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</li> <li id="toc-Dependence_of_the_F-score_on_class_imbalance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dependence_of_the_F-score_on_class_imbalance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Dependence of the F-score on class imbalance</span> </div> </a> <ul id="toc-Dependence_of_the_F-score_on_class_imbalance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Criticism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Criticism"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Criticism</span> </div> </a> <ul id="toc-Criticism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Difference_from_Fowlkes–Mallows_index" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Difference_from_Fowlkes–Mallows_index"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Difference from Fowlkes–Mallows index</span> </div> </a> <ul id="toc-Difference_from_Fowlkes–Mallows_index-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_to_multi-class_classification" class="vector-toc-list-item vector-toc-level-1 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</div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">F-score</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://ca.wikipedia.org/wiki/F-score" title="F-score – Catalan" lang="ca" hreflang="ca" data-title="F-score" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Beurteilung_eines_Klassifikators#Kombinierte_Maße" title="Beurteilung eines Klassifikators – German" lang="de" hreflang="de" data-title="Beurteilung eines Klassifikators" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Valor-F" title="Valor-F – Spanish" lang="es" hreflang="es" data-title="Valor-F" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/F_puntuazio" title="F puntuazio – Basque" lang="eu" hreflang="eu" data-title="F puntuazio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%85%D8%AA%DB%8C%D8%A7%D8%B2_%D8%A7%D9%81_%DB%B1" title="امتیاز اف ۱ – Persian" lang="fa" hreflang="fa" data-title="امتیاز اف ۱" 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/F-mesure" title="F-mesure – French" lang="fr" hreflang="fr" data-title="F-mesure" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Statistical measure of a test's accuracy</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the significance test, see <a href="/wiki/F-test" title="F-test">F-test</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Precisionrecall.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Precisionrecall.svg/350px-Precisionrecall.svg.png" decoding="async" width="350" height="636" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Precisionrecall.svg/525px-Precisionrecall.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Precisionrecall.svg/700px-Precisionrecall.svg.png 2x" data-file-width="440" data-file-height="800" /></a><figcaption>Precision and recall</figcaption></figure> <p>In <a href="/wiki/Statistics" title="Statistics">statistical</a> analysis of <a href="/wiki/Binary_classification" title="Binary classification">binary classification</a> and <a href="/wiki/Information_retrieval" title="Information retrieval">information retrieval</a> systems, the <b>F-score</b> or <b>F-measure</b> is a measure of predictive performance. It is calculated from the <a href="/wiki/Precision_(information_retrieval)" class="mw-redirect" title="Precision (information retrieval)">precision</a> and <a href="/wiki/Recall_(information_retrieval)" class="mw-redirect" title="Recall (information retrieval)">recall</a> of the test, where the precision is the number of true positive results divided by the number of all samples predicted to be positive, including those not identified correctly, and the recall is the number of true positive results divided by the number of all samples that should have been identified as positive. Precision is also known as <a href="/wiki/Positive_predictive_value" class="mw-redirect" title="Positive predictive value">positive predictive value</a>, and recall is also known as <a href="/wiki/Sensitivity_and_specificity" title="Sensitivity and specificity">sensitivity</a> in diagnostic binary classification. </p><p>The <b>F<sub>1</sub></b> score is the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> of the precision and recall. It thus symmetrically represents both precision and recall in one metric. The more generic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb39f5958eb83bc3c28cd8b1a5397d3bb14fecdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.669ex; height:2.843ex;" alt="{\displaystyle F_{\beta }}"></span> score applies additional weights, valuing one of precision or recall more than the other. </p><p>The highest possible value of an F-score is 1.0, indicating perfect precision and recall, and the lowest possible value is 0, if precision and recall are zero. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=1" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The name F-measure is believed to be named after a different F function in Van Rijsbergen's book, when introduced to the Fourth <a href="/wiki/Message_Understanding_Conference" title="Message Understanding Conference">Message Understanding Conference</a> (MUC-4, 1992).