T{\text{ and }}A}=\overbrace {\mathrm {H} (T)} ^{\text{entropy (parent)}}-\overbrace {\mathrm {H} (T\mid A)} ^{\text{weighted sum of entropies (children)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>IG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>expected information gain</mtext> </mrow> </mover> <mo>=</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>;</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>mutual information between </mtext> </mrow> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>A</mi> </mrow> </mover> <mo>=</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>entropy (parent)</mtext> </mrow> </mover> <mo>−</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>∣</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>weighted sum of entropies (children)</mtext> </mrow> </mover> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \overbrace {E_{A}(\operatorname {IG} (T,a))} ^{\text{expected information gain}}=\overbrace {I(T;A)} ^{{\text{mutual information between }}T{\text{ and }}A}=\overbrace {\mathrm {H} (T)} ^{\text{entropy (parent)}}-\overbrace {\mathrm {H} (T\mid A)} ^{\text{weighted sum of entropies (children)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524ebf09a1d2675e3629f18fa12ed4e5ac2ea83d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:93.671ex; height:6.509ex;" alt="{\displaystyle \overbrace {E_{A}(\operatorname {IG} (T,a))} ^{\text{expected information gain}}=\overbrace {I(T;A)} ^{{\text{mutual information between }}T{\text{ and }}A}=\overbrace {\mathrm {H} (T)} ^{\text{entropy (parent)}}-\overbrace {\mathrm {H} (T\mid A)} ^{\text{weighted sum of entropies (children)}}}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =-\sum _{i=1}^{J}p_{i}\log _{2}p_{i}-\sum _{a}p(a)\sum _{i=1}^{J}-\Pr(i\mid a)\log _{2}\Pr(i\mid a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <mo>−</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>∣</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>∣</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =-\sum _{i=1}^{J}p_{i}\log _{2}p_{i}-\sum _{a}p(a)\sum _{i=1}^{J}-\Pr(i\mid a)\log _{2}\Pr(i\mid a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c52c58a17d690d341bbfae868ec8d05bd59ee627" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.591ex; height:7.343ex;" alt="{\displaystyle =-\sum _{i=1}^{J}p_{i}\log _{2}p_{i}-\sum _{a}p(a)\sum _{i=1}^{J}-\Pr(i\mid a)\log _{2}\Pr(i\mid a)}"></span></dd> <dd>Where weighted sum of entropies is given by,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathrm {H} (T\mid A)}=\sum _{a}p(a)\sum _{i=1}^{J}-\Pr(i\mid a)\log _{2}\Pr(i\mid a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>∣</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </munderover> <mo>−</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>∣</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>∣</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {H} (T\mid A)}=\sum _{a}p(a)\sum _{i=1}^{J}-\Pr(i\mid a)\log _{2}\Pr(i\mid a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7403a37c98d146168b24fd5b2c1c59f8fec158ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.589ex; height:7.343ex;" alt="{\displaystyle {\mathrm {H} (T\mid A)}=\sum _{a}p(a)\sum _{i=1}^{J}-\Pr(i\mid a)\log _{2}\Pr(i\mid a)}"></span></dd></dl> <p>That is, the expected information gain is the <a href="/wiki/Mutual_information" title="Mutual information">mutual information</a>, meaning that on average, the reduction in the entropy of <i>T</i> is the mutual information. </p><p>Information gain is used to decide which feature to split on at each step in building the tree. Simplicity is best, so we want to keep our tree small. To do so, at each step we should choose the split that results in the most consistent child nodes. A commonly used measure of consistency is called <a href="/wiki/Information_theory" title="Information theory">information</a> which is measured in <a href="/wiki/Bit" title="Bit">bits</a>. For each node of the tree, the information value "represents the expected amount of information that would be needed to specify whether a new instance should be classified yes or no, given that the example reached that node".<sup id="cite_ref-Witten_2011_102–103_27-1" class="reference"><a href="#cite_note-Witten_2011_102–103-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>Consider an example data set with four attributes: <i>outlook</i> (sunny, overcast, rainy), <i>temperature</i> (hot, mild, cool), <i>humidity</i> (high, normal), and <i>windy</i> (true, false), with a binary (yes or no) target variable, <i>play</i>, and 14 data points. To construct a decision tree on this data, we need to compare the information gain of each of four trees, each split on one of the four features. The split with the highest information gain will be taken as the first split and the process will continue until all children nodes each have consistent data, or until the information gain is 0. </p><p>To find the information gain of the split using <i>windy</i>, we must first calculate the information in the data before the split. The original data contained nine yes's and five no's. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{E}([9,5])=-{\frac {9}{14}}\log _{2}{\frac {9}{14}}-{\frac {5}{14}}\log _{2}{\frac {5}{14}}=0.94}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>9</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>14</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>14</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>14</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>14</mn> </mfrac> </mrow> <mo>=</mo> <mn>0.94</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{E}([9,5])=-{\frac {9}{14}}\log _{2}{\frac {9}{14}}-{\frac {5}{14}}\log _{2}{\frac {5}{14}}=0.94}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c46c2ae65097e48149aec84b9e4723001227cb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.197ex; height:5.176ex;" alt="{\displaystyle I_{E}([9,5])=-{\frac {9}{14}}\log _{2}{\frac {9}{14}}-{\frac {5}{14}}\log _{2}{\frac {5}{14}}=0.94}"></span></dd></dl> <p>The split using the feature <i>windy</i> results in two children nodes, one for a <i>windy</i> value of true and one for a <i>windy</i> value of false. In this data set, there are six data points with a true <i>windy</i> value, three of which have a <i>play</i> (where <i>play</i> is the target variable) value of yes and three with a <i>play</i> value of no. The eight remaining data points with a <i>windy</i> value of false contain two no's and six yes's. The information of the <i>windy</i>=true node is calculated using the entropy equation above. Since there is an equal number of yes's and no's in this node, we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{E}([3,3])=-{\frac {3}{6}}\log _{2}{\frac {3}{6}}-{\frac {3}{6}}\log _{2}{\frac {3}{6}}=-{\frac {1}{2}}\log _{2}{\frac {1}{2}}-{\frac {1}{2}}\log _{2}{\frac {1}{2}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{E}([3,3])=-{\frac {3}{6}}\log _{2}{\frac {3}{6}}-{\frac {3}{6}}\log _{2}{\frac {3}{6}}=-{\frac {1}{2}}\log _{2}{\frac {1}{2}}-{\frac {1}{2}}\log _{2}{\frac {1}{2}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c344c5ef64a208878afa465bb82b70588dce4a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.917ex; height:5.176ex;" alt="{\displaystyle I_{E}([3,3])=-{\frac {3}{6}}\log _{2}{\frac {3}{6}}-{\frac {3}{6}}\log _{2}{\frac {3}{6}}=-{\frac {1}{2}}\log _{2}{\frac {1}{2}}-{\frac {1}{2}}\log _{2}{\frac {1}{2}}=1}"></span></dd></dl> <p>For the node where <i>windy</i>=false there were eight data points, six yes's and two no's. Thus we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{E}([6,2])=-{\frac {6}{8}}\log _{2}{\frac {6}{8}}-{\frac {2}{8}}\log _{2}{\frac {2}{8}}=-{\frac {3}{4}}\log _{2}{\frac {3}{4}}-{\frac {1}{4}}\log _{2}{\frac {1}{4}}=0.81}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>8</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>8</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>8</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>8</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>0.81</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{E}([6,2])=-{\frac {6}{8}}\log _{2}{\frac {6}{8}}-{\frac {2}{8}}\log _{2}{\frac {2}{8}}=-{\frac {3}{4}}\log _{2}{\frac {3}{4}}-{\frac {1}{4}}\log _{2}{\frac {1}{4}}=0.81}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6543aa2df6c4c940aaea3922c36112ed1dc7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:66.889ex; height:5.176ex;" alt="{\displaystyle I_{E}([6,2])=-{\frac {6}{8}}\log _{2}{\frac {6}{8}}-{\frac {2}{8}}\log _{2}{\frac {2}{8}}=-{\frac {3}{4}}\log _{2}{\frac {3}{4}}-{\frac {1}{4}}\log _{2}{\frac {1}{4}}=0.81}"></span></dd></dl> <p>To find the information of the split, we take the weighted average of these two numbers based on how many observations fell into which node. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{E}([3,3],[6,2])=I_{E}({\text{windy or not}})={\frac {6}{14}}\cdot 1+{\frac {8}{14}}\cdot 0.81=0.89}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>windy or not</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>14</mn> </mfrac> </mrow> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>14</mn> </mfrac> </mrow> <mo>⋅</mo> <mn>0.81</mn> <mo>=</mo> <mn>0.89</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{E}([3,3],[6,2])=I_{E}({\text{windy or not}})={\frac {6}{14}}\cdot 1+{\frac {8}{14}}\cdot 0.81=0.89}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb49253c4ec56b6d9dc422d26e77370b0e7cc65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:62.959ex; height:5.176ex;" alt="{\displaystyle I_{E}([3,3],[6,2])=I_{E}({\text{windy or not}})={\frac {6}{14}}\cdot 1+{\frac {8}{14}}\cdot 0.81=0.89}"></span></dd></dl> <p>Now we can calculate the information gain achieved by splitting on the <i>windy</i> feature. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {IG} ({\text{windy}})=I_{E}([9,5])-I_{E}([3,3],[6,2])=0.94-0.89=0.05}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>IG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>windy</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>9</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.94</mn> <mo>−</mo> <mn>0.89</mn> <mo>=</mo> <mn>0.05</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {IG} ({\text{windy}})=I_{E}([9,5])-I_{E}([3,3],[6,2])=0.94-0.89=0.05}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a11c2c97209089893bc6241012533e922abbb5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.622ex; height:2.843ex;" alt="{\displaystyle \operatorname {IG} ({\text{windy}})=I_{E}([9,5])-I_{E}([3,3],[6,2])=0.94-0.89=0.05}"></span></dd></dl> <p>To build the tree, the information gain of each possible first split would need to be calculated. The best first split is the one that provides the most information gain. This process is repeated for each impure node until the tree is complete. This example is adapted from the example appearing in Witten et al.