Random forest - Wikipedia

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id="toc-Random_forests_for_high-dimensional_data-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Variable_importance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variable_importance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Variable importance</span> </div> </a> <ul id="toc-Variable_importance-sublist" class="vector-toc-list"> <li id="toc-Permutation_importance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Permutation_importance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Permutation importance</span> </div> </a> <ul id="toc-Permutation_importance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mean_decrease_in_impurity_feature_importance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mean_decrease_in_impurity_feature_importance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Mean decrease in impurity feature importance</span> </div> </a> <ul id="toc-Mean_decrease_in_impurity_feature_importance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relationship_to_nearest_neighbors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_nearest_neighbors"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Relationship to nearest neighbors</span> </div> </a> <ul id="toc-Relationship_to_nearest_neighbors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Unsupervised_learning" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Unsupervised_learning"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Unsupervised learning</span> </div> </a> <ul id="toc-Unsupervised_learning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variants" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Variants"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Variants</span> </div> </a> <ul id="toc-Variants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kernel_random_forest" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Kernel_random_forest"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Kernel random forest</span> </div> </a> <button aria-controls="toc-Kernel_random_forest-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 Kernel random forest subsection</span> </button> <ul id="toc-Kernel_random_forest-sublist" class="vector-toc-list"> <li id="toc-History_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>History</span> </div> </a> <ul id="toc-History_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notations_and_definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notations_and_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Notations and definitions</span> </div> </a> <ul id="toc-Notations_and_definitions-sublist" class="vector-toc-list"> <li id="toc-Preliminaries:_Centered_forests" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Preliminaries:_Centered_forests"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Preliminaries: Centered forests</span> </div> </a> <ul id="toc-Preliminaries:_Centered_forests-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniform_forest" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Uniform_forest"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.2</span> <span>Uniform forest</span> </div> </a> <ul id="toc-Uniform_forest-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-From_random_forest_to_KeRF" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#From_random_forest_to_KeRF"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.3</span> <span>From random forest to KeRF</span> </div> </a> <ul id="toc-From_random_forest_to_KeRF-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Centered_KeRF" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Centered_KeRF"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.4</span> <span>Centered KeRF</span> </div> </a> <ul id="toc-Centered_KeRF-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniform_KeRF" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Uniform_KeRF"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.5</span> <span>Uniform KeRF</span> </div> </a> <ul id="toc-Uniform_KeRF-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Properties</span> </div> </a> <ul id="toc-Properties_2-sublist" class="vector-toc-list"> <li id="toc-Relation_between_KeRF_and_random_forest" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relation_between_KeRF_and_random_forest"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3.1</span> <span>Relation between KeRF and random forest</span> </div> </a> <ul id="toc-Relation_between_KeRF_and_random_forest-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_between_infinite_KeRF_and_infinite_random_forest" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relation_between_infinite_KeRF_and_infinite_random_forest"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3.2</span> <span>Relation between infinite KeRF and infinite random forest</span> </div> </a> <ul id="toc-Relation_between_infinite_KeRF_and_infinite_random_forest-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Consistency_results" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Consistency_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Consistency results</span> </div> </a> <ul id="toc-Consistency_results-sublist" class="vector-toc-list"> <li id="toc-Consistency_of_centered_KeRF" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Consistency_of_centered_KeRF"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.1</span> <span>Consistency of centered KeRF</span> </div> </a> <ul id="toc-Consistency_of_centered_KeRF-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Consistency_of_uniform_KeRF" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Consistency_of_uniform_KeRF"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.2</span> <span>Consistency of uniform KeRF</span> </div> </a> <ul id="toc-Consistency_of_uniform_KeRF-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Disadvantages" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Disadvantages"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Disadvantages</span> </div> </a> <ul id="toc-Disadvantages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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Available in 25 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-25" 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">25 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%BA%D8%A7%D8%A8%D8%A9_%D8%B9%D8%B4%D9%88%D8%A7%D8%A6%D9%8A%D8%A9" title="غابة عشوائية – Arabic" lang="ar" hreflang="ar" data-title="غابة عشوائية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%B0%D0%B4_%D0%B2%D1%8B%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B0%D0%B3%D0%B0_%D0%BB%D0%B5%D1%81%D1%83" title="Метад выпадковага лесу – Belarusian" lang="be" hreflang="be" data-title="Метад выпадковага лесу" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Random_forest" title="Random forest – Catalan" lang="ca" hreflang="ca" data-title="Random forest" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/N%C3%A1hodn%C3%BD_les" title="Náhodný les – Czech" lang="cs" hreflang="cs" data-title="Náhodný les" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Random_Forest" title="Random Forest – German" lang="de" hreflang="de" data-title="Random Forest" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Otsustusmets" title="Otsustusmets – Estonian" lang="et" hreflang="et" data-title="Otsustusmets" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Random_forest" title="Random forest – Spanish" lang="es" hreflang="es" data-title="Random forest" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D9%86%DA%AF%D9%84_%D8%AA%D8%B5%D8%A7%D8%AF%D9%81%DB%8C" title="جنگل تصادفی – Persian" lang="fa" hreflang="fa" data-title="جنگل تصادفی" 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/For%C3%AAt_d%27arbres_d%C3%A9cisionnels" title="Forêt d'arbres décisionnels – French" lang="fr" hreflang="fr" data-title="Forêt d'arbres décisionnels" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Random_Forest" title="Random Forest – Galician" lang="gl" hreflang="gl" data-title="Random Forest" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%9E%9C%EB%8D%A4_%ED%8F%AC%EB%A0%88%EC%8A%A4%ED%8A%B8" title="랜덤 포레스트 – Korean" lang="ko" hreflang="ko" data-title="랜덤 포레스트" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Random_forest" title="Random forest – Indonesian" lang="id" hreflang="id" data-title="Random forest" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Foresta_casuale" title="Foresta casuale – Italian" lang="it" hreflang="it" data-title="Foresta casuale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%A2%D7%A8_%D7%90%D7%A7%D7%A8%D7%90%D7%99" title="יער אקראי – Hebrew" lang="he" hreflang="he" data-title="יער אקראי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%B3%E3%83%80%E3%83%A0%E3%83%95%E3%82%A9%E3%83%AC%E3%82%B9%E3%83%88" title="ランダムフォレスト – Japanese" lang="ja" hreflang="ja" data-title="ランダムフォレスト" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tilfeldig_skog" title="Tilfeldig skog – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tilfeldig skog" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Las_losowy" title="Las losowy – Polish" lang="pl" hreflang="pl" data-title="Las losowy" 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/Floresta_aleat%C3%B3ria" title="Floresta aleatória – Portuguese" lang="pt" hreflang="pt" data-title="Floresta aleatória" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BB%D0%B5%D1%81%D0%B0" title="Метод случайного леса – Russian" lang="ru" hreflang="ru" data-title="Метод случайного леса" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Random_forest" title="Random forest – Simple English" lang="en-simple" hreflang="en-simple" data-title="Random forest" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AF%D8%A7%D8%B1%D8%B3%D8%AA%D8%A7%D9%86%DB%8C_%D8%A8%DB%95%DA%95%DB%8E%DA%A9%DB%95%D9%88%D8%AA" title="دارستانی بەڕێکەوت – Central Kurdish" lang="ckb" hreflang="ckb" data-title="دارستانی بەڕێکەوت" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Rastgele_orman" title="Rastgele orman – Turkish" lang="tr" hreflang="tr" data-title="Rastgele orman" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/Random_forest" title="Random forest – Ukrainian" lang="uk" hreflang="uk" data-title="Random forest" 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/%E9%9A%A8%E6%A9%9F%E6%A3%AE%E6%9E%97" 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.mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488" /><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a><br />and <a href="/wiki/Data_mining" title="Data mining">data mining</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Paradigms</div><div class="sidebar-list-content mw-collapsible-content hlist"> <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 href="/wiki/Decision_tree_learning" title="Decision tree learning">Decision trees</a></li> <li><a href="/wiki/Ensemble_learning" title="Ensemble learning">Ensembles</a> <ul><li><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bagging</a></li> <li><a href="/wiki/Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a></li> <li><a class="mw-selflink selflink">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> <li><a href="/wiki/Topological_deep_learning" title="Topological deep learning">Topological deep learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Journals and conferences</div><div class="sidebar-list-content mw-collapsible-content hlist"> <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>Random forests</b> or <b>random decision forests</b> is an <a href="/wiki/Ensemble_learning" title="Ensemble learning">ensemble learning</a> method for <a href="/wiki/Statistical_classification" title="Statistical classification">classification</a>, <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> and other tasks that works by creating a multitude of <a href="/wiki/Decision_tree_learning" title="Decision tree learning">decision trees</a> during training. For classification tasks, the output of the random forest is the class selected by most trees. For regression tasks, the output is the average of the predictions of the trees.