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id="toc-Probabilistic_Definition-sublist" class="vector-toc-list"> <li id="toc-No-Skill_Classifiers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#No-Skill_Classifiers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>No-Skill Classifiers</span> </div> </a> <ul id="toc-No-Skill_Classifiers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Imbalanced_data" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Imbalanced_data"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Imbalanced data</span> </div> </a> <ul id="toc-Imbalanced_data-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-F-measure" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#F-measure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>F-measure</span> </div> </a> <ul id="toc-F-measure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitations_as_goals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limitations_as_goals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Limitations as goals</span> </div> </a> <ul id="toc-Limitations_as_goals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multi-class_evaluation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Multi-class_evaluation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Multi-class evaluation</span> </div> </a> <ul id="toc-Multi-class_evaluation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav 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id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Precision and recall</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://az.wikipedia.org/wiki/D%C9%99qiqlik_v%C9%99_tan%C4%B1ma" title="Dəqiqlik və tanıma – Azerbaijani" lang="az" hreflang="az" data-title="Dəqiqlik və tanıma" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%B8%E0%A6%BF%E0%A6%B6%E0%A6%A8_%E0%A6%93_%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A6%B2" title="প্রিসিশন ও রিকল – Bangla" lang="bn" hreflang="bn" data-title="প্রিসিশন ও রিকল" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Precisi%C3%B3_i_reclam" title="Precisió i reclam – Catalan" lang="ca" hreflang="ca" data-title="Precisió i reclam" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Precision_und_Recall" title="Precision und Recall – German" lang="de" hreflang="de" data-title="Precision und Recall" 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/T%C3%A4psus_ja_saagis" title="Täpsus ja saagis – Estonian" lang="et" hreflang="et" data-title="Täpsus ja saagis" 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/Precisi%C3%B3n_y_exhaustividad" title="Precisión y exhaustividad – Spanish" lang="es" hreflang="es" data-title="Precisión y exhaustividad" 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/Doitasun_eta_estaldura" title="Doitasun eta estaldura – Basque" lang="eu" hreflang="eu" data-title="Doitasun eta estaldura" 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%AF%D9%82%D8%AA_%D9%88_%D8%A8%D8%A7%D8%B2%DB%8C%D8%A7%D8%A8%DB%8C" title="دقت و بازیابی – Persian" lang="fa" hreflang="fa" data-title="دقت و بازیابی" 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/Pr%C3%A9cision_et_rappel" title="Précision et rappel – French" lang="fr" hreflang="fr" data-title="Précision et rappel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EB%B0%80%EB%8F%84%EC%99%80_%EC%9E%AC%ED%98%84%EC%9C%A8" title="정밀도와 재현율 – Korean" lang="ko" hreflang="ko" data-title="정밀도와 재현율" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Precisione_e_recupero" title="Precisione e recupero – Italian" lang="it" hreflang="it" data-title="Precisione e recupero" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Presisjon_og_dekning" title="Presisjon og dekning – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Presisjon og dekning" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Precis%C3%A3o_e_revoca%C3%A7%C3%A3o" title="Precisão e revocação – Portuguese" lang="pt" hreflang="pt" data-title="Precisão e revocação" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%BB%D1%83%D1%87%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D1%82%D0%B0_%D0%BF%D0%BE%D0%B2%D0%BD%D0%BE%D1%82%D0%B0" title="Влучність та повнота – Ukrainian" 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item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button 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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 pattern recognition, information <a href="/wiki/Information_retrieval" title="Information retrieval">retrieval</a>, <a href="/wiki/Object_detection" title="Object detection">object detection</a> and <a href="/wiki/Classification_(machine_learning)" class="mw-redirect" title="Classification (machine learning)">classification (machine learning)</a>, <b>precision</b> and <b>recall</b> are performance metrics that apply to data retrieved from a <a href="/wiki/Data_store" title="Data store">collection</a>, <a href="/wiki/Text_corpus" title="Text corpus">corpus</a> or <a href="/wiki/Sample_space" title="Sample space">sample space</a>. </p><p><b>Precision</b> (also called <a href="/wiki/Positive_predictive_value" class="mw-redirect" title="Positive predictive value">positive