<sup id="cite_ref-Witten_2011_102–103_27-2" class="reference"><a href="#cite_note-Witten_2011_102–103-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>Information gain is also known as <a href="/wiki/Diversity_index#Shannon_index" title="Diversity index">Shannon index</a> in bio diversity research. </p> <div class="mw-heading mw-heading3"><h3 id="Variance_reduction">Variance reduction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=7" title="Edit section: Variance reduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Introduced in CART,<sup id="cite_ref-bfos_7-3" class="reference"><a href="#cite_note-bfos-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> variance reduction is often employed in cases where the target variable is continuous (regression tree), meaning that use of many other metrics would first require discretization before being applied. The variance reduction of a node <span class="texhtml mvar" style="font-style:italic;">N</span> is defined as the total reduction of the variance of the target variable <span class="texhtml mvar" style="font-style:italic;">Y</span> due to the split at this node: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{V}(N)={\frac {1}{|S|^{2}}}\sum _{i\in S}\sum _{j\in S}{\frac {1}{2}}(y_{i}-y_{j})^{2}-\left({\frac {|S_{t}|^{2}}{|S|^{2}}}{\frac {1}{|S_{t}|^{2}}}\sum _{i\in S_{t}}\sum _{j\in S_{t}}{\frac {1}{2}}(y_{i}-y_{j})^{2}+{\frac {|S_{f}|^{2}}{|S|^{2}}}{\frac {1}{|S_{f}|^{2}}}\sum _{i\in S_{f}}\sum _{j\in S_{f}}{\frac {1}{2}}(y_{i}-y_{j})^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{V}(N)={\frac {1}{|S|^{2}}}\sum _{i\in S}\sum _{j\in S}{\frac {1}{2}}(y_{i}-y_{j})^{2}-\left({\frac {|S_{t}|^{2}}{|S|^{2}}}{\frac {1}{|S_{t}|^{2}}}\sum _{i\in S_{t}}\sum _{j\in S_{t}}{\frac {1}{2}}(y_{i}-y_{j})^{2}+{\frac {|S_{f}|^{2}}{|S|^{2}}}{\frac {1}{|S_{f}|^{2}}}\sum _{i\in S_{f}}\sum _{j\in S_{f}}{\frac {1}{2}}(y_{i}-y_{j})^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7195f78d2525a8d5615ad0890517510bd54fcb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:105.578ex; height:8.509ex;" alt="{\displaystyle I_{V}(N)={\frac {1}{|S|^{2}}}\sum _{i\in S}\sum _{j\in S}{\frac {1}{2}}(y_{i}-y_{j})^{2}-\left({\frac {|S_{t}|^{2}}{|S|^{2}}}{\frac {1}{|S_{t}|^{2}}}\sum _{i\in S_{t}}\sum _{j\in S_{t}}{\frac {1}{2}}(y_{i}-y_{j})^{2}+{\frac {|S_{f}|^{2}}{|S|^{2}}}{\frac {1}{|S_{f}|^{2}}}\sum _{i\in S_{f}}\sum _{j\in S_{f}}{\frac {1}{2}}(y_{i}-y_{j})^{2}\right)}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2391e6e796fbf718be3828080775ac2ac3d3d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.251ex; height:2.509ex;" alt="{\displaystyle S_{t}}"></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 S_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd92398c43428bd0f8edadae8ed9655f9476bf24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.561ex; height:2.843ex;" alt="{\displaystyle S_{f}}"></span> are the set of presplit sample indices, set of sample indices for which the split test is true, and set of sample indices for which the split test is false, respectively. Each of the above summands are indeed <a href="/wiki/Variance" title="Variance">variance</a> estimates, though, written in a form without directly referring to the mean. </p><p>By replacing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{i}-y_{j})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y_{i}-y_{j})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccff77f70542f11883602aa370b775f251fd1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.692ex; height:3.343ex;" alt="{\displaystyle (y_{i}-y_{j})^{2}}"></span> in the formula above with the dissimilarity <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 d_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0116346390c7dcd72d9ed714177a9b6c81e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.686ex; height:2.843ex;" alt="{\displaystyle d_{ij}}"></span> between two objects <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> 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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>, the variance reduction criterion applies to any kind of object for which pairwise dissimilarities can be computed.