<sup id="cite_ref-ho1995_1-0" class="reference"><a href="#cite_note-ho1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ho1998_2-0" class="reference"><a href="#cite_note-ho1998-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Random forests correct for decision trees' habit of <a href="/wiki/Overfitting" title="Overfitting">overfitting</a> to their <a href="/wiki/Test_set" class="mw-redirect" title="Test set">training set</a>.<sup id="cite_ref-elemstatlearn_3-0" class="reference"><a href="#cite_note-elemstatlearn-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 587–588">: 587–588 </span></sup> </p><p>The first algorithm for random decision forests was created in 1995 by <a href="/wiki/Tin_Kam_Ho" title="Tin Kam Ho">Tin Kam Ho</a><sup id="cite_ref-ho1995_1-1" class="reference"><a href="#cite_note-ho1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> using the <a href="/wiki/Random_subspace_method" title="Random subspace method">random subspace method</a>,<sup id="cite_ref-ho1998_2-1" class="reference"><a href="#cite_note-ho1998-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> which, in Ho's formulation, is a way to implement the "stochastic discrimination" approach to classification proposed by Eugene Kleinberg.<sup id="cite_ref-kleinberg1990_4-0" class="reference"><a href="#cite_note-kleinberg1990-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kleinberg1996_5-0" class="reference"><a href="#cite_note-kleinberg1996-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kleinberg2000_6-0" class="reference"><a href="#cite_note-kleinberg2000-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>An extension of the algorithm was developed by <a href="/wiki/Leo_Breiman" title="Leo Breiman">Leo Breiman</a><sup id="cite_ref-breiman2001_7-0" class="reference"><a href="#cite_note-breiman2001-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Adele_Cutler" title="Adele Cutler">Adele Cutler</a>,<sup id="cite_ref-rpackage_8-0" class="reference"><a href="#cite_note-rpackage-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> who registered<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> "Random Forests" as a <a href="/wiki/Trademark" title="Trademark">trademark</a> in 2006 (as of 2019<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Random_forest&action=edit">[update]</a></sup>, owned by <a href="/wiki/Minitab" title="Minitab">Minitab, Inc.</a>).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The extension combines Breiman's "<a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">bagging</a>" idea and random selection of features, introduced first by Ho<sup id="cite_ref-ho1995_1-2" class="reference"><a href="#cite_note-ho1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and later independently by Amit and <a href="/wiki/Donald_Geman" title="Donald Geman">Geman</a><sup id="cite_ref-amitgeman1997_11-0" class="reference"><a href="#cite_note-amitgeman1997-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> in order to construct a collection of decision trees with controlled variance. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The general method of random decision forests was first proposed by Salzberg and Heath in 1993,<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> with a method that used a randomized decision tree algorithm to create multiple trees and then combine them using majority voting. This idea was developed further by Ho in 1995.<sup id="cite_ref-ho1995_1-3" class="reference"><a href="#cite_note-ho1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Ho established that forests of trees splitting with oblique hyperplanes can gain accuracy as they grow without suffering from overtraining, as long as the forests are randomly restricted to be sensitive to only selected <a href="/wiki/Feature_(machine_learning)" title="Feature (machine learning)">feature</a> dimensions. A subsequent work along the same lines<sup id="cite_ref-ho1998_2-2" class="reference"><a href="#cite_note-ho1998-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> concluded that other splitting methods behave similarly, as long as they are randomly forced to be insensitive to some feature dimensions. This observation that a more complex classifier (a larger forest) gets more accurate nearly monotonically is in sharp contrast to the common belief that the complexity of a classifier can only grow to a certain level of accuracy before being hurt by overfitting. The explanation of the forest method's resistance to overtraining can be found in Kleinberg's theory of stochastic discrimination.<sup id="cite_ref-kleinberg1990_4-1" class="reference"><a href="#cite_note-kleinberg1990-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kleinberg1996_5-1" class="reference"><a href="#cite_note-kleinberg1996-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kleinberg2000_6-1" class="reference"><a href="#cite_note-kleinberg2000-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The early development of Breiman's notion of random forests was influenced by the work of Amit and Geman<sup id="cite_ref-amitgeman1997_11-1" class="reference"><a href="#cite_note-amitgeman1997-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> who introduced the idea of searching over a random subset of the available decisions when splitting a node, in the context of growing a single <a href="/wiki/Decision_tree" title="Decision tree">tree</a>. The idea of random subspace selection from Ho<sup id="cite_ref-ho1998_2-3" class="reference"><a href="#cite_note-ho1998-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> was also influential in the design of random forests. This method grows a forest of trees, and introduces variation among the trees by projecting the training data into a randomly chosen <a href="/wiki/Linear_subspace" title="Linear subspace">subspace</a> before fitting each tree or each node. Finally, the idea of randomized node optimization, where the decision at each node is selected by a randomized procedure, rather than a deterministic optimization was first introduced by <a href="/wiki/Thomas_G._Dietterich" title="Thomas G. Dietterich">Thomas G. Dietterich</a>.<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> </p><p>The proper introduction of random forests was made in a paper by <a href="/wiki/Leo_Breiman" title="Leo Breiman">Leo Breiman</a>.<sup id="cite_ref-breiman2001_7-1" class="reference"><a href="#cite_note-breiman2001-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This paper describes a method of building a forest of uncorrelated trees using a <a href="/wiki/Classification_and_regression_tree" class="mw-redirect" title="Classification and regression tree">CART</a> like procedure, combined with randomized node optimization and <a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">bagging</a>. In addition, this paper combines several ingredients, some previously known and some novel, which form the basis of the modern practice of random forests, in particular: </p> <ol><li>Using <a href="/wiki/Out-of-bag_error" title="Out-of-bag error">out-of-bag error</a> as an estimate of the <a href="/wiki/Generalization_error" title="Generalization error">generalization error</a>.</li> <li>Measuring variable importance through permutation.</li></ol> <p>The report also offers the first theoretical result for random forests in the form of a bound on the <a href="/wiki/Generalization_error" title="Generalization error">generalization error</a> which depends on the strength of the trees in the forest and their <a href="/wiki/Correlation" title="Correlation">correlation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=2" title="Edit section: Algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Preliminaries:_decision_tree_learning">Preliminaries: decision tree learning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=3" title="Edit section: Preliminaries: decision tree learning"><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/Decision_tree_learning" title="Decision tree learning">Decision tree learning</a></div> <p>Decision trees are a popular method for various machine learning tasks. Tree learning is almost "an off-the-shelf procedure for data mining", say <a href="/wiki/Trevor_Hastie" title="Trevor Hastie">Hastie</a> <i>et al.</i>, "because it is invariant under scaling and various other transformations of feature values, is robust to inclusion of irrelevant features, and produces inspectable models. However, they are seldom accurate".<sup id="cite_ref-elemstatlearn_3-1" class="reference"><a href="#cite_note-elemstatlearn-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 352">: 352 </span></sup> </p><p>In particular, trees that are grown very deep tend to learn highly irregular patterns: they <a href="/wiki/Overfitting" title="Overfitting">overfit</a> their training sets, i.e. have <a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">low bias, but very high variance</a>. Random forests are a way of averaging multiple deep decision trees, trained on different parts of the same training set, with the goal of reducing the variance.<sup id="cite_ref-elemstatlearn_3-2" class="reference"><a href="#cite_note-elemstatlearn-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 587–588">: 587–588 </span></sup> This comes at the expense of a small increase in the bias and some loss of interpretability, but generally greatly boosts the performance in the final model. </p> <div class="mw-heading mw-heading3"><h3 id="Bagging">Bagging</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=4" title="Edit section: Bagging"><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/Bootstrap_aggregating" title="Bootstrap aggregating">Bootstrap aggregating</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Random_Forest_Bagging_Illustration.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Random_Forest_Bagging_Illustration.png/220px-Random_Forest_Bagging_Illustration.png" decoding="async" width="220" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Random_Forest_Bagging_Illustration.png/330px-Random_Forest_Bagging_Illustration.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Random_Forest_Bagging_Illustration.png/440px-Random_Forest_Bagging_Illustration.png 2x" data-file-width="1138" data-file-height="932" /></a><figcaption>Illustration of training a Random Forest model. The training dataset (in this case, of 250 rows and 100 columns) is randomly sampled with replacement <i>n</i> times. Then, a decision tree is trained on each sample. Finally, for prediction, the results of all <i>n</i> trees are aggregated to produce a final decision.