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The traditional F-measure or balanced F-score (<b>F<sub>1</sub> score</b>) is the <a href="/wiki/Harmonic_mean#Harmonic_mean_of_two_numbers" title="Harmonic mean">harmonic mean</a> of precision and recall:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}={\frac {2}{\mathrm {recall} ^{-1}+\mathrm {precision} ^{-1}}}=2{\frac {\mathrm {precision} \cdot \mathrm {recall} }{\mathrm {precision} +\mathrm {recall} }}={\frac {2\mathrm {TP} }{2\mathrm {TP} +\mathrm {FP} +\mathrm {FN} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">P</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">P</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">N</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}={\frac {2}{\mathrm {recall} ^{-1}+\mathrm {precision} ^{-1}}}=2{\frac {\mathrm {precision} \cdot \mathrm {recall} }{\mathrm {precision} +\mathrm {recall} }}={\frac {2\mathrm {TP} }{2\mathrm {TP} +\mathrm {FP} +\mathrm {FN} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9b2235daf4402960538e73d6310bc8c2b8de52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:71.06ex; height:6.343ex;" alt="{\displaystyle F_{1}={\frac {2}{\mathrm {recall} ^{-1}+\mathrm {precision} ^{-1}}}=2{\frac {\mathrm {precision} \cdot \mathrm {recall} }{\mathrm {precision} +\mathrm {recall} }}={\frac {2\mathrm {TP} }{2\mathrm {TP} +\mathrm {FP} +\mathrm {FN} }}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Fβ_score"><span id="F.CE.B2_score"></span>F<sub>β</sub> score</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=3" title="Edit section: Fβ score"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A more general F score, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb39f5958eb83bc3c28cd8b1a5397d3bb14fecdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.669ex; height:2.843ex;" alt="{\displaystyle F_{\beta }}"></span>, that uses a positive real factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> is chosen such that recall is considered <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> times as important as precision, is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\beta }=(1+\beta ^{2})\cdot {\frac {\mathrm {precision} \cdot \mathrm {recall} }{(\beta ^{2}\cdot \mathrm {precision} )+\mathrm {recall} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }=(1+\beta ^{2})\cdot {\frac {\mathrm {precision} \cdot \mathrm {recall} }{(\beta ^{2}\cdot \mathrm {precision} )+\mathrm {recall} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136f45612c08805f4254f63d2f2524bc25075fff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.572ex; height:6.176ex;" alt="{\displaystyle F_{\beta }=(1+\beta ^{2})\cdot {\frac {\mathrm {precision} \cdot \mathrm {recall} }{(\beta ^{2}\cdot \mathrm {precision} )+\mathrm {recall} }}}"></span>.</dd></dl> <p>In terms of <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">Type I and type II errors</a> this becomes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\beta }={\frac {(1+\beta ^{2})\cdot \mathrm {true\ positive} }{(1+\beta ^{2})\cdot \mathrm {true\ positive} +\beta ^{2}\cdot \mathrm {false\ negative} +\mathrm {false\ positive} }}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }={\frac {(1+\beta ^{2})\cdot \mathrm {true\ positive} }{(1+\beta ^{2})\cdot \mathrm {true\ positive} +\beta ^{2}\cdot \mathrm {false\ negative} +\mathrm {false\ positive} }}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbeb471033fdd63a2c2ca7830afc7abdf8b8134" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:65.873ex; height:6.676ex;" alt="{\displaystyle F_{\beta }={\frac {(1+\beta ^{2})\cdot \mathrm {true\ positive} }{(1+\beta ^{2})\cdot \mathrm {true\ positive} +\beta ^{2}\cdot \mathrm {false\ negative} +\mathrm {false\ positive} }}\,}"></span>.</dd></dl> <p>Two commonly used values for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> are 2, which weighs recall higher than precision, and 0.5, which weighs recall lower than precision. </p><p>The F-measure was derived so 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 F_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb39f5958eb83bc3c28cd8b1a5397d3bb14fecdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.669ex; height:2.843ex;" alt="{\displaystyle F_{\beta }}"></span> "measures the effectiveness of retrieval with respect to a user who attaches <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> times as much importance to recall as precision".<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It is based on <a href="/wiki/C._J._van_Rijsbergen" title="C. J. van Rijsbergen">Van Rijsbergen</a>'s effectiveness measure </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=1-\left({\frac {\alpha }{p}}+{\frac {1-\alpha }{r}}\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>p</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=1-\left({\frac {\alpha }{p}}+{\frac {1-\alpha }{r}}\right)^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf2de0adedf04b55032e75ccc9a09cb39ab7bd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.122ex; height:6.509ex;" alt="{\displaystyle E=1-\left({\frac {\alpha }{p}}+{\frac {1-\alpha }{r}}\right)^{-1}}"></span>.