<sup id="cite_ref-:1_1-1" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Measure_of_"goodness""><span id="Measure_of_.22goodness.22"></span>Measure of "goodness"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=8" title="Edit section: Measure of "goodness""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Used by CART in 1984,<sup id="cite_ref-ll_28-0" class="reference"><a href="#cite_note-ll-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> the measure of "goodness" is a function that seeks to optimize the balance of a candidate split's capacity to create pure children with its capacity to create equally-sized children. This process is repeated for each impure node until the tree is complete. The function <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 \varphi (s\mid t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>∣</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (s\mid t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a82b6f7a49f06fb905eb590aa7a0fe5aaf7bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.197ex; height:2.843ex;" alt="{\displaystyle \varphi (s\mid t)}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is a candidate split at node <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, is defined as below </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (s\mid t)=2P_{L}P_{R}\sum _{j=1}^{\text{class count}}|P(j\mid t_{L})-P(j\mid t_{R})|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>∣</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>class count</mtext> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>∣</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>∣</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (s\mid t)=2P_{L}P_{R}\sum _{j=1}^{\text{class count}}|P(j\mid t_{L})-P(j\mid t_{R})|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5d16bcf7cdca1f1b5d5b0c31e865fb7ae498fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.497ex; height:7.676ex;" alt="{\displaystyle \varphi (s\mid t)=2P_{L}P_{R}\sum _{j=1}^{\text{class count}}|P(j\mid t_{L})-P(j\mid t_{R})|}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e7c65e125b1bf99382d350dbc87119390d686d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.191ex; height:2.343ex;" alt="{\displaystyle t_{L}}"></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 t_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2718919868fbbc98a8fcfdd17e4c04c3f3a77985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.343ex;" alt="{\displaystyle t_{R}}"></span> are the left and right children of node <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> using split <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>, respectively; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbfa4c6ed25929d86a4f6d3e7ec3f37abc65b4b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.844ex; height:2.509ex;" alt="{\displaystyle P_{L}}"></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 P_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea3ad403ba0657b26fe3634c927faf5a14c6f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle P_{R}}"></span> are the proportions of records in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e7c65e125b1bf99382d350dbc87119390d686d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.191ex; height:2.343ex;" alt="{\displaystyle t_{L}}"></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 t_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2718919868fbbc98a8fcfdd17e4c04c3f3a77985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.343ex;" alt="{\displaystyle t_{R}}"></span>, respectively; 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 P(j\mid t_{L})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>∣</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(j\mid t_{L})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8269ba3fd961b4706fb5374d321e4c310e1efcbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.641ex; height:2.843ex;" alt="{\displaystyle P(j\mid t_{L})}"></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 P(j\mid t_{R})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>∣</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(j\mid t_{R})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef711882653817e4516fc011547faf74c5ea180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.769ex; height:2.843ex;" alt="{\displaystyle P(j\mid t_{R})}"></span> are the proportions of class <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span> records in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e7c65e125b1bf99382d350dbc87119390d686d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.191ex; height:2.343ex;" alt="{\displaystyle t_{L}}"></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 t_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2718919868fbbc98a8fcfdd17e4c04c3f3a77985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.343ex;" alt="{\displaystyle t_{R}}"></span>, respectively. </p><p>Consider an example data set with three attributes: <i>savings</i>(low, medium, high), <i>assets</i>(low, medium, high), <i>income</i>(numerical value), and a binary target variable <i>credit risk</i>(good, bad) and 8 data points.