</figcaption></figure> <p>The training algorithm for random forests applies the general technique of <a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">bootstrap aggregating</a>, or bagging, to tree learners. Given a training set <span class="texhtml mvar" style="font-style:italic;">X</span> = <span class="texhtml mvar" style="font-style:italic;">x<sub>1</sub></span>, ..., <span class="texhtml mvar" style="font-style:italic;">x<sub>n</sub></span> with responses <span class="texhtml mvar" style="font-style:italic;">Y</span> = <span class="texhtml mvar" style="font-style:italic;">y<sub>1</sub></span>, ..., <span class="texhtml mvar" style="font-style:italic;">y<sub>n</sub></span>, bagging repeatedly (<i>B</i> times) selects a <a href="/wiki/Sampling_(statistics)#Replacement_of_selected_units" title="Sampling (statistics)">random sample with replacement</a> of the training set and fits trees to these samples: </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;">For <span class="texhtml mvar" style="font-style:italic;">b</span> = 1, ..., <span class="texhtml mvar" style="font-style:italic;">B</span>: <ol><li>Sample, with replacement, <span class="texhtml mvar" style="font-style:italic;">n</span> training examples from <span class="texhtml mvar" style="font-style:italic;">X</span>, <span class="texhtml mvar" style="font-style:italic;">Y</span>; call these <span class="texhtml mvar" style="font-style:italic;">X<sub>b</sub></span>, <span class="texhtml mvar" style="font-style:italic;">Y<sub>b</sub></span>.</li> <li>Train a classification or regression tree <span class="texhtml mvar" style="font-style:italic;">f<sub>b</sub></span> on <span class="texhtml mvar" style="font-style:italic;">X<sub>b</sub></span>, <span class="texhtml mvar" style="font-style:italic;">Y<sub>b</sub></span>.</li></ol></div> <p>After training, predictions for unseen samples <span class="texhtml mvar" style="font-style:italic;">x'</span> can be made by averaging the predictions from all the individual regression trees on <span class="texhtml mvar" style="font-style:italic;">x'</span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}={\frac {1}{B}}\sum _{b=1}^{B}f_{b}(x')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>B</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}={\frac {1}{B}}\sum _{b=1}^{B}f_{b}(x')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b54befce12aefdb29442bfc71cb5ad452364e8d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.427ex; height:7.343ex;" alt="{\displaystyle {\hat {f}}={\frac {1}{B}}\sum _{b=1}^{B}f_{b}(x')}" /></span> </p><p>or by taking the plurality vote in the case of classification trees. </p><p>This bootstrapping procedure leads to better model performance because it decreases the <a href="/wiki/Bias%E2%80%93variance_dilemma" class="mw-redirect" title="Bias–variance dilemma">variance</a> of the model, without increasing the bias. This means that while the predictions of a single tree are highly sensitive to noise in its training set, the average of many trees is not, as long as the trees are not correlated. Simply training many trees on a single training set would give strongly correlated trees (or even the same tree many times, if the training algorithm is deterministic); bootstrap sampling is a way of de-correlating the trees by showing them different training sets. </p><p>Additionally, an estimate of the uncertainty of the prediction can be made as the standard deviation of the predictions from all the individual regression trees on <span class="texhtml mvar" style="font-style:italic;">x′</span>: <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 \sigma ={\sqrt {\frac {\sum _{b=1}^{B}(f_{b}(x')-{\hat {f}})^{2}}{B-1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>B</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\sqrt {\frac {\sum _{b=1}^{B}(f_{b}(x')-{\hat {f}})^{2}}{B-1}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e64a440f8625c492d848b38785f833a5882432" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.03ex; height:7.676ex;" alt="{\displaystyle \sigma ={\sqrt {\frac {\sum _{b=1}^{B}(f_{b}(x')-{\hat {f}})^{2}}{B-1}}}.}" /></span> </p><p>The number <span class="texhtml mvar" style="font-style:italic;">B</span> of samples (equivalently, of trees) is a free parameter. Typically, a few hundred to several thousand trees are used, depending on the size and nature of the training set. <span class="texhtml mvar" style="font-style:italic;">B</span> can be optimized using <a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">cross-validation</a>, or by observing the <i><a href="/wiki/Out-of-bag_error" title="Out-of-bag error">out-of-bag error</a></i>: the mean prediction error on each training sample <span class="texhtml mvar" style="font-style:italic;">x<sub>i</sub></span>, using only the trees that did not have <span class="texhtml mvar" style="font-style:italic;">x<sub>i</sub></span> in their bootstrap sample.<sup id="cite_ref-islr_14-0" class="reference"><a href="#cite_note-islr-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>The training and test error tend to level off after some number of trees have been fit. </p> <div class="mw-heading mw-heading3"><h3 id="From_bagging_to_random_forests">From bagging to random forests</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=5" title="Edit section: From bagging to random forests"><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/Random_subspace_method" title="Random subspace method">Random subspace method</a></div> <p>The above procedure describes the original bagging algorithm for trees. Random forests also include another type of bagging scheme: they use a modified tree learning algorithm that selects, at each candidate split in the learning process, a <a href="/wiki/Random_subspace_method" title="Random subspace method">random subset of the features</a>. This process is sometimes called "feature bagging". The reason for doing this is the correlation of the trees in an ordinary bootstrap sample: if one or a few <a href="/wiki/Feature_(machine_learning)" title="Feature (machine learning)">features</a> are very strong predictors for the response variable (target output), these features will be selected in many of the <span class="texhtml mvar" style="font-style:italic;">B</span> trees, causing them to become correlated. An analysis of how bagging and random subspace projection contribute to accuracy gains under different conditions is given by Ho.<sup id="cite_ref-ho2002_15-0" class="reference"><a href="#cite_note-ho2002-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Typically, for a classification problem with <span class="texhtml mvar" style="font-style:italic;">p</span> features, <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><span class="texhtml mvar" style="font-style:italic;">p</span></span></span> (rounded down) features are used in each split.<sup id="cite_ref-elemstatlearn_3-3" class="reference"><a href="#cite_note-elemstatlearn-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 592">: 592 </span></sup> For regression problems the inventors recommend <span class="texhtml"><i>p</i>/3</span> (rounded down) with a minimum node size of 5 as the default.<sup id="cite_ref-elemstatlearn_3-4" class="reference"><a href="#cite_note-elemstatlearn-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 592">: 592 </span></sup> In practice, the best values for these parameters should be tuned on a case-to-case basis for every problem.<sup id="cite_ref-elemstatlearn_3-5" class="reference"><a href="#cite_note-elemstatlearn-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 592">: 592 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="ExtraTrees">ExtraTrees</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=6" title="Edit section: ExtraTrees"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Adding one further step of randomization yields <i>extremely randomized trees</i>, or ExtraTrees. As with ordinary random forests, they are an ensemble of individual trees, but there are two main differences: (1) each tree is trained using the whole learning sample (rather than a bootstrap sample), and (2) the top-down splitting is randomized: for each feature under consideration, a number of <i>random</i> cut-points are selected, instead of computing the locally <i>optimal</i> cut-point (based on, e.g., <a href="/wiki/Information_gain" class="mw-redirect" title="Information gain">information gain</a> or the <a href="/wiki/Gini_impurity" class="mw-redirect" title="Gini impurity">Gini impurity</a>). The values are chosen from a uniform distribution within the feature's empirical range (in the tree's training set). Then, of all the randomly chosen splits, the split that yields the highest score is chosen to split the node. </p><p>Similar to ordinary random forests, the number of randomly selected features to be considered at each node can be specified. Default values for this parameter are <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 {\sqrt {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>p</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0527785cd1ad7fa60789e172c720affdcdb28b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.105ex; height:3.009ex;" alt="{\displaystyle {\sqrt {p}}}" /></span> for classification 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span> for regression, 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span> is the number of features in the model.<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> <div class="mw-heading mw-heading3"><h3 id="Random_forests_for_high-dimensional_data">Random forests for high-dimensional data</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=7" title="Edit section: Random forests for high-dimensional data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The basic random forest procedure may not work well in situations where there are a large number of features but only a small proportion of these features are informative with respect to sample classification. This can be addressed by encouraging the procedure to focus mainly on features and trees that are informative. Some methods for accomplishing this are: </p> <ul><li>Prefiltering: Eliminate features that are mostly just noise.<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><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></li> <li>Enriched random forest (ERF): Use weighted random sampling instead of simple random sampling at each node of each tree, giving greater weight to features that appear to be more informative.<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>Tree-weighted random forest (TWRF): Give more weight to more accurate trees.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=8" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Variable_importance">Variable importance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=9" title="Edit section: Variable importance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Random forests can be used to rank the importance of variables in a regression or classification problem in a natural way. The following technique was described in Breiman's original paper<sup id="cite_ref-breiman2001_7-2" class="reference"><a href="#cite_note-breiman2001-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and is implemented in the <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> package <code>randomForest</code>.