</dd></dl> <p>Their relationship is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\beta }=1-E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }=1-E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588a56674a39e59ba281d71e792a2a702326ecec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.546ex; height:2.843ex;" alt="{\displaystyle F_{\beta }=1-E}"></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 \alpha ={\frac {1}{1+\beta ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {1}{1+\beta ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db630a6a3c5f1561bd101535d7cba0512de48b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.816ex; height:5.843ex;" alt="{\displaystyle \alpha ={\frac {1}{1+\beta ^{2}}}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Diagnostic_testing">Diagnostic testing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=4" title="Edit section: Diagnostic testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This is related to the field of <a href="/wiki/Binary_classification" title="Binary classification">binary classification</a> where recall is often termed "sensitivity". </p> <table class="wikitable" align="center" style="border:none; background:transparent; text-align:center;"> <tbody><tr> <td style="border:none;" rowspan="2"> </td> <td style="border:none;"> </td> <td style="background:#bbeeee;" colspan="2"><b>Predicted condition</b> </td> <td style="border:none; text-align:right;" colspan="2"><sup>Sources: </sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <sup><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul 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dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><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 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hlist"><ul><li class="nv-view"><a href="/wiki/Template:Diagnostic_testing_diagram" title="Template:Diagnostic testing diagram"><span title="View this template">view</span></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Diagnostic_testing_diagram" title="Template talk:Diagnostic testing diagram"><span title="Discuss this template">talk</span></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Diagnostic_testing_diagram" title="Special:EditPage/Template:Diagnostic testing diagram"><span title="Edit this template">edit</span></a></li></ul></div></sup> </td></tr> <tr> <td style="background:#eeeeee;"><a href="/wiki/Statistical_population" title="Statistical population">Total population</a> <br /><span class="texhtml">= P + N</span> </td> <td style="background:#ccffff;"><b>Predicted positive (PP)</b> </td> <td style="background:#aadddd;"><b>Predicted negative (PN)</b> </td> <td style="border-left:double silver;background:#ffffff"><a href="/wiki/Youden%27s_J_statistic" title="Youden's J statistic">Informedness</a>, <span style="font-size:85%;">bookmaker informedness (BM)</span> <br /><span class="texhtml">= TPR + TNR − 1</span> </td> <td style="background:#ffffff;"><a href="/wiki/Prevalence_threshold" class="mw-redirect" title="Prevalence threshold">Prevalence threshold</a> (PT) <br /><span class="texhtml">= <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">TPR × FPR</span></span> - FPR</span><span class="sr-only">/</span><span class="den">TPR - FPR</span></span>⁠</span></span> </td></tr> <tr> <td rowspan="2" class="nowrap ts-vertical-header is-valign-middle" style="background:#eeeebb;"><div style=""><style data-mw-deduplicate="TemplateStyles:r1221560606">@supports(writing-mode:vertical-rl){.mw-parser-output .ts-vertical-header{line-height:1;max-width:1em;padding:0.4em;vertical-align:bottom;width:1em}html.client-js .mw-parser-output .sortable:not(.jquery-tablesorter) .ts-vertical-header:not(.unsortable),html.client-js .mw-parser-output .ts-vertical-header.headerSort{background-position:50%.4em;padding-right:0.4em;padding-top:21px}.mw-parser-output .ts-vertical-header.is-valign-top{vertical-align:top}.mw-parser-output .ts-vertical-header.is-valign-middle{vertical-align:middle}.mw-parser-output .ts-vertical-header.is-normal{font-weight:normal}.mw-parser-output .ts-vertical-header>*{display:inline-block;transform:rotate(180deg);writing-mode:vertical-rl}@supports(writing-mode:sideways-lr){.mw-parser-output .ts-vertical-header>*{transform:none;writing-mode:sideways-lr}}}</style><b>Actual condition</b></div> </td> <td style="background:#ffffcc;"><b>Positive (P)</b> <sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </td> <td style="background:#ccffcc;"><b><a href="/wiki/True_positive" title="True positive">True positive</a></b> (TP), <br /><span style="font-size:85%;">hit</span><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> </td> <td style="background:#ffdddd;"><b><a href="/wiki/False_negative" class="mw-redirect" title="False negative">False negative</a></b> (FN), <br /><span style="font-size:85%;">miss, underestimation</span> </td> <td style="background:#eeffee;"><a href="/wiki/True_positive_rate" class="mw-redirect" title="True positive rate">True positive rate</a> (TPR), <a href="/wiki/Recall_(information_retrieval)" class="mw-redirect" title="Recall (information retrieval)">recall</a>, <a href="/wiki/Sensitivity_(tests)" class="mw-redirect" title="Sensitivity (tests)">sensitivity</a> (SEN), <span style="font-size:85%;">probability of detection, hit rate, <a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">power</a></span> <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">TP</span><span class="sr-only">/</span><span class="den">P</span></span>⁠</span></span> <span class="texhtml">= 1 − FNR</span> </td> <td style="background:#ffeeee;"><a href="/wiki/False_negative_rate" class="mw-redirect" title="False negative rate">False negative rate</a> (FNR), <br /><span style="font-size:85%;">miss rate</span> <br /><span style="font-size:85%;"><a href="/wiki/Type_II_error" class="mw-redirect" title="Type II error">type II error</a></span> <sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">FN</span><span class="sr-only">/</span><span class="den">P</span></span>⁠</span></span> <span class="texhtml">= 1 − TPR</span> </td></tr> <tr> <td style="background:#ddddaa;"><b>Negative (N)</b><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> </td> <td style="background:#ffcccc;"><b><a href="/wiki/False_positive" class="mw-redirect" title="False positive">False positive</a></b> (FP), <br /><span style="font-size:85%;">false alarm, overestimation</span> </td> <td style="background:#bbeebb;"><b><a href="/wiki/True_negative" class="mw-redirect" title="True negative">True negative</a></b> (TN), <br /><span style="font-size:85%;">correct rejection</span><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>e<span class="cite-bracket">]</span></a></sup> </td> <td style="background:#eedddd;"><a href="/wiki/False_positive_rate" title="False positive rate">False positive rate</a> (FPR), <br /><span style="font-size:85%;">probability of false alarm, <a href="/wiki/Evaluation_measures_(information_retrieval)#Fall-out" title="Evaluation measures (information retrieval)"><span class="nowrap">fall-out</span></a></span> <br /><span style="font-size:85%;"><a href="/wiki/Type_I_error" class="mw-redirect" title="Type I error">type I error</a></span> <sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>f<span class="cite-bracket">]</span></a></sup> <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">FP</span><span class="sr-only">/</span><span class="den">N</span></span>⁠</span></span> <span class="texhtml">= 1 − TNR</span> </td> <td style="background:#ddeedd;"><a href="/wiki/True_negative_rate" class="mw-redirect" title="True negative rate">True negative rate</a> (TNR), <br /><span style="font-size:85%;"><a href="/wiki/Specificity_(tests)" class="mw-redirect" title="Specificity (tests)">specificity</a> (SPC), selectivity</span> <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">TN</span><span class="sr-only">/</span><span class="den">N</span></span>⁠</span></span> <span class="texhtml">= 1 − FPR</span> </td></tr> <tr> <td style="border:none;" rowspan="3"> </td> <td style="border-top:double silver; border-right:double silver;background:#ffffff;"><a href="/wiki/Prevalence" title="Prevalence">Prevalence</a> <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">P</span><span class="sr-only">/</span><span class="den">P + N</span></span>⁠</span></span> </td> <td style="background:#eeffee;"><span class="nowrap"><a href="/wiki/Positive_predictive_value" class="mw-redirect" title="Positive predictive value">Positive predictive value</a> (PPV),</span> <span style="font-size:85%;"><a href="/wiki/Precision_(information_retrieval)" class="mw-redirect" title="Precision (information retrieval)">precision</a></span> <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">TP</span><span class="sr-only">/</span><span class="den">PP</span></span>⁠</span></span> <span class="texhtml">= 1 − FDR</span> </td> <td style="background:#ffeeee;border-right:double silver;"><a href="/wiki/False_omission_rate" class="mw-redirect" title="False omission rate">False omission rate</a> (FOR) <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">FN</span><span class="sr-only">/</span><span class="den">PN</span></span>⁠</span></span> <span class="texhtml">= 1 − NPV</span> </td> <td style="background:#eeeeff;"><a href="/wiki/Positive_likelihood_ratio" class="mw-redirect" title="Positive likelihood ratio">Positive likelihood ratio</a> (LR+) <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">TPR</span><span class="sr-only">/</span><span class="den">FPR</span></span>⁠</span></span> </td> <td style="background:#eeeeff;"><a href="/wiki/Negative_likelihood_ratio" class="mw-redirect" title="Negative likelihood ratio">Negative likelihood ratio</a> (LR−) <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">FNR</span><span class="sr-only">/</span><span class="den">TNR</span></span>⁠</span></span> </td></tr> <tr> <td style="border-right:double silver;background:#ffffff;"><a href="/wiki/Accuracy_and_precision#In_classification" title="Accuracy and precision">Accuracy</a> (ACC) <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">TP + TN</span><span class="sr-only">/</span><span class="den">P + N</span></span>⁠</span></span> </td> <td style="background:#eedddd;"><a href="/wiki/False_discovery_rate" title="False discovery rate">False discovery rate</a> (FDR) <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">FP</span><span class="sr-only">/</span><span class="den">PP</span></span>⁠</span></span> <span class="texhtml">= 1 − PPV</span> </td> <td style="background:#ddeedd;"><a href="/wiki/Negative_predictive_value" class="mw-redirect" title="Negative predictive value">Negative predictive value</a> (NPV) <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">TN</span><span class="sr-only">/</span><span class="den">PN</span></span>⁠</span></span> <span class="texhtml">= 1 − FOR</span> </td> <td