<sup id="cite_ref-ll_28-1" class="reference"><a href="#cite_note-ll-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> The full data is presented in the table below. To start a decision tree, we will calculate the maximum value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (s\mid t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>∣</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (s\mid t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a82b6f7a49f06fb905eb590aa7a0fe5aaf7bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.197ex; height:2.843ex;" alt="{\displaystyle \varphi (s\mid t)}"></span> using each feature to find which one will split the root node. This process will continue until all children are pure or all <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 \varphi (s\mid t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>∣</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (s\mid t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a82b6f7a49f06fb905eb590aa7a0fe5aaf7bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.197ex; height:2.843ex;" alt="{\displaystyle \varphi (s\mid t)}"></span> values are below a set threshold. </p> <table class="wikitable"> <tbody><tr> <th>Customer</th> <th>Savings</th> <th>Assets</th> <th>Income ($1000s)</th> <th>Credit risk </th></tr> <tr> <td>1</td> <td>Medium</td> <td>High</td> <td>75</td> <td>Good </td></tr> <tr> <td>2</td> <td>Low</td> <td>Low</td> <td>50</td> <td>Bad </td></tr> <tr> <td>3</td> <td>High</td> <td>Medium</td> <td>25</td> <td>Bad </td></tr> <tr> <td>4</td> <td>Medium</td> <td>Medium</td> <td>50</td> <td>Good </td></tr> <tr> <td>5</td> <td>Low</td> <td>Medium</td> <td>100</td> <td>Good </td></tr> <tr> <td>6</td> <td>High</td> <td>High</td> <td>25</td> <td>Good </td></tr> <tr> <td>7</td> <td>Low</td> <td>Low</td> <td>25</td> <td>Bad </td></tr> <tr> <td>8</td> <td>Medium</td> <td>Medium</td> <td>75</td> <td>Good </td></tr></tbody></table> <p>To find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (s\mid t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>∣</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (s\mid t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a82b6f7a49f06fb905eb590aa7a0fe5aaf7bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.197ex; height:2.843ex;" alt="{\displaystyle \varphi (s\mid t)}"></span> of the feature <i>savings</i>, we need to note the quantity of each value. The original data contained three low's, three medium's, and two high's. Out of the low's, one had a good <i>credit risk</i> while out of the medium's and high's, 4 had a good <i>credit risk</i>. Assume a candidate split <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> such that records with a low <i>savings</i> will be put in the left child and all other records will be put into the right child. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (s\mid {\text{root}})=2\cdot {\frac {3}{8}}\cdot {\frac {5}{8}}\cdot \left(\left|\left({\frac {1}{3}}-{\frac {4}{5}}\right)\right|+\left|\left({\frac {2}{3}}-{\frac {1}{5}}\right)\right|\right)=0.44}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>root</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>8</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.44</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (s\mid {\text{root}})=2\cdot {\frac {3}{8}}\cdot {\frac {5}{8}}\cdot \left(\left|\left({\frac {1}{3}}-{\frac {4}{5}}\right)\right|+\left|\left({\frac {2}{3}}-{\frac {1}{5}}\right)\right|\right)=0.44}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47debc7ce555ff334552ec123ce46970afb7f849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.393ex; height:6.176ex;" alt="{\displaystyle \varphi (s\mid {\text{root}})=2\cdot {\frac {3}{8}}\cdot {\frac {5}{8}}\cdot \left(\left|\left({\frac {1}{3}}-{\frac {4}{5}}\right)\right|+\left|\left({\frac {2}{3}}-{\frac {1}{5}}\right)\right|\right)=0.44}"></span></dd></dl> <p>To build the tree, the "goodness" of all candidate splits for the root node need to be calculated. The candidate with the maximum value will split the root node, and the process will continue for each impure node until the tree is complete. </p><p>Compared to other metrics such as information gain, the measure of "goodness" will attempt to create a more balanced tree, leading to more-consistent decision time. However, it sacrifices some priority for creating pure children which can lead to additional splits that are not present with other metrics. </p> <div class="mw-heading mw-heading2"><h2 id="Uses">Uses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=9" title="Edit section: Uses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Advantages">Advantages</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=10" title="Edit section: Advantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Amongst other data mining methods, decision trees have various advantages: </p> <ul><li><b>Simple to understand and interpret.</b> People are able to understand decision tree models after a brief explanation. Trees can also be displayed graphically in a way that is easy for non-experts to interpret.<sup id="cite_ref-:0_29-0" class="reference"><a href="#cite_note-:0-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li> <li><b>Able to handle both numerical and <a href="/wiki/Categorical_variable" title="Categorical variable">categorical</a> data.</b><sup id="cite_ref-:0_29-1" class="reference"><a href="#cite_note-:0-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Other techniques are usually specialized in analyzing datasets that have only one type of variable. (For example, relation rules can be used only with nominal variables while neural networks can be used only with numerical variables or categoricals converted to 0-1 values.) Early decision trees were only capable of handling categorical variables, but more recent versions, such as C4.5, do not have this limitation.