<sup id="cite_ref-rpackage_8-1" class="reference"><a href="#cite_note-rpackage-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Permutation_importance">Permutation importance</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=10" title="Edit section: Permutation importance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To measure a feature's importance in a data set <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 {\mathcal {D}}_{n}=\{(X_{i},Y_{i})\}_{i=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{n}=\{(X_{i},Y_{i})\}_{i=1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee98824c21c5539b16d0c560a1445101f74898ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.051ex; height:3.009ex;" alt="{\displaystyle {\mathcal {D}}_{n}=\{(X_{i},Y_{i})\}_{i=1}^{n}}" /></span>, first a random forest is trained on the data. During training, the <a href="/wiki/Out-of-bag_error" title="Out-of-bag error">out-of-bag error</a> for each data point is recorded and averaged over the forest. (If bagging is not used during training, we can instead compute errors on an independent test set.) </p><p>After training, the values of the feature are permuted in the out-of-bag samples and the out-of-bag error is again computed on this perturbed data set. The importance for the feature is computed by averaging the difference in out-of-bag error before and after the permutation over all trees. The score is normalized by the standard deviation of these differences. </p><p>Features which produce large values for this score are ranked as more important than features which produce small values. The statistical definition of the variable importance measure was given and analyzed by Zhu <i>et al.</i><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>This method of determining variable importance has some drawbacks: </p> <ul><li>When features have different numbers of values, random forests favor features with more values. Solutions to this problem include <a href="/wiki/Partial_permutation" title="Partial permutation">partial permutations</a><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:02_26-0" class="reference"><a href="#cite_note-:02-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and growing unbiased trees.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></li> <li>If the data contain groups of correlated features of similar relevance, then smaller groups are favored over large groups.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li> <li>If there are collinear features, the procedure may fail to identify important features. A solution is to permute groups of correlated features together.<sup id="cite_ref-:2_30-0" class="reference"><a href="#cite_note-:2-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading4"><h4 id="Mean_decrease_in_impurity_feature_importance">Mean decrease in impurity feature importance</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=11" title="Edit section: Mean decrease in impurity feature importance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This approach to feature importance for random forests considers as important the variables which decrease a lot the impurity during splitting.<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 is described in the book <i>Classification and Regression Trees</i> by Leo Breiman<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> and is the default implementation in <a href="/wiki/Scikit-learn" title="Scikit-learn"><code>sci-kit learn</code></a> and <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>. The definition is:<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 {\text{unormalized average importance}}(x)={\frac {1}{n_{T}}}\sum _{i=1}^{n_{T}}\sum _{{\text{node }}j\in T_{i}|{\text{split variable}}(j)=x}p_{T_{i}}(j)\Delta i_{T_{i}}(j),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>unormalized average importance</mtext> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </munderover> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>node </mtext> </mrow> <mi>j</mi> <mo>∈</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>split variable</mtext> </mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{unormalized average importance}}(x)={\frac {1}{n_{T}}}\sum _{i=1}^{n_{T}}\sum _{{\text{node }}j\in T_{i}|{\text{split variable}}(j)=x}p_{T_{i}}(j)\Delta i_{T_{i}}(j),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5b9e74f891d6049b82bc70ad12aa30cfb554ce0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:82.096ex; height:7.509ex;" alt="{\displaystyle {\text{unormalized average importance}}(x)={\frac {1}{n_{T}}}\sum _{i=1}^{n_{T}}\sum _{{\text{node }}j\in T_{i}|{\text{split variable}}(j)=x}p_{T_{i}}(j)\Delta i_{T_{i}}(j),}" /></span>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> is a feature</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1972fb23b45426057388210f0927fb46d6ff64ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.784ex; height:2.009ex;" alt="{\displaystyle n_{T}}" /></span> is the number of trees in the forest</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8dd1c50cb9436474f83624c3f679ccf3eebbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.157ex; height:2.509ex;" alt="{\displaystyle T_{i}}" /></span> is tree <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></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{T_{i}}(j)={\frac {n_{j}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{T_{i}}(j)={\frac {n_{j}}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9b544909fc0fbe2a859aef500749359f53ec41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:12.082ex; height:5.009ex;" alt="{\displaystyle p_{T_{i}}(j)={\frac {n_{j}}{n}}}" /></span> is the fraction of samples reaching 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 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></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta i_{T_{i}}(j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta i_{T_{i}}(j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9766461fceef705ca580069d807ece0cff9b89f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.323ex; height:3.009ex;" alt="{\displaystyle \Delta i_{T_{i}}(j)}" /></span> is the change in impurity in tree <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> 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 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>.</li></ul> <p>As impurity measure for samples falling in a node e.g. the following statistics can be used: </p> <ul><li><a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li> <li><a href="/wiki/Gini_coefficient" title="Gini coefficient">Gini coefficient</a></li> <li><a href="/wiki/Mean_squared_error" title="Mean squared error">Mean squared error</a></li></ul> <p>The normalized importance is then obtained by normalizing over all features, so that the sum of normalized feature importances is 1. </p><p>The <code>sci-kit learn</code> default implementation can report misleading feature importance:<sup id="cite_ref-:2_30-1" class="reference"><a href="#cite_note-:2-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <ul><li>it favors high cardinality features</li> <li>it uses training statistics and so does not reflect a feature's usefulness for predictions on a test set<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></li></ul> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_nearest_neighbors">Relationship to nearest neighbors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=12" title="Edit section: Relationship to nearest neighbors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A relationship between random forests and the <a href="/wiki/K-nearest_neighbor_algorithm" class="mw-redirect" title="K-nearest neighbor algorithm"><span class="texhtml mvar" style="font-style:italic;">k</span>-nearest neighbor algorithm</a> (<span class="texhtml mvar" style="font-style:italic;">k</span>-NN) was pointed out by Lin and Jeon in 2002.<sup id="cite_ref-linjeon02_34-0" class="reference"><a href="#cite_note-linjeon02-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Both can be viewed as so-called <i>weighted neighborhoods schemes</i>. These are models built from a training set <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_{i},y_{i})\}_{i=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(x_{i},y_{i})\}_{i=1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a9e0a59cd73d8fb11a10caa787512bf93af04b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.137ex; height:3.009ex;" alt="{\displaystyle \{(x_{i},y_{i})\}_{i=1}^{n}}" /></span> that make predictions <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 {\hat {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc8de3d8ea01304329ef9518fad7a6d196c4c01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\hat {y}}}" /></span> for new points <span class="texhtml mvar" style="font-style:italic;">x'</span> by looking at the "neighborhood" of the point, formalized by a weight function <span class="texhtml mvar" style="font-style:italic;">W</span>:<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 {\hat {y}}=\sum _{i=1}^{n}W(x_{i},x')\,y_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </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>n</mi> </mrow> </munderover> <mi>W</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {y}}=\sum _{i=1}^{n}W(x_{i},x')\,y_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9cd87e3168f0200bd67d04530ab9124dcb8cafc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.538ex; height:6.843ex;" alt="{\displaystyle {\hat {y}}=\sum _{i=1}^{n}W(x_{i},x')\,y_{i}.}" /></span>Here, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(x_{i},x')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(x_{i},x')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87df356c5a2445516fca56d2df6eb73e64e48ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.422ex; height:3.009ex;" alt="{\displaystyle W(x_{i},x')}" /></span> is the non-negative weight of the <span class="texhtml mvar" style="font-style:italic;">i</span>'th training point relative to the new point <span class="texhtml mvar" style="font-style:italic;">x'</span> in the same tree. For any <span class="texhtml mvar" style="font-style:italic;">x'</span>, the weights for points <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" /></span> must sum to 1. Weight functions are as follows: </p> <ul><li>In <span class="texhtml mvar" style="font-style:italic;">k</span>-NN, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(x_{i},x')={\frac {1}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(x_{i},x')={\frac {1}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3098adcbde36fd1253a0ee85a3868d67b1861f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.568ex; height:5.343ex;" alt="{\displaystyle W(x_{i},x')={\frac {1}{k}}}" /></span> if <span class="texhtml mvar" style="font-style:italic;">x<sub>i</sub></span> is one of the <span class="texhtml mvar" style="font-style:italic;">k</span> points closest to <span class="texhtml mvar" style="font-style:italic;">x'</span>, and zero otherwise.</li> <li>In a tree, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(x_{i},x')={\frac {1}{k'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mo>′</mo> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(x_{i},x')={\frac {1}{k'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d80899c24f8d4488f56290ae99cfe61558667873" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.253ex; height:5.343ex;" alt="{\displaystyle W(x_{i},x')={\frac {1}{k'}}}" /></span> if <span class="texhtml mvar" style="font-style:italic;">x<sub>i</sub></span> is one of the <span class="texhtml mvar" style="font-style:italic;">k'</span> points in the same leaf as <span class="texhtml mvar" style="font-style:italic;">x'</span>, and zero otherwise.