style="border-top:double silver;border-right:double silver;background:#ffffff;"><a href="/wiki/Markedness" title="Markedness">Markedness</a> (MK), <span style="font-size:85%;">deltaP (Δp)</span> <br /><span class="texhtml">= PPV + NPV − 1</span> </td> <td style="background:#eeeeff;"><a href="/wiki/Diagnostic_odds_ratio" title="Diagnostic odds ratio">Diagnostic <span class="nowrap">odds ratio</span></a> (DOR) <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">LR+</span><span class="sr-only">/</span><span class="den">LR−</span></span>⁠</span></span> </td></tr> <tr> <td style="background:#ffffff;">Balanced accuracy (BA) <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">TPR + TNR</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> </td> <td style="border-top:double silver;background:#ffffff;"><a href="/wiki/F1_score" class="mw-redirect" title="F1 score">F<sub>1</sub> score</a> <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2 PPV × TPR</span><span class="sr-only">/</span><span class="den">PPV + TPR</span></span>⁠</span></span> <span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2 TP</span><span class="sr-only">/</span><span class="den">2 TP + FP + FN</span></span>⁠</span></span> </td> <td style="border-top:double silver;background:#ffffff;"><a href="/wiki/Fowlkes%E2%80%93Mallows_index" title="Fowlkes–Mallows index">Fowlkes–Mallows index</a> (FM) <br /><span class="texhtml">= <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">PPV × TPR</span></span></span> </td> <td style="border-top:double silver;background:#ffffff;"><a href="/wiki/Matthews_correlation_coefficient" class="mw-redirect" title="Matthews correlation coefficient">Matthews correlation coefficient</a> (MCC) <br /><span class="texhtml texhtml-big" style="font-size:80%;">= <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">TPR × TNR × PPV × NPV</span></span></span> <span class="texhtml texhtml-big" style="font-size:80%;">- <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">FNR × FPR × FOR × FDR</span></span></span> </td> <td style="border-top:double silver;background:#ffffff;" colspan="2">Threat score (TS), critical success index (CSI), <a href="/wiki/Jaccard_index#Jaccard_index_in_binary_classification_confusion_matrices" title="Jaccard index">Jaccard index</a> <br /><span class="texhtml">= <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">TP</span><span class="sr-only">/</span><span class="den">TP + FN + FP</span></span>⁠</span></span> </td></tr></tbody></table> <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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">the number of real positive cases in the data</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">A test result that correctly indicates the presence of a condition or characteristic</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Type II error: A test result which wrongly indicates that a particular condition or attribute is absent</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">the number of real negative cases in the data</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">A test result that correctly indicates the absence of a condition or characteristic</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Type I error: A test result which wrongly indicates that a particular condition or attribute is present</span> </li> </ol></div></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Harmonic_mean_3D_plot_from_0_to_100.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Harmonic_mean_3D_plot_from_0_to_100.png/300px-Harmonic_mean_3D_plot_from_0_to_100.png" decoding="async" width="300" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Harmonic_mean_3D_plot_from_0_to_100.png/450px-Harmonic_mean_3D_plot_from_0_to_100.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Harmonic_mean_3D_plot_from_0_to_100.png/600px-Harmonic_mean_3D_plot_from_0_to_100.png 2x" data-file-width="1578" data-file-height="1491" /></a><figcaption>Normalised harmonic mean plot where <i>x</i> is precision, <i>y</i> is recall and the vertical axis is F<sub>1</sub> score, in percentage points</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PR_curve_with_optimal_fscore.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/PR_curve_with_optimal_fscore.png/220px-PR_curve_with_optimal_fscore.png" decoding="async" width="220" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/PR_curve_with_optimal_fscore.png/330px-PR_curve_with_optimal_fscore.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/PR_curve_with_optimal_fscore.png/440px-PR_curve_with_optimal_fscore.png 2x" data-file-width="653" data-file-height="547" /></a><figcaption>Precision-Recall Curve: points from different thresholds are color coded, the point with optimal F-score is highlighted in red</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Dependence_of_the_F-score_on_class_imbalance">Dependence of the F-score on class imbalance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=5" title="Edit section: Dependence of the F-score on class imbalance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Precision-recall curve, and thus 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 F_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb39f5958eb83bc3c28cd8b1a5397d3bb14fecdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.669ex; height:2.843ex;" alt="{\displaystyle F_{\beta }}"></span> score, explicitly depends on the ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> of positive to negative test cases.