<sup id="cite_ref-tdidt_3-1" class="reference"><a href="#cite_note-tdidt-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li><b>Requires little data preparation.</b> Other techniques often require data normalization. Since trees can handle qualitative predictors, there is no need to create <a href="/wiki/Dummy_variable_(statistics)" title="Dummy variable (statistics)">dummy variables</a>.<sup id="cite_ref-:0_29-2" class="reference"><a href="#cite_note-:0-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li> <li><b>Uses a <a href="/wiki/White_box_(software_engineering)" title="White box (software engineering)">white box</a> or open-box<sup id="cite_ref-tdidt_3-2" class="reference"><a href="#cite_note-tdidt-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> model.</b> If a given situation is observable in a model the explanation for the condition is easily explained by <a href="/wiki/Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a>. By contrast, in a <a href="/wiki/Black_box" title="Black box">black box</a> model, the explanation for the results is typically difficult to understand, for example with an <a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">artificial neural network</a>.</li> <li><b>Possible to validate a model using statistical tests.</b> That makes it possible to account for the reliability of the model.</li> <li>Non-parametric approach that makes no assumptions of the training data or prediction residuals; e.g., no distributional, independence, or constant variance assumptions</li> <li><b>Performs well with large datasets.</b> Large amounts of data can be analyzed using standard computing resources in reasonable time.</li> <li><b>Accuracy with flexible modeling</b>. These methods may be applied to healthcare research with increased accuracy.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></li> <li><b>Mirrors human decision making more closely than other approaches.</b><sup id="cite_ref-:0_29-3" class="reference"><a href="#cite_note-:0-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> This could be useful when modeling human decisions/behavior.</li> <li><b>Robust against co-linearity, particularly boosting.</b></li> <li><b>In built</b> <b><a href="/wiki/Feature_selection" title="Feature selection">feature selection</a></b>. Additional irrelevant feature will be less used so that they can be removed on subsequent runs. The hierarchy of attributes in a decision tree reflects the importance of attributes.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> It means that the features on top are the most informative.<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></li> <li><b>Decision trees can approximate any <a href="/wiki/Boolean_function" title="Boolean function">Boolean function</a> e.g. <a href="/wiki/Exclusive_or" title="Exclusive or">XOR</a>.<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></b></li></ul> <div class="mw-heading mw-heading3"><h3 id="Limitations">Limitations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=11" title="Edit section: Limitations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Trees can be very non-robust. A small change in the <a href="/wiki/Training,_test,_and_validation_sets" class="mw-redirect" title="Training, test, and validation sets">training data</a> can result in a large change in the tree and consequently the final predictions.<sup id="cite_ref-:0_29-4" class="reference"><a href="#cite_note-:0-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li> <li>The problem of learning an optimal decision tree is known to be <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> under several aspects of optimality and even for simple concepts.<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><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> Consequently, practical decision-tree learning algorithms are based on heuristics such as the <a href="/wiki/Greedy_algorithm" title="Greedy algorithm">greedy algorithm</a> where locally optimal decisions are made at each node. Such algorithms cannot guarantee to return the globally optimal decision tree. To reduce the greedy effect of local optimality, some methods such as the dual information distance (DID) tree were proposed.<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></li> <li>Decision-tree learners can create over-complex trees that do not generalize well from the training data. (This is known as <a href="/wiki/Overfitting" title="Overfitting">overfitting</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup>) Mechanisms such as <a href="/wiki/Pruning_(decision_trees)" class="mw-redirect" title="Pruning (decision trees)">pruning</a> are necessary to avoid this problem (with the exception of some algorithms such as the Conditional Inference approach, that does not require pruning).<sup id="cite_ref-Hothorn2006_21-1" class="reference"><a href="#cite_note-Hothorn2006-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Strobl2009_22-1" class="reference"><a href="#cite_note-Strobl2009-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li> <li>The average depth of the tree that is defined by the number of nodes or tests till classification is not guaranteed to be minimal or small under various splitting criteria.