</li></ul> <p>Since a forest averages the predictions of a set of <span class="texhtml mvar" style="font-style:italic;">m</span> trees with individual weight functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa98874e6beb16373e8d0e056ba550cf653676a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.103ex; height:2.843ex;" alt="{\displaystyle W_{j}}" /></span>, its predictions are<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 {\hat {y}}={\frac {1}{m}}\sum _{j=1}^{m}\sum _{i=1}^{n}W_{j}(x_{i},x')\,y_{i}=\sum _{i=1}^{n}\left({\frac {1}{m}}\sum _{j=1}^{m}W_{j}(x_{i},x')\right)\,y_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>y</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>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {y}}={\frac {1}{m}}\sum _{j=1}^{m}\sum _{i=1}^{n}W_{j}(x_{i},x')\,y_{i}=\sum _{i=1}^{n}\left({\frac {1}{m}}\sum _{j=1}^{m}W_{j}(x_{i},x')\right)\,y_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b819eff4f3bb5ee472825d9996c3fd5f6b18ed7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:58.543ex; height:7.676ex;" alt="{\displaystyle {\hat {y}}={\frac {1}{m}}\sum _{j=1}^{m}\sum _{i=1}^{n}W_{j}(x_{i},x')\,y_{i}=\sum _{i=1}^{n}\left({\frac {1}{m}}\sum _{j=1}^{m}W_{j}(x_{i},x')\right)\,y_{i}.}" /></span> </p><p>This shows that the whole forest is again a weighted neighborhood scheme, with weights that average those of the individual trees. The neighbors of <span class="texhtml mvar" style="font-style:italic;">x'</span> in this interpretation are the points <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" /></span> sharing the same leaf in any tree <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>. In this way, the neighborhood of <span class="texhtml mvar" style="font-style:italic;">x'</span> depends in a complex way on the structure of the trees, and thus on the structure of the training set. Lin and Jeon show that the shape of the neighborhood used by a random forest adapts to the local importance of each feature.<sup id="cite_ref-linjeon02_34-1" class="reference"><a href="#cite_note-linjeon02-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Unsupervised_learning">Unsupervised learning</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=13" title="Edit section: Unsupervised learning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As part of their construction, random forest predictors naturally lead to a dissimilarity measure among observations. One can analogously define dissimilarity between unlabeled data, by training a forest to distinguish original "observed" data from suitably generated synthetic data drawn from a reference distribution.<sup id="cite_ref-breiman2001_7-3" class="reference"><a href="#cite_note-breiman2001-7"><span class="cite-bracket">[</span>7<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> A random forest dissimilarity is attractive because it handles mixed variable types very well, is invariant to monotonic transformations of the input variables, and is robust to outlying observations. Random forest dissimilarity easily deals with a large number of semi-continuous variables due to its intrinsic variable selection; for example, the "Addcl 1" random forest dissimilarity weighs the contribution of each variable according to how dependent it is on other variables. Random forest dissimilarity has been used in a variety of applications, e.g. to find clusters of patients based on tissue marker data.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Variants">Variants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=14" title="Edit section: Variants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Instead of decision trees, linear models have been proposed and evaluated as base estimators in random forests, in particular <a href="/wiki/Multinomial_logistic_regression" title="Multinomial logistic regression">multinomial logistic regression</a> and <a href="/wiki/Naive_Bayes_classifier" title="Naive Bayes classifier">naive Bayes classifiers</a>.<sup id="cite_ref-:0_37-0" class="reference"><a href="#cite_note-:0-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><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> In cases that the relationship between the predictors and the target variable is linear, the base learners may have an equally high accuracy as the ensemble learner.<sup id="cite_ref-:1_40-0" class="reference"><a href="#cite_note-:1-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_37-1" class="reference"><a href="#cite_note-:0-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Kernel_random_forest">Kernel random forest</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=15" title="Edit section: Kernel random forest"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In machine learning, kernel random forests (KeRF) establish the connection between random forests and <a href="/wiki/Kernel_method" title="Kernel method">kernel methods</a>. By slightly modifying their definition, random forests can be rewritten as <a href="/wiki/Kernel_method" title="Kernel method">kernel methods</a>, which are more interpretable and easier to analyze.<sup id="cite_ref-scornet2015random_41-0" class="reference"><a href="#cite_note-scornet2015random-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="History_2">History</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=16" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Leo_Breiman" title="Leo Breiman">Leo Breiman</a><sup id="cite_ref-breiman2000some_42-0" class="reference"><a href="#cite_note-breiman2000some-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> was the first person to notice the link between random forest and <a href="/wiki/Kernel_methods" class="mw-redirect" title="Kernel methods">kernel methods</a>. He pointed out that random forests trained using <a href="/wiki/I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> random vectors in the tree construction are equivalent to a kernel acting on the true margin. Lin and Jeon<sup id="cite_ref-lin2006random_43-0" class="reference"><a href="#cite_note-lin2006random-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> established the connection between random forests and adaptive nearest neighbor, implying that random forests can be seen as adaptive kernel estimates. Davies and Ghahramani<sup id="cite_ref-davies2014random_44-0" class="reference"><a href="#cite_note-davies2014random-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> proposed Kernel Random Forest (KeRF) and showed that it can empirically outperform state-of-art kernel methods. Scornet<sup id="cite_ref-scornet2015random_41-1" class="reference"><a href="#cite_note-scornet2015random-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> first defined KeRF estimates and gave the explicit link between KeRF estimates and random forest. He also gave explicit expressions for kernels based on centered random forest<sup id="cite_ref-breiman2004consistency_45-0" class="reference"><a href="#cite_note-breiman2004consistency-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> and uniform random forest,<sup id="cite_ref-arlot2014analysis_46-0" class="reference"><a href="#cite_note-arlot2014analysis-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> two simplified models of random forest. He named these two KeRFs Centered KeRF and Uniform KeRF, and proved upper bounds on their rates of consistency. </p> <div class="mw-heading mw-heading3"><h3 id="Notations_and_definitions">Notations and definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=17" title="Edit section: Notations and definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Preliminaries:_Centered_forests">Preliminaries: Centered forests</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=18" title="Edit section: Preliminaries: Centered forests"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Centered forest<sup id="cite_ref-breiman2004consistency_45-1" class="reference"><a href="#cite_note-breiman2004consistency-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> is a simplified model for Breiman's original random forest, which uniformly selects an attribute among all attributes and performs splits at the center of the cell along the pre-chosen attribute. The algorithm stops when a fully binary tree of level <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> is built, 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 k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" /></span> is a parameter of the algorithm. </p> <div class="mw-heading mw-heading4"><h4 id="Uniform_forest">Uniform forest</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=19" title="Edit section: Uniform forest"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uniform forest<sup id="cite_ref-arlot2014analysis_46-1" class="reference"><a href="#cite_note-arlot2014analysis-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> is another simplified model for Breiman's original random forest, which uniformly selects a feature among all features and performs splits at a point uniformly drawn on the side of the cell, along the preselected feature. </p> <div class="mw-heading mw-heading4"><h4 id="From_random_forest_to_KeRF">From random forest to KeRF</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=20" title="Edit section: From random forest to KeRF"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a training sample <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 {\mathcal {D}}_{n}=\{(\mathbf {X} _{i},Y_{i})\}_{i=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{n}=\{(\mathbf {X} _{i},Y_{i})\}_{i=1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c67c8cf7d7485be2a566d02284d9bc27a3f61241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.147ex; height:3.009ex;" alt="{\displaystyle {\mathcal {D}}_{n}=\{(\mathbf {X} _{i},Y_{i})\}_{i=1}^{n}}" /></span> 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 [0,1]^{p}\times \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]^{p}\times \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6724c5c806dd7cee08c1f2133fc89f94ed0c2e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.23ex; height:2.843ex;" alt="{\displaystyle [0,1]^{p}\times \mathbb {R} }" /></span>-valued independent random variables distributed as the independent prototype pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {X} ,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {X} ,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f50eb05dacdb567fc019a3bbcd44b596a1c59a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.636ex; height:2.843ex;" alt="{\displaystyle (\mathbf {X} ,Y)}" /></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 \operatorname {E} [Y^{2}]<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [Y^{2}]<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a863a618762e9fecf30fadce90c3f6484db196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.253ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [Y^{2}]<\infty }" /></span>. We aim at predicting the response <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>, associated with the random 