<sup id="cite_ref-brabec2020-model_18-0" class="reference"><a href="#cite_note-brabec2020-model-18"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This means that comparison of the F-score across different problems with differing class ratios is problematic. One way to address this issue (see e.g., Siblini et al., 2020<sup id="cite_ref-siblini-20_19-0" class="reference"><a href="#cite_note-siblini-20-19"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> ) is to use a standard class ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb12fcfddb65e3d1e6a044215f6e833f0cd4337b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{0}}"></span> when making such comparisons. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=6" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The F-score is often used in the field of <a href="/wiki/Information_retrieval" title="Information retrieval">information retrieval</a> for measuring <a href="/wiki/Web_search" class="mw-redirect" title="Web search">search</a>, <a href="/wiki/Document_classification" title="Document classification">document classification</a>, and <a href="/wiki/Query_classification" class="mw-redirect" title="Query classification">query classification</a> performance.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> It is particularly relevant in applications which are primarily concerned with the positive class and where the positive class is rare relative to the negative class. </p><p>Earlier works focused primarily on the F<sub>1</sub> score, but with the proliferation of large scale search engines, performance goals changed to place more emphasis on either precision or recall<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb39f5958eb83bc3c28cd8b1a5397d3bb14fecdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.669ex; height:2.843ex;" alt="{\displaystyle F_{\beta }}"></span> is seen in wide application. </p><p>The F-score is also used in <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> However, the F-measures do not take true negatives into account, hence measures such as the <a href="/wiki/Matthews_correlation_coefficient" class="mw-redirect" title="Matthews correlation coefficient">Matthews correlation coefficient</a>, <a href="/wiki/Informedness" class="mw-redirect" title="Informedness">Informedness</a> or <a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a> may be preferred to assess the performance of a binary classifier.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The F-score has been widely used in the natural language processing literature,<sup id="cite_ref-Derczynski2016_24-0" class="reference"><a href="#cite_note-Derczynski2016-24"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> such as in the evaluation of <a href="/wiki/Named_entity_recognition" class="mw-redirect" title="Named entity recognition">named entity recognition</a> and <a href="/wiki/Word_segmentation" class="mw-redirect" title="Word segmentation">word segmentation</a>. </p> <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=F-score&action=edit&section=7" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The F<sub>1</sub> score is the <a href="/wiki/Dice_coefficient" class="mw-redirect" title="Dice coefficient">Dice coefficient</a> of the set of retrieved items and the set of relevant items.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <ul><li>The F<sub>1</sub>-score of a classifier which always predicts the positive class converges to 1 as the probability of the positive class increases.</li> <li>The F<sub>1</sub>-score of a classifier which always predicts the positive class is equal to 2 * proportion_of_positive_class / ( 1 + proportion_of_positive_class ), since the recall is 1, and the precision is equal to the proportion of the positive class.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li> <li>If the scoring model is uninformative (cannot distinguish between the positive and negative class) then the optimal threshold is 0 so that the positive class is always predicted.</li> <li>F<sub>1</sub> score is <a href="/wiki/Concave_function" title="Concave function">concave</a> in the true positive rate.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Criticism">Criticism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=8" title="Edit section: Criticism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/David_Hand_(statistician)" title="David Hand (statistician)">David Hand</a> and others criticize the widespread use of the F<sub>1</sub> score since it gives equal importance to precision and recall. In practice, different types of mis-classifications incur different costs. In other words, the relative importance of precision and recall is an aspect of the problem.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>According to Davide Chicco and Giuseppe Jurman, the F<sub>1</sub> score is less truthful and informative than the <a href="/wiki/Matthews_correlation_coefficient" class="mw-redirect" title="Matthews correlation coefficient">Matthews correlation coefficient (MCC)</a> in binary evaluation classification.