<sup id="cite_ref-Tris_38-0" class="reference"><a href="#cite_note-Tris-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup></li> <li>For data including categorical variables with different numbers of levels, <a href="/wiki/Information_gain_in_decision_trees" class="mw-redirect" title="Information gain in decision trees">information gain in decision trees</a> is biased in favor of attributes with more levels.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> To counter this problem, instead of choosing the attribute with highest <a href="/wiki/Information_gain" class="mw-redirect" title="Information gain">information gain</a>, one can choose the attribute with the highest <a href="/wiki/Information_gain_ratio" title="Information gain ratio">information gain ratio</a> among the attributes whose information gain is greater than the mean information gain.<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> This biases the decision tree against considering attributes with a large number of distinct values, while not giving an unfair advantage to attributes with very low information gain. Alternatively, the issue of biased predictor selection can be avoided by the Conditional Inference approach,<sup id="cite_ref-Hothorn2006_21-2" class="reference"><a href="#cite_note-Hothorn2006-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> a two-stage approach,<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> or adaptive leave-one-out feature selection.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Implementations">Implementations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=12" title="Edit section: Implementations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many data mining software packages provide implementations of one or more decision tree algorithms (e.g. random forest). </p><p>Open source examples include: </p> <ul><li><a href="/wiki/ALGLIB" title="ALGLIB">ALGLIB</a>, a C++, C# and Java numerical analysis library with data analysis features (random forest)</li> <li><a href="/wiki/KNIME" title="KNIME">KNIME</a>, a free and open-source data analytics, reporting and integration platform (decision trees, random forest)</li> <li><a href="/wiki/Orange_(software)" title="Orange (software)">Orange</a>, an open-source data visualization, machine learning and data mining toolkit (random forest)</li> <li><a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> (an open-source software environment for statistical computing, which includes several CART implementations such as rpart, party and randomForest packages),</li> <li class="mw-empty-elt"></li> <li><a href="/wiki/Scikit-learn" title="Scikit-learn">scikit-learn</a> (a free and open-source machine learning library for the <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> programming language).</li> <li><a href="/wiki/Weka_(machine_learning)" class="mw-redirect" title="Weka (machine learning)">Weka</a> (a free and open-source data-mining suite, contains many decision tree algorithms),</li></ul> <p>Notable commercial software: </p> <ul><li><a href="/wiki/MATLAB" title="MATLAB">MATLAB</a>,</li> <li><a href="/wiki/Microsoft_SQL_Server" title="Microsoft SQL Server">Microsoft SQL Server</a>, and</li> <li><a href="/wiki/RapidMiner" title="RapidMiner">RapidMiner</a>,</li> <li class="mw-empty-elt"></li> <li><a href="/wiki/SAS_(software)#Components" title="SAS (software)">SAS Enterprise Miner</a>,</li> <li><a href="/wiki/SPSS_Modeler" title="SPSS Modeler">IBM SPSS Modeler</a>,</li></ul> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=13" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Decision_graphs">Decision graphs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=14" title="Edit section: Decision graphs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a decision tree, all paths from the root node to the leaf node proceed by way of conjunction, or <i>AND</i>. In a decision graph, it is possible to use disjunctions (ORs) to join two more paths together using <a href="/wiki/Minimum_message_length" title="Minimum message length">minimum message length</a> (MML).<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> Decision graphs have been further extended to allow for previously unstated new attributes to be learnt dynamically and used at different places within the graph.<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> The more general coding scheme results in better predictive accuracy and log-loss probabilistic scoring.<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. (January 2012)">citation needed</span></a></i>]</sup> In general, decision graphs infer models with fewer leaves than decision trees. </p> <div class="mw-heading mw-heading3"><h3 id="Alternative_search_methods">Alternative search methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=15" title="Edit section: Alternative search methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Evolutionary algorithms have been used to avoid local optimal decisions and search the decision tree space with little <i>a priori</i> bias.<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><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p>It is also possible for a tree to be sampled using <a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">MCMC</a>.