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 \mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" /></span>, by estimating the regression 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 m(\mathbf {x} )=\operatorname {E} [Y\mid \mathbf {X} =\mathbf {x} ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(\mathbf {x} )=\operatorname {E} [Y\mid \mathbf {X} =\mathbf {x} ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c097ce105048693746d5524ac0ef9fc514da53b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.475ex; height:2.843ex;" alt="{\displaystyle m(\mathbf {x} )=\operatorname {E} [Y\mid \mathbf {X} =\mathbf {x} ]}" /></span>. A random regression forest is an ensemble 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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> randomized regression trees. Denote <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 m_{n}(\mathbf {x} ,\mathbf {\Theta } _{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{n}(\mathbf {x} ,\mathbf {\Theta } _{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ad3dfa3a0ed6fc135df01b527d6be634ff9d07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.5ex; height:3.009ex;" alt="{\displaystyle m_{n}(\mathbf {x} ,\mathbf {\Theta } _{j})}" /></span> the predicted value at point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" /></span> by the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>-th tree, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Theta } _{1},\ldots ,\mathbf {\Theta } _{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\Theta } _{1},\ldots ,\mathbf {\Theta } _{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8588793690df49c9b10f8c44499b1ee82106867f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.347ex; height:2.509ex;" alt="{\displaystyle \mathbf {\Theta } _{1},\ldots ,\mathbf {\Theta } _{M}}" /></span> are independent random variables, distributed as a generic random 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 \mathbf {\Theta } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Θ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\Theta } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a15b5576ec407e52c5fdfc21c5d7bf45402406a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.078ex; height:2.176ex;" alt="{\displaystyle \mathbf {\Theta } }" /></span>, independent of the sample <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 {\mathcal {D}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba7d8da2372a68d034e848dbd973b7f173769fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.01ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}_{n}}" /></span>. This random variable can be used to describe the randomness induced by node splitting and the sampling procedure for tree construction. The trees are combined to form the finite forest 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 m_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{M}}\sum _{j=1}^{M}m_{n}(\mathbf {x} ,\Theta _{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{M}}\sum _{j=1}^{M}m_{n}(\mathbf {x} ,\Theta _{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1cd03b3dba46782bb60f083359fd8e77250024c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.242ex; height:7.676ex;" alt="{\displaystyle m_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{M}}\sum _{j=1}^{M}m_{n}(\mathbf {x} ,\Theta _{j})}" /></span>. For regression trees, we have <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 m_{n}=\sum _{i=1}^{n}{\frac {Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}{N_{n}(\mathbf {x} ,\Theta _{j})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</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>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{n}=\sum _{i=1}^{n}{\frac {Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}{N_{n}(\mathbf {x} ,\Theta _{j})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ad1f4a5e0e849bfbf10492b08e5e9b55b6c711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.729ex; height:7.009ex;" alt="{\displaystyle m_{n}=\sum _{i=1}^{n}{\frac {Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}{N_{n}(\mathbf {x} ,\Theta _{j})}}}" /></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 A_{n}(\mathbf {x} ,\Theta _{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}(\mathbf {x} ,\Theta _{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1811d33726c0ea70d83e6f5ee5228d82e6229300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.934ex; height:3.009ex;" alt="{\displaystyle A_{n}(\mathbf {x} ,\Theta _{j})}" /></span> is the cell containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" /></span>, designed with randomness <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 \Theta _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b9d5bab99a44d98171cce92f04d0e700060512" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.718ex; height:2.843ex;" alt="{\displaystyle \Theta _{j}}" /></span> and dataset <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 {\mathcal {D}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba7d8da2372a68d034e848dbd973b7f173769fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.01ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}_{n}}" /></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 N_{n}(\mathbf {x} ,\Theta _{j})=\sum _{i=1}^{n}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{n}(\mathbf {x} ,\Theta _{j})=\sum _{i=1}^{n}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c13629db1ca0ba725280a4f2bbf7adfc5c9f8a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.54ex; height:6.843ex;" alt="{\displaystyle N_{n}(\mathbf {x} ,\Theta _{j})=\sum _{i=1}^{n}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}" /></span>. </p><p>Thus random forest estimates satisfy, for 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 \mathbf {x} \in [0,1]^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in [0,1]^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691c9727c059400e1d15031c7e0e08fbd6774282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.996ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} \in [0,1]^{d}}" /></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 m_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{M}}\sum _{j=1}^{M}\left(\sum _{i=1}^{n}{\frac {Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}{N_{n}(\mathbf {x} ,\Theta _{j})}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{M}}\sum _{j=1}^{M}\left(\sum _{i=1}^{n}{\frac {Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}{N_{n}(\mathbf {x} ,\Theta _{j})}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671426d343ccaba3944706fd2517e5f7035f674c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:54.064ex; height:7.676ex;" alt="{\displaystyle m_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{M}}\sum _{j=1}^{M}\left(\sum _{i=1}^{n}{\frac {Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}{N_{n}(\mathbf {x} ,\Theta _{j})}}\right)}" /></span>. Random regression forest has two levels of averaging, first over the samples in the target cell of a tree, then over all trees. Thus the contributions of observations that are in cells with a high density of data points are smaller than that of observations which belong to less populated cells. In order to improve the random forest methods and compensate the misestimation, Scornet<sup id="cite_ref-scornet2015random_41-2" class="reference"><a href="#cite_note-scornet2015random-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> defined KeRF by <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 {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{\sum _{j=1}^{M}N_{n}(\mathbf {x} ,\Theta _{j})}}\sum _{j=1}^{M}\sum _{i=1}^{n}Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{\sum _{j=1}^{M}N_{n}(\mathbf {x} ,\Theta _{j})}}\sum _{j=1}^{M}\sum _{i=1}^{n}Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565e75beaa1dd3b92753899bfa390abba19af5fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:63.659ex; height:7.843ex;" alt="{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{\sum _{j=1}^{M}N_{n}(\mathbf {x} ,\Theta _{j})}}\sum _{j=1}^{M}\sum _{i=1}^{n}Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})},}" /></span> which is equal to the mean of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}" /></span>'s falling in the cells containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" /></span> in the forest. If we define the connection function of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> finite forest as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{M,n}(\mathbf {x} ,\mathbf {z} )={\frac {1}{M}}\sum _{j=1}^{M}\mathbf {1} _{\mathbf {z} \in A_{n}(\mathbf {x} ,\Theta _{j})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{M,n}(\mathbf {x} ,\mathbf {z} )={\frac {1}{M}}\sum _{j=1}^{M}\mathbf {1} _{\mathbf {z} \in A_{n}(\mathbf {x} ,\Theta _{j})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7723dd64358a146a8bc819a79f9c15266b4d37dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.754ex; height:7.676ex;" alt="{\displaystyle K_{M,n}(\mathbf {x} ,\mathbf {z} )={\frac {1}{M}}\sum _{j=1}^{M}\mathbf {1} _{\mathbf {z} \in A_{n}(\mathbf {x} ,\Theta _{j})}}" /></span>, i.e. the proportion of cells shared between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82eca5d0928078d5a61b9e7e98cc73db31070909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {z} }" /></span>, then almost surely we have <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 {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {\sum _{i=1}^{n}Y_{i}K_{M,n}(\mathbf {x} ,\mathbf {x} _{i})}{\sum _{\ell =1}^{n}K_{M,n}(\mathbf {x} ,\mathbf {x} _{\ell })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {\sum _{i=1}^{n}Y_{i}K_{M,n}(\mathbf {x} ,\mathbf {x} _{i})}{\sum _{\ell =1}^{n}K_{M,n}(\mathbf {x} ,\mathbf {x} _{\ell })}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea4edc4c4ad2b30acfe0b5898862397424612eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:45.172ex; height:6.843ex;" alt="{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {\sum _{i=1}^{n}Y_{i}K_{M,n}(\mathbf {x} ,\mathbf {x} _{i})}{\sum _{\ell =1}^{n}K_{M,n}(\mathbf {x} ,\mathbf {x} _{\ell })}}}" /></span>, which defines the KeRF. </p> <div class="mw-heading mw-heading4"><h4 id="Centered_KeRF">Centered KeRF</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=21" title="Edit section: Centered KeRF"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The construction of Centered KeRF of level <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> is the same as for centered forest, except that predictions are made by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3939edfe84d8ece5969248f08b77f0b23843ad3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.505ex; height:3.009ex;" alt="{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})}" /></span>, the corresponding kernel function, or connection function is <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 K_{k}^{cc}(\mathbf {x} ,\mathbf {z} )=\sum _{k_{1},\ldots ,k_{d},\sum _{j=1}^{d}k_{j}=k}{\frac {k!}{k_{1}!\cdots k_{d}!}}\left({\frac {1}{d}}\right)^{k}\prod _{j=1}^{d}\mathbf {1} _{\lceil 2^{k_{j}}x_{j}\rceil =\lceil 2^{k_{j}}z_{j}\rceil },\qquad {\text{ for all }}\mathbf {x} ,\mathbf {z} \in [0,1]^{d}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>c</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>,</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>!