predictive value</a>) is the fraction of relevant instances among the retrieved instances. Written as a formula: </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 {\text{Precision}}={\frac {\text{Relevant retrieved instances}}{{\text{All }}{\textbf {retrieved}}{\text{ instances}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Precision</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>Relevant retrieved instances</mtext> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>All </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">retrieved</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> instances</mtext> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Precision}}={\frac {\text{Relevant retrieved instances}}{{\text{All }}{\textbf {retrieved}}{\text{ instances}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf4ff05845a63d65d7a43394f708e0882dba7c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.518ex; height:5.509ex;" alt="{\displaystyle {\text{Precision}}={\frac {\text{Relevant retrieved instances}}{{\text{All }}{\textbf {retrieved}}{\text{ instances}}}}}"></span> </p><p><b>Recall</b> (also known as <a href="/wiki/Sensitivity_and_specificity" title="Sensitivity and specificity">sensitivity</a>) is the fraction of relevant instances that were retrieved. Written as a formula: </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 {\text{Recall}}={\frac {\text{Relevant retrieved instances}}{{\text{All }}{\textbf {relevant}}{\text{ instances}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Recall</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>Relevant retrieved instances</mtext> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>All </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">relevant</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> instances</mtext> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Recall}}={\frac {\text{Relevant retrieved instances}}{{\text{All }}{\textbf {relevant}}{\text{ instances}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/541bb1ab5d962d6dfcbc9298d9c7da2a97133dbe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.525ex; height:5.509ex;" alt="{\displaystyle {\text{Recall}}={\frac {\text{Relevant retrieved instances}}{{\text{All }}{\textbf {relevant}}{\text{ instances}}}}}"></span> </p><p>Both precision and recall are therefore based on <a href="/wiki/Relevance_(information_retrieval)" title="Relevance (information retrieval)">relevance</a>. </p><p>Consider a computer program for recognizing dogs (the <b>relevant</b> element) in a digital photograph. Upon processing a picture which contains ten cats and twelve dogs, the program identifies eight dogs. Of the eight elements identified as dogs, only five actually are dogs (<a href="/wiki/True_positive" class="mw-redirect" title="True positive">true positives</a>), while the other three are cats (<a href="/wiki/False_positives_and_false_negatives" title="False positives and false negatives">false positives</a>). Seven dogs were missed (<a href="/wiki/False_positives_and_false_negatives" title="False positives and false negatives">false negatives</a>), and seven cats were correctly excluded (<a href="/wiki/False_positives_and_false_negatives" title="False positives and false negatives">true negatives</a>). The program's precision is then 5/8 (true positives / selected elements) while its recall is 5/12 (true positives / relevant elements). </p><p>Adopting a <a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">hypothesis-testing</a> approach, where in this case, the <a href="/wiki/Null_hypothesis" title="Null hypothesis">null hypothesis</a> is that a given item is <i>irrelevant</i> (not a dog), absence of <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">type I and type II errors</a> (perfect <a href="/wiki/Specificity_and_sensitivity" class="mw-redirect" title="Specificity and sensitivity">specificity and sensitivity</a>) corresponds respectively to perfect precision (no false positives) and perfect recall (no false negatives). </p><p>More generally, recall is simply the complement of the type II error rate (i.e., one minus the type II error rate). Precision is related to the type I error rate, but in a slightly more complicated way, as it also depends upon the <a href="/wiki/Prior_distribution" class="mw-redirect" title="Prior distribution">prior distribution</a> of seeing a relevant vs. an irrelevant item. </p><p>The above cat and dog example contained 8 − 5 = 3 type I errors (false positives) out of 10 total cats (true negatives), for a type I error rate of 3/10, and 12 − 5 = 7 type II errors (false negatives), for a type II error rate of 7/12. Precision can be seen as a measure of quality, and recall as a measure of quantity. Higher precision means that an algorithm returns more relevant results than irrelevant ones, and high recall means that an algorithm returns most of the relevant results (whether or not irrelevant ones are also returned). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precision_and_recall&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a <a href="/wiki/Classification_(machine_learning)" class="mw-redirect" title="Classification (machine learning)">classification</a> task, the precision for a class is the <i>number of true positives</i> (i.e. the number of items correctly labelled as belonging to the positive class) <i>divided by the total number of elements labelled as belonging to the positive class</i> (i.e. the sum of true positives and <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">false positives</a>, which are items incorrectly labelled as belonging to the class). Recall in this context is defined as the <i>number of true positives divided by the total number of elements that actually belong to the positive class</i> (i.e. the sum of true positives and <a href="/wiki/Type_I_and_type_II_errors" title="Type I and type II errors">false negatives</a>, which are items which were not labelled as belonging to the positive class but should have been). </p><p>Precision and recall are not particularly useful metrics when used in isolation. For instance, it is possible to have perfect recall by simply retrieving every single item. Likewise, it is possible to achieve perfect precision by selecting only a very small number of extremely likely items. </p><p>In a classification task, a precision score of 1.0 for a class C means that every item labelled as belonging to class C does indeed belong to class C (but says nothing about the number of items from class C that were not labelled correctly) whereas a recall of 1.0 means that every item from class C was labelled as belonging to class C (but says nothing about how many items from other classes were incorrectly also labelled as belonging to class C). </p><p>Often, there is an inverse relationship between precision and recall, where it is possible to increase one at the cost of reducing the other, but context may dictate if one is more valued in a given situation: </p><p>A smoke detector is generally designed to commit many Type I errors (to alert in many situations when there is no danger), because the cost of a Type II error (failing to sound an alarm during a major fire) is prohibitively high. As such, smoke detectors are designed with recall in mind (to catch all real danger), even while giving little weight to the losses in precision (and making many false alarms). In the other direction, <a href="/wiki/Blackstone%27s_ratio" title="Blackstone's ratio">Blackstone's ratio</a>, "It is better that ten guilty persons escape than that one innocent suffer," emphasizes the costs of a Type I error (convicting an innocent person). As such, the criminal justice system is geared toward precision (not convicting innocents), even at the cost of losses in recall (letting more guilty people go free). </p><p>A brain surgeon removing a cancerous tumor from a patient's brain illustrates the tradeoffs as well: The surgeon needs to remove all of the tumor cells since any remaining cancer cells will regenerate the tumor. Conversely, the surgeon must not remove healthy brain cells since that would leave the patient with impaired brain function. The surgeon may be more liberal in the area of the brain they remove to ensure they have extracted all the cancer cells. This decision increases recall but reduces precision. On the other hand, the surgeon may be more conservative in the brain cells they remove to ensure they extracts only cancer cells. This decision increases precision but reduces recall. That is to say, greater recall increases the chances of removing healthy cells (negative outcome) and increases the chances of removing all cancer cells (positive outcome). Greater precision decreases the chances of removing healthy cells (positive outcome) but also decreases the chances of removing all cancer cells (negative outcome). </p><p>Usually, precision and recall scores are not discussed in isolation. A <i>precision-recall curve</i> plots precision as a function of recall; usually precision will decrease as the recall increases. Alternatively, values for one measure can be compared for a fixed level at the other measure (e.g. <i>precision at a recall level of 0.75</i>) or both are combined into a single measure. Examples of measures that are a combination of precision and recall are the <a href="#F-measure">F-measure</a> (the weighted <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> of precision and recall), or the <a href="/wiki/Matthews_correlation_coefficient" class="mw-redirect" title="Matthews correlation coefficient">Matthews correlation coefficient</a>, which is a <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> of the chance-corrected variants: the <a href="/wiki/Regression_coefficient" class="mw-redirect" title="Regression coefficient">regression coefficients</a> <a href="/wiki/Informedness" class="mw-redirect" title="Informedness">Informedness</a> (DeltaP') and <a href="/wiki/Markedness" title="Markedness">Markedness</a> (DeltaP).<sup id="cite_ref-Powers2011_1-0" class="reference"><a href="#cite_note-Powers2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> <a href="/wiki/Accuracy_(binary_classification)" class="mw-redirect" title="Accuracy (binary classification)">Accuracy</a> is a weighted arithmetic mean of Precision and Inverse Precision (weighted by Bias) as well as a weighted arithmetic mean of Recall and Inverse Recall (weighted by Prevalence).