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p><a href="/w/index.php?title=David_M_W_Powers&action=edit&redlink=1" class="new" title="David M W Powers (page does not exist)">David M W Powers</a> has pointed out that F<sub>1</sub> ignores the True Negatives and thus is misleading for unbalanced classes, while kappa and correlation measures are symmetric and assess both directions of predictability - the classifier predicting the true class and the true class predicting the classifier prediction, proposing separate multiclass measures <a href="/wiki/Informedness" class="mw-redirect" title="Informedness">Informedness</a> and <a href="/wiki/Markedness" title="Markedness">Markedness</a> for the two directions, noting that their geometric mean is correlation.<sup id="cite_ref-Powers2007_30-0" class="reference"><a href="#cite_note-Powers2007-30"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Another source of critique of F<sub>1</sub> is its lack of symmetry. It means it may change its value when dataset labeling is changed - the "positive" samples are named "negative" and vice versa. This criticism is met by the <a href="/wiki/P4-metric" title="P4-metric">P4 metric</a> definition, which is sometimes indicated as a symmetrical extension of F<sub>1</sub>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Difference_from_Fowlkes–Mallows_index"><span id="Difference_from_Fowlkes.E2.80.93Mallows_index"></span>Difference from Fowlkes–Mallows index</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=9" title="Edit section: Difference from Fowlkes–Mallows index"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While the F-measure is the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> of recall and precision, the <a href="/wiki/Fowlkes%E2%80%93Mallows_index" title="Fowlkes–Mallows index">Fowlkes–Mallows index</a> is their <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Extension_to_multi-class_classification">Extension to multi-class classification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=10" title="Edit section: Extension to multi-class classification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The F-score is also used for evaluating classification problems with more than two classes (<a href="/wiki/Multiclass_classification" title="Multiclass classification">Multiclass classification</a>). A common method is to average the F-score over each class, aiming at a balanced measurement of performance.<sup id="cite_ref-op24_33-0" class="reference"><a href="#cite_note-op24-33"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Macro_F1">Macro F1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=11" title="Edit section: Macro F1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Macro F1</i> is a macro-averaged F1 score aiming at a balanced performance measurement. To calculate macro F1, two different averaging-formulas have been used: the F1 score of (arithmetic) class-wise precision and recall means or the arithmetic mean of class-wise F1 scores, where the latter exhibits more desirable properties.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Micro_F1">Micro F1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=12" title="Edit section: Micro F1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Micro F1</i> is the harmonic mean of <i>micro precision</i> (number of correct predictions normalized by false positives) and <i>micro recall</i> (number of correct predictions normalized by false negatives). Since in multi-class evaluation the overall amount of false positives equals the amount of false negatives, <i>micro F1</i> is equivalent to <i>Accuracy</i>.<sup id="cite_ref-op24_33-1" class="reference"><a href="#cite_note-op24-33"><span class="cite-bracket">[</span>27<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=F-score&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/BLEU" title="BLEU">BLEU</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li> <li><a href="/wiki/Evaluation_of_binary_classifiers#Single_metrics" title="Evaluation of binary classifiers">Hypothesis tests for accuracy</a></li> <li><a href="/wiki/METEOR" title="METEOR">METEOR</a></li> <li><a href="/wiki/NIST_(metric)" title="NIST (metric)">NIST (metric)</a></li> <li><a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">Receiver operating characteristic</a></li> <li><a href="/wiki/ROUGE_(metric)" title="ROUGE (metric)">ROUGE (metric)</a></li> <li><a href="/wiki/Uncertainty_coefficient" title="Uncertainty coefficient">Uncertainty coefficient</a>, aka Proficiency</li> <li><a href="/wiki/Word_error_rate" title="Word error rate">Word error rate</a></li> <li><a href="/wiki/LEPOR" title="LEPOR">LEPOR</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Machine_learning_evaluation_metrics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Machine_learning_evaluation_metrics" title="Template:Machine learning evaluation metrics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Machine_learning_evaluation_metrics" title="Template talk:Machine learning evaluation metrics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Machine_learning_evaluation_metrics" title="Special:EditPage/Template:Machine learning evaluation metrics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Machine_learning_evaluation_metrics" style="font-size:114%;margin:0 