</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <mo>⋯</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌈</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⌉</mo> <mo>=</mo> <mo fence="false" stretchy="false">⌈</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⌉</mo> </mrow> </msub> <mo>,</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{k}^{cc}(\mathbf {x} ,\mathbf {z} )=\sum _{k_{1},\ldots ,k_{d},\sum _{j=1}^{d}k_{j}=k}{\frac {k!}{k_{1}!\cdots k_{d}!}}\left({\frac {1}{d}}\right)^{k}\prod _{j=1}^{d}\mathbf {1} _{\lceil 2^{k_{j}}x_{j}\rceil =\lceil 2^{k_{j}}z_{j}\rceil },\qquad {\text{ for all }}\mathbf {x} ,\mathbf {z} \in [0,1]^{d}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b7a16a2bcfb63020d9081379559d5fcd82a7a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:87.29ex; height:8.676ex;" alt="{\displaystyle K_{k}^{cc}(\mathbf {x} ,\mathbf {z} )=\sum _{k_{1},\ldots ,k_{d},\sum _{j=1}^{d}k_{j}=k}{\frac {k!}{k_{1}!\cdots k_{d}!}}\left({\frac {1}{d}}\right)^{k}\prod _{j=1}^{d}\mathbf {1} _{\lceil 2^{k_{j}}x_{j}\rceil =\lceil 2^{k_{j}}z_{j}\rceil },\qquad {\text{ for all }}\mathbf {x} ,\mathbf {z} \in [0,1]^{d}.}" /></span> </p> <div class="mw-heading mw-heading4"><h4 id="Uniform_KeRF">Uniform KeRF</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=22" title="Edit section: Uniform KeRF"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uniform KeRF is built in the same way as uniform forest, except that predictions are made by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3939edfe84d8ece5969248f08b77f0b23843ad3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.505ex; height:3.009ex;" alt="{\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})}" /></span>, the corresponding kernel function, or connection function is <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 K_{k}^{uf}(\mathbf {0} ,\mathbf {x} )=\sum _{k_{1},\ldots ,k_{d},\sum _{j=1}^{d}k_{j}=k}{\frac {k!}{k_{1}!\ldots k_{d}!}}\left({\frac {1}{d}}\right)^{k}\prod _{m=1}^{d}\left(1-|x_{m}|\sum _{j=0}^{k_{m}-1}{\frac {\left(-\ln |x_{m}|\right)^{j}}{j!}}\right){\text{ for all }}\mathbf {x} \in [0,1]^{d}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>f</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>,</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </munderover> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>!</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <mo>…</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{k}^{uf}(\mathbf {0} ,\mathbf {x} )=\sum _{k_{1},\ldots ,k_{d},\sum _{j=1}^{d}k_{j}=k}{\frac {k!}{k_{1}!\ldots k_{d}!}}\left({\frac {1}{d}}\right)^{k}\prod _{m=1}^{d}\left(1-|x_{m}|\sum _{j=0}^{k_{m}-1}{\frac {\left(-\ln |x_{m}|\right)^{j}}{j!}}\right){\text{ for all }}\mathbf {x} \in [0,1]^{d}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e06f4c0b6697dda0b20263456b511570143a7785" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:96.713ex; height:8.676ex;" alt="{\displaystyle K_{k}^{uf}(\mathbf {0} ,\mathbf {x} )=\sum _{k_{1},\ldots ,k_{d},\sum _{j=1}^{d}k_{j}=k}{\frac {k!}{k_{1}!\ldots k_{d}!}}\left({\frac {1}{d}}\right)^{k}\prod _{m=1}^{d}\left(1-|x_{m}|\sum _{j=0}^{k_{m}-1}{\frac {\left(-\ln |x_{m}|\right)^{j}}{j!}}\right){\text{ for all }}\mathbf {x} \in [0,1]^{d}.}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Properties_2">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=23" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Relation_between_KeRF_and_random_forest">Relation between KeRF and random forest</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=24" title="Edit section: Relation between KeRF and random forest"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Predictions given by KeRF and random forests are close if the number of points in each cell is controlled: </p> <blockquote> <p>Assume that there exist sequences <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_{n}),(b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n}),(b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8157b9b64afbd714b16ed6b8de06c32afb696a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.317ex; height:2.843ex;" alt="{\displaystyle (a_{n}),(b_{n})}" /></span> such that, almost surely, <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 a_{n}\leq N_{n}(\mathbf {x} ,\Theta )\leq b_{n}{\text{ and }}a_{n}\leq {\frac {1}{M}}\sum _{m=1}^{M}N_{n}{\mathbf {x} ,\Theta _{m}}\leq b_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\leq N_{n}(\mathbf {x} ,\Theta )\leq b_{n}{\text{ and }}a_{n}\leq {\frac {1}{M}}\sum _{m=1}^{M}N_{n}{\mathbf {x} ,\Theta _{m}}\leq b_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139dc876ec018ffab1c1d499f269b95a6e8c245a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.034ex; height:7.343ex;" alt="{\displaystyle a_{n}\leq N_{n}(\mathbf {x} ,\Theta )\leq b_{n}{\text{ and }}a_{n}\leq {\frac {1}{M}}\sum _{m=1}^{M}N_{n}{\mathbf {x} ,\Theta _{m}}\leq b_{n}.}" /></span> Then almost surely, <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 |m_{M,n}(\mathbf {x} )-{\tilde {m}}_{M,n}(\mathbf {x} )|\leq {\frac {b_{n}-a_{n}}{a_{n}}}{\tilde {m}}_{M,n}(\mathbf {x} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |m_{M,n}(\mathbf {x} )-{\tilde {m}}_{M,n}(\mathbf {x} )|\leq {\frac {b_{n}-a_{n}}{a_{n}}}{\tilde {m}}_{M,n}(\mathbf {x} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad592965613268c7026f8f57701caba977fd34ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.21ex; height:5.676ex;" alt="{\displaystyle |m_{M,n}(\mathbf {x} )-{\tilde {m}}_{M,n}(\mathbf {x} )|\leq {\frac {b_{n}-a_{n}}{a_{n}}}{\tilde {m}}_{M,n}(\mathbf {x} ).}" /></span> </p> </blockquote> <div class="mw-heading mw-heading4"><h4 id="Relation_between_infinite_KeRF_and_infinite_random_forest">Relation between infinite KeRF and infinite random forest</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=25" title="Edit section: Relation between infinite KeRF and infinite random forest"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When the number of trees <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> goes to infinity, then we have infinite random forest and infinite KeRF. Their estimates are close if the number of observations in each cell is bounded: </p> <blockquote> <p>Assume that there exist sequences <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 (\varepsilon _{n}),(a_{n}),(b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varepsilon _{n}),(a_{n}),(b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e5f769151263bad5a8d716a2c6af2686687855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.462ex; height:2.843ex;" alt="{\displaystyle (\varepsilon _{n}),(a_{n}),(b_{n})}" /></span> such that, almost surely </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [N_{n}(\mathbf {x} ,\Theta )]\geq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>≥</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [N_{n}(\mathbf {x} ,\Theta )]\geq 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f5beb5ef7860bc9f527fb5b01fbed3dc0fc716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.931ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [N_{n}(\mathbf {x} ,\Theta )]\geq 1,}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} [a_{n}\leq N_{n}(\mathbf {x} ,\Theta )\leq b_{n}\mid {\mathcal {D}}_{n}]\geq 1-\varepsilon _{n}/2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>≥</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} [a_{n}\leq N_{n}(\mathbf {x} ,\Theta )\leq b_{n}\mid {\mathcal {D}}_{n}]\geq 1-\varepsilon _{n}/2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bad78155a4139322247009072e48d6b7bf8c5b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.207ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} [a_{n}\leq N_{n}(\mathbf {x} ,\Theta )\leq b_{n}\mid {\mathcal {D}}_{n}]\geq 1-\varepsilon _{n}/2,}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} [a_{n}\leq \operatorname {E} _{\Theta }[N_{n}(\mathbf {x} ,\Theta )]\leq b_{n}\mid {\mathcal {D}}_{n}]\geq 1-\varepsilon _{n}/2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Θ</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>≥</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} [a_{n}\leq \operatorname {E} _{\Theta }[N_{n}(\mathbf {x} ,\Theta )]\leq b_{n}\mid {\mathcal {D}}_{n}]\geq 1-\varepsilon _{n}/2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71711062ff59f3d2075162074e8f94f917a0cdad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.595ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} [a_{n}\leq \operatorname {E} _{\Theta }[N_{n}(\mathbf {x} ,\Theta )]\leq b_{n}\mid {\mathcal {D}}_{n}]\geq 1-\varepsilon _{n}/2,}" /></span></li></ul> <p>Then almost surely, <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 |m_{\infty ,n}(\mathbf {x} )-{\tilde {m}}_{\infty ,n}(\mathbf {x} )|\leq {\frac {b_{n}-a_{n}}{a_{n}}}{\tilde {m}}_{\infty ,n}(\mathbf {x} )+n\varepsilon _{n}\left(\max _{1\leq i\leq n}Y_{i}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>n</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |m_{\infty ,n}(\mathbf {x} )-{\tilde {m}}_{\infty ,n}(\mathbf {x} )|\leq {\frac {b_{n}-a_{n}}{a_{n}}}{\tilde {m}}_{\infty ,n}(\mathbf {x} )+n\varepsilon _{n}\left(\max _{1\leq i\leq n}Y_{i}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26197c8d5ca305f9a7a922c4227c5daa11350e76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.162ex; height:6.176ex;" alt="{\displaystyle |m_{\infty ,n}(\mathbf {x} )-{\tilde {m}}_{\infty ,n}(\mathbf {x} )|\leq {\frac {b_{n}-a_{n}}{a_{n}}}{\tilde {m}}_{\infty ,n}(\mathbf {x} )+n\varepsilon _{n}\left(\max _{1\leq i\leq n}Y_{i}\right).}" /></span> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Consistency_results">Consistency results</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=26" title="Edit section: Consistency results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Assume that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=m(\mathbf {X} )+\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=m(\mathbf {X} )+\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8617bcdd783f680a6c45c9d8c627a5730a878e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.665ex; height:2.843ex;" alt="{\displaystyle Y=m(\mathbf {X} )+\varepsilon }" /></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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" /></span> is a centered Gaussian noise, independent of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" /></span>, with finite variance <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 \sigma ^{2}<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c01c50d5825dbb5b11add552a422af72a9664f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.807ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}<\infty }" /></span>. Moreover, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" /></span> is uniformly distributed on <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 [0,1]^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13ae4917276744b214714a20b3cb8ee305e309d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.745ex; height:3.176ex;" alt="{\displaystyle [0,1]^{d}}" /></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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /></span> is <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz</a>. Scornet<sup id="cite_ref-scornet2015random_41-3" class="reference"><a