<sup id="cite_ref-Powers2011_1-1" class="reference"><a href="#cite_note-Powers2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Inverse Precision and Inverse Recall are simply the Precision and Recall of the inverse problem where positive and negative labels are exchanged (for both real classes and prediction labels). <a href="/wiki/True_Positive_Rate" class="mw-redirect" title="True Positive Rate">True Positive Rate</a> and <a href="/wiki/False_Positive_Rate" class="mw-redirect" title="False Positive Rate">False Positive Rate</a>, or equivalently Recall and 1 - Inverse Recall, are frequently plotted against each other as <a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">ROC</a> curves and provide a principled mechanism to explore operating point tradeoffs. Outside of Information Retrieval, the application of Recall, Precision and F-measure are argued to be flawed as they ignore the true negative cell of the <a href="/wiki/Contingency_table" title="Contingency table">contingency table</a>, and they are easily manipulated by biasing the predictions.<sup id="cite_ref-Powers2011_1-2" class="reference"><a href="#cite_note-Powers2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The first problem is 'solved' by using <a href="/wiki/Accuracy_(binary_classification)" class="mw-redirect" title="Accuracy (binary classification)">Accuracy</a> and the second problem is 'solved' by discounting the chance component and renormalizing to <a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a>, but this no longer affords the opportunity to explore tradeoffs graphically. However, <a href="/wiki/Informedness" class="mw-redirect" title="Informedness">Informedness</a> and <a href="/wiki/Markedness" title="Markedness">Markedness</a> are Kappa-like renormalizations of Recall and 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> and their geometric mean <a href="/wiki/Matthews_correlation_coefficient" class="mw-redirect" title="Matthews correlation coefficient">Matthews correlation coefficient</a> thus acts like a debiased F-measure. </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=Precision_and_recall&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For classification tasks, the terms <i>true positives</i>, <i>true negatives</i>, <i>false positives</i>, and <i>false negatives</i> compare the results of the classifier under test with trusted external judgments. The terms <i>positive</i> and <i>negative</i> refer to the classifier's prediction (sometimes known as the <i>expectation</i>), and the terms <i>true</i> and <i>false</i> refer to whether that prediction corresponds to the external judgment (sometimes known as the <i>observation</i>). </p><p>Let us define an experiment from <i>P</i> positive instances and <i>N</i> negative instances for some condition. The four outcomes can be formulated in a 2×2 <a href="/wiki/Contingency_table" title="Contingency table">contingency table</a> or <a href="/wiki/Confusion_matrix" title="Confusion matrix">confusion matrix</a>, as follows: </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 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dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd 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 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.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"><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</b> </td> <td style="background:#aadddd;"><b>Predicted negative</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" class="mw-redirect" 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">TP + FP</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">TN + FN</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">TP + FP</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">TN + FN</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> <p><br /> Precision and recall are then defined as:<sup id="cite_ref-OlsonDelen_18-0" class="reference"><a href="#cite_note-OlsonDelen-18"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}{\text{Precision}}&={\frac {tp}{tp+fp}}\\{\text{Recall}}&={\frac {tp}{tp+fn}}\,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Precision</mtext> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mi>p</mi> </mrow> <mrow> <mi>t</mi> <mi>p</mi> <mo>+</mo> <mi>f</mi> <mi>p</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Recall</mtext> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mi>p</mi> </mrow> <mrow> <mi>t</mi> <mi>p</mi> <mo>+</mo> <mi>f</mi> <mi>n</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{Precision}}&={\frac {tp}{tp+fp}}\\{\text{Recall}}&={\frac {tp}{tp+fn}}\,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37e557b5bfc8de22afa8aad1c187a357ac81bdb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:21.82ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}{\text{Precision}}&={\frac {tp}{tp+fp}}\\{\text{Recall}}&={\frac {tp}{tp+fn}}\,\end{aligned}}}"></span> </p><p>Recall in this context is also referred to as the true positive rate or <a href="/wiki/Sensitivity_and_specificity" title="Sensitivity and specificity">sensitivity</a>, and precision is also referred to as <a href="/wiki/Positive_predictive_value" class="mw-redirect" title="Positive predictive value">positive predictive value</a> (PPV); other related measures used in classification include true negative rate and <a href="/wiki/Accuracy_(binary_classification)" class="mw-redirect" title="Accuracy (binary classification)">accuracy</a>.