4em"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a> evaluation metrics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean_squared_error" title="Mean squared error">MSE</a></li> <li><a href="/wiki/Mean_absolute_error" title="Mean absolute error">MAE</a></li> <li><a href="/wiki/Symmetric_mean_absolute_percentage_error" title="Symmetric mean absolute percentage error">sMAPE</a></li> <li><a href="/wiki/Mean_absolute_percentage_error" title="Mean absolute percentage error">MAPE</a></li> <li><a href="/wiki/Mean_absolute_scaled_error" title="Mean absolute scaled error">MASE</a></li> <li><a href="/wiki/Mean_squared_prediction_error" title="Mean squared prediction error">MSPE</a></li> <li><a href="/wiki/Root_mean_square" title="Root mean square">RMS</a></li> <li><a href="/wiki/Root-mean-square_deviation" class="mw-redirect" title="Root-mean-square deviation">RMSE/RMSD</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">R<sup>2</sup></a></li> <li><a href="/wiki/Mean_directional_accuracy" title="Mean directional accuracy">MDA</a></li> <li><a href="/wiki/Median_absolute_deviation" title="Median absolute deviation">MAD</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">F-score</a></li> <li><a href="/wiki/P4-metric" title="P4-metric">P4</a></li> <li><a href="/wiki/Accuracy_and_precision" title="Accuracy and precision">Accuracy</a></li> <li><a href="/wiki/Precision_and_recall" title="Precision and recall">Precision</a></li> <li><a href="/wiki/Precision_and_recall" title="Precision and recall">Recall</a></li> <li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Kappa</a></li> <li><a href="/wiki/Phi_coefficient" title="Phi coefficient">MCC</a></li> <li><a href="/wiki/Receiver_operating_characteristic#Area_under_the_curve" title="Receiver operating characteristic">AUC</a></li> <li><a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">ROC</a></li> <li><a href="/wiki/Sensitivity_and_specificity" title="Sensitivity and specificity">Sensitivity and specificity</a></li> <li><a href="/wiki/Cross-entropy#Cross-entropy_loss_function_and_logistic_regression" title="Cross-entropy">Logarithmic Loss</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Silhouette_(clustering)" title="Silhouette (clustering)">Silhouette</a></li> <li><a href="/wiki/Calinski-Harabasz_index" class="mw-redirect" title="Calinski-Harabasz index">Calinski-Harabasz index</a></li> <li><a href="/wiki/Davies%E2%80%93Bouldin_index" title="Davies–Bouldin index">Davies-Bouldin</a></li> <li><a href="/wiki/Dunn_index" title="Dunn index">Dunn index</a></li> <li><a href="/wiki/Hopkins_statistic" title="Hopkins statistic">Hopkins statistic</a></li> <li><a href="/wiki/Jaccard_index" title="Jaccard index">Jaccard index</a></li> <li><a href="/wiki/Rand_index" title="Rand index">Rand index</a></li> <li><a href="/wiki/Similarity_measure" title="Similarity measure">Similarity measure</a></li> <li><a href="/wiki/Simple_matching_coefficient" title="Simple matching coefficient">SMC</a></li> <li><a href="/wiki/SimHash" title="SimHash">SimHash</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ranking_(information_retrieval)" title="Ranking (information retrieval)">Ranking</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean_reciprocal_rank" title="Mean reciprocal rank">MRR</a></li> <li><a href="/wiki/NDCG" class="mw-redirect" title="NDCG">NDCG</a></li> <li><a href="/wiki/Average_precision" class="mw-redirect" title="Average precision">AP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_Vision" class="mw-redirect" title="Computer Vision">Computer Vision</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/PSNR" class="mw-redirect" title="PSNR">PSNR</a></li> <li><a href="/wiki/SSIM" class="mw-redirect" title="SSIM">SSIM</a></li> <li><a href="/wiki/Intersection_over_union" class="mw-redirect" title="Intersection over union">IoU</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Natural_language_processing" title="Natural language processing">NLP</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Perplexity" title="Perplexity">Perplexity</a></li> <li><a href="/wiki/BLEU" title="BLEU">BLEU</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Deep Learning Related Metrics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Inception_score" title="Inception score">Inception score</a></li> <li><a href="/wiki/Fr%C3%A9chet_inception_distance" title="Fréchet inception distance">FID</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Recommender_system" title="Recommender system">Recommender system</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coverage_probability" title="Coverage probability">Coverage</a></li> <li><a href="/w/index.php?title=Intra-list_Similarity&action=edit&redlink=1" class="new" title="Intra-list Similarity (page does not exist)">Intra-list Similarity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Similarity</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cosine_similarity" title="Cosine similarity">Cosine similarity</a></li> <li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson correlation coefficient</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li></ul> </div></td></tr></tbody></table></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F-score&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a 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