href="#cite_note-scornet2015random-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> proved upper bounds on the rates of consistency for centered KeRF and uniform KeRF. </p> <div class="mw-heading mw-heading4"><h4 id="Consistency_of_centered_KeRF">Consistency of centered KeRF</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=27" title="Edit section: Consistency of centered KeRF"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Providing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612a3ec99f1c9f12de1cfab011e306ae799858ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="{\displaystyle k\rightarrow \infty }" /></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 n/2^{k}\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n/2^{k}\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbec163fbc2ac671e4c9f78bdc70a985599dc19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.746ex; height:3.176ex;" alt="{\displaystyle n/2^{k}\rightarrow \infty }" /></span>, there exists a constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f2ff302742806f4db7411f29e54944b4d9bda7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.977ex; height:2.509ex;" alt="{\displaystyle C_{1}>0}" /></span> such that, for 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>, <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 \mathbb {E} [{\tilde {m}}_{n}^{cc}(\mathbf {X} )-m(\mathbf {X} )]^{2}\leq C_{1}n^{-1/(3+d\log 2)}(\log n)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>c</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mi>d</mi> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [{\tilde {m}}_{n}^{cc}(\mathbf {X} )-m(\mathbf {X} )]^{2}\leq C_{1}n^{-1/(3+d\log 2)}(\log n)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd76dd80031cfdb1b8042faeab38ecaafefa2e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.051ex; height:3.343ex;" alt="{\displaystyle \mathbb {E} [{\tilde {m}}_{n}^{cc}(\mathbf {X} )-m(\mathbf {X} )]^{2}\leq C_{1}n^{-1/(3+d\log 2)}(\log n)^{2}}" /></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Consistency_of_uniform_KeRF">Consistency of uniform KeRF</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=28" title="Edit section: Consistency of uniform KeRF"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Providing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612a3ec99f1c9f12de1cfab011e306ae799858ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="{\displaystyle k\rightarrow \infty }" /></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 n/2^{k}\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n/2^{k}\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbec163fbc2ac671e4c9f78bdc70a985599dc19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.746ex; height:3.176ex;" alt="{\displaystyle n/2^{k}\rightarrow \infty }" /></span>, there exists a constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84d4126c6df243734f9355927c026df6b0d3859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C>0}" /></span> such that, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [{\tilde {m}}_{n}^{uf}(\mathbf {X} )-m(\mathbf {X} )]^{2}\leq Cn^{-2/(6+3d\log 2)}(\log n)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>f</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mi>C</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo>+</mo> <mn>3</mn> <mi>d</mi> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [{\tilde {m}}_{n}^{uf}(\mathbf {X} )-m(\mathbf {X} )]^{2}\leq Cn^{-2/(6+3d\log 2)}(\log n)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51b3c9f4980f936fc01a73621cc56e9857059da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.344ex; height:3.343ex;" alt="{\displaystyle \mathbb {E} [{\tilde {m}}_{n}^{uf}(\mathbf {X} )-m(\mathbf {X} )]^{2}\leq Cn^{-2/(6+3d\log 2)}(\log n)^{2}}" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Disadvantages">Disadvantages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=29" title="Edit section: Disadvantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While random forests often achieve higher accuracy than a single decision tree, they sacrifice the intrinsic <a href="/wiki/Interpretability" title="Interpretability">interpretability</a> of decision trees. Decision trees are among a fairly small family of machine learning models that are easily interpretable along with linear models, <a href="/wiki/Rule-based_machine_learning" title="Rule-based machine learning">rule-based</a> models, and <a href="/wiki/Attention_(machine_learning)" title="Attention (machine learning)">attention</a>-based models. This interpretability is one of the main advantages of decision trees. It allows developers to confirm that the model has learned realistic information from the data and allows end-users to have trust and confidence in the decisions made by the model.<sup id="cite_ref-:0_37-2" class="reference"><a href="#cite_note-:0-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-elemstatlearn_3-6" class="reference"><a href="#cite_note-elemstatlearn-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> For example, following the path that a decision tree takes to make its decision is quite trivial, but following the paths of tens or hundreds of trees is much harder. To achieve both performance and interpretability, some model compression techniques allow transforming a random forest into a minimal "born-again" decision tree that faithfully reproduces the same decision function.<sup id="cite_ref-:0_37-3" class="reference"><a href="#cite_note-:0-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><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><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> </p><p>Another limitation of random forests is that if features are linearly correlated with the target, random forest may not enhance the accuracy of the base learner.<sup id="cite_ref-:0_37-4" class="reference"><a href="#cite_note-:0-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_40-1" class="reference"><a href="#cite_note-:1-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Likewise in problems with multiple categorical variables.<sup id="cite_ref-:3_49-0" class="reference"><a href="#cite_note-:3-49"><span class="cite-bracket">[</span>49<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=Random_forest&action=edit&section=30" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a> – Method in machine learning</li> <li><a href="/wiki/Decision_tree_learning" title="Decision tree learning">Decision tree learning</a> – Machine learning algorithm</li> <li><a href="/wiki/Ensemble_learning" title="Ensemble learning">Ensemble learning</a> – Statistics and machine learning technique</li> <li><a href="/wiki/Gradient_boosting" title="Gradient boosting">Gradient boosting</a> – Machine learning technique</li> <li><a href="/wiki/Non-parametric_statistics" class="mw-redirect" title="Non-parametric statistics">Non-parametric statistics</a> – Type of statistical analysis<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a> – Algorithm that employs a degree of randomness as part of its logic or procedure</li></ul> <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=Random_forest&action=edit&section=31" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 32em;"> <ol class="references"> <li id="cite_note-ho1995-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-ho1995_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ho1995_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-ho1995_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-ho1995_1-3"><sup><i><b>d</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="CITEREFHo1995" class="citation conference cs1">Ho, Tin Kam (1995). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160417030218/http://ect.bell-labs.com/who/tkh/publications/papers/odt.pdf"><i>Random Decision Forests</i></a> <span 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reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_forest&action=edit&section=32" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output 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ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPrinziePoel2007" class="citation conference cs1">Prinzie A, Poel D (2007). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/225175169">"Random Multiclass Classification: Generalizing Random Forests to Random MNL and Random NB"</a>. <i>Database and Expert Systems Applications</i>. <a href="/wiki/Lecture_Notes_in_Computer_Science" title="Lecture Notes in Computer Science">Lecture Notes in Computer Science</a>. Vol. 4653. p. 349. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-74469-6_35">10.1007/978-3-540-74469-6_35</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-74467-2" title="Special:BookSources/978-3-540-74467-2"><bdi>978-3-540-74467-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Random+Multiclass+Classification%3A+Generalizing+Random+Forests+to+Random+MNL+and+Random+NB&rft.btitle=Database+and+Expert+Systems+Applications&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=349&rft.date=2007&rft_id=info%3Adoi%2F10.1007%2F978-3-540-74469-6_35&rft.isbn=978-3-540-74467-2&rft.aulast=Prinzie&rft.aufirst=Anita&rft.au=Poel%2C+Dirk&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F225175169&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+forest" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDeniskoHoffman2018" class="citation journal cs1">Denisko D, Hoffman MM (February 2018). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5828645">"Classification and interaction in random forests"</a>. <i>Proceedings of the National Academy of Sciences of the United States of America</i>. <b>115</b> (8): <span class="nowrap">1690–</span>1692. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018PNAS..115.1690D">2018PNAS..115.1690D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.1800256115">10.1073/pnas.1800256115</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5828645">5828645</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/29440440">29440440</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=Classification+and+interaction+in+random+forests&rft.volume=115&rft.issue=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1690-%3C%2Fspan%3E1692&rft.date=2018-02&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5828645%23id-name%3DPMC&rft_id=info%3Apmid%2F29440440&rft_id=info%3Adoi%2F10.1073%2Fpnas.1800256115&rft_id=info%3Abibcode%2F2018PNAS..115.1690D&rft.aulast=Denisko&rft.aufirst=D&rft.au=Hoffman%2C+MM&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5828645&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+forest" class="Z3988"></span></li></ul> </div> <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=Random_forest&action=edit&section=33" 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.stat.berkeley.edu/~breiman/RandomForests/cc_home.htm">Random Forests classifier description</a> (Leo Breiman's site)</li> <li><a rel="nofollow" class="external text" href="https://cran.r-project.org/doc/Rnews/Rnews_2002-3.pdf">Liaw, Andy & Wiener, Matthew "Classification and Regression by randomForest" R News (2002) Vol. 2/3 p. 18</a> (Discussion of the use of the random forest package for <a href="/wiki/R_programming_language" class="mw-redirect" title="R programming language">R</a>)</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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