<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> </p><p>The tree can be searched for in a bottom-up fashion.<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> Or several trees can be constructed parallelly to reduce the expected number of tests till classification.<sup id="cite_ref-Tris_38-1" class="reference"><a href="#cite_note-Tris-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p> <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=Decision_tree_learning&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Decision-tree_pruning" class="mw-redirect" title="Decision-tree pruning">Decision tree pruning</a></li> <li><a href="/wiki/Binary_decision_diagram" title="Binary decision diagram">Binary decision diagram</a></li> <li><a href="/wiki/CHAID" class="mw-redirect" title="CHAID">CHAID</a></li> <li><a href="/wiki/Predictive_analytics#Classification_and_regression_trees_(CART)" title="Predictive analytics">CART</a></li> <li><a href="/wiki/ID3_algorithm" title="ID3 algorithm">ID3 algorithm</a></li> <li><a href="/wiki/C4.5_algorithm" title="C4.5 algorithm">C4.5 algorithm</a></li> <li><a href="/wiki/Decision_stump" title="Decision stump">Decision stumps</a>, used in e.g. <a href="/wiki/AdaBoost" title="AdaBoost">AdaBoosting</a></li> <li><a href="/wiki/Decision_list" title="Decision list">Decision list</a></li> <li><a href="/wiki/Incremental_decision_tree" title="Incremental decision tree">Incremental decision tree</a></li> <li><a href="/wiki/Alternating_decision_tree" title="Alternating decision tree">Alternating decision tree</a></li> <li><a href="/wiki/Structured_data_analysis_(statistics)" title="Structured data analysis (statistics)">Structured data analysis (statistics)</a></li> <li><a href="/wiki/Logistic_model_tree" title="Logistic model tree">Logistic model tree</a></li> <li><a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">Hierarchical clustering</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=Decision_tree_learning&action=edit&section=17" 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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:1-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_1-1"><sup><i><b>b</b></i></sup></a></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="CITEREFStuderRitschardGabadinhoMüller2011" class="citation journal cs1">Studer, Matthias; Ritschard, Gilbert; Gabadinho, Alexis; Müller, Nicolas S. 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"A bottom-up oblique decision tree induction algorithm". <i>Proceedings of the 11th International Conference on Intelligent Systems Design and Applications (ISDA 2011)</i>. pp. 450–456. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FISDA.2011.6121697">10.1109/ISDA.2011.6121697</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4577-1676-8" title="Special:BookSources/978-1-4577-1676-8"><bdi>978-1-4577-1676-8</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15574923">15574923</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+bottom-up+oblique+decision+tree+induction+algorithm&rft.btitle=Proceedings+of+the+11th+International+Conference+on+Intelligent+Systems+Design+and+Applications+%28ISDA+2011%29&rft.pages=450-456&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15574923%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FISDA.2011.6121697&rft.isbn=978-1-4577-1676-8&rft.aulast=Barros&rft.aufirst=R.+C.&rft.au=Cerri%2C+R.&rft.au=Jaskowiak%2C+P.+A.&rft.au=Carvalho%2C+A.+C.+P.+L.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecision+tree+learning" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decision_tree_learning&action=edit&section=18" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJamesWittenHastieTibshirani2017" class="citation book cs1">James, Gareth; Witten, Daniela; Hastie, Trevor; Tibshirani, Robert (2017). <a rel="nofollow" class="external text" href="https://www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf#page=317">"Tree-Based Methods"</a> <span class="cs1-format">(PDF)</span>. <i>An Introduction to Statistical Learning: with Applications in R</i>. New York: Springer. pp. 303–336. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-7137-0" title="Special:BookSources/978-1-4614-7137-0"><bdi>978-1-4614-7137-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tree-Based+Methods&rft.btitle=An+Introduction+to+Statistical+Learning%3A+with+Applications+in+R&rft.place=New+York&rft.pages=303-336&rft.pub=Springer&rft.date=2017&rft.isbn=978-1-4614-7137-0&rft.aulast=James&rft.aufirst=Gareth&rft.au=Witten%2C+Daniela&rft.au=Hastie%2C+Trevor&rft.au=Tibshirani%2C+Robert&rft_id=https%3A%2F%2Fwww-bcf.usc.edu%2F~gareth%2FISL%2FISLR%2520Seventh%2520Printing.pdf%23page%3D317&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADecision+tree+learning" 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=Decision_tree_learning&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.cs.kent.ac.uk/people/staff/mg483/code/evoldectrees/">Evolutionary Learning of Decision Trees in C++</a></li> <li><a rel="nofollow" class="external text" href="http://christianherta.de/lehre/dataScience/machineLearning/decision-trees.html">A very detailed explanation of information gain as splitting criterion</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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