<sup id="cite_ref-OlsonDelen_18-1" class="reference"><a href="#cite_note-OlsonDelen-18"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> True negative rate is also called <a href="/wiki/Specificity_(tests)#Specificity" class="mw-redirect" title="Specificity (tests)">specificity</a>. </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 {\text{True negative rate}}={\frac {tn}{tn+fp}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>True negative rate</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mi>n</mi> </mrow> <mrow> <mi>t</mi> <mi>n</mi> <mo>+</mo> <mi>f</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{True negative rate}}={\frac {tn}{tn+fp}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af37c5d528612c6dce5987718585ae5313abf79a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.393ex; height:5.676ex;" alt="{\displaystyle {\text{True negative rate}}={\frac {tn}{tn+fp}}\,}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Precision_vs._Recall">Precision vs. Recall</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precision_and_recall&action=edit&section=3" title="Edit section: Precision vs. Recall"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Both precision and recall may be useful in cases where there is imbalanced data. However, it may be valuable to prioritize one metric over the other in cases where the outcome of a false positive or false negative is costly. For example, in medical diagnosis, a false positive test can lead to unnecessary treatment and expenses. In this situation, it is useful to value precision over recall. In other cases, the cost of a false negative is high, and recall may be a more valuable metric. For instance, the cost of a false negative in fraud detection is high, as failing to detect a fraudulent transaction can result in significant financial loss.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Probabilistic_Definition">Probabilistic Definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precision_and_recall&action=edit&section=4" title="Edit section: Probabilistic Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Precision and recall can be interpreted as (estimated) <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probabilities</a>:<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> Precision is given 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 \mathbb {P} (C=P|{\hat {C}}=P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (C=P|{\hat {C}}=P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d7191f6cc960525459ffc8325649ac04002f97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.097ex; height:3.343ex;" alt="{\displaystyle \mathbb {P} (C=P|{\hat {C}}=P)}"></span> while recall is given 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 \mathbb {P} ({\hat {C}}=P|C=P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ({\hat {C}}=P|C=P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aebeedf10af01d84586e09bea105fe22c9500f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.097ex; height:3.343ex;" alt="{\displaystyle \mathbb {P} ({\hat {C}}=P|C=P)}"></span>,<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> 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 {\hat {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7691287b1a9fdf0a9b86dbc296679667e836892c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.843ex;" alt="{\displaystyle {\hat {C}}}"></span> is the predicted class and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> is the actual class (i.e. <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=P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59df237dd6bd01dd6087dbdffe245b9857ba4ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.61ex; height:2.176ex;" alt="{\displaystyle C=P}"></span> means the actual class is positive). Both quantities are, therefore, connected by <a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a>. </p> <div class="mw-heading mw-heading4"><h4 id="No-Skill_Classifiers">No-Skill Classifiers</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precision_and_recall&action=edit&section=5" title="Edit section: No-Skill Classifiers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The probabilistic interpretation allows to easily derive how a no-skill classifier would perform. A no-skill classifiers is defined by the property that the joint probability <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 {P} (C=P,{\hat {C}}=P)=\mathbb {P} (C=P)\mathbb {P} ({\hat {C}}=P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (C=P,{\hat {C}}=P)=\mathbb {P} (C=P)\mathbb {P} ({\hat {C}}=P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9117c4e45615a0f2b1277509dcd146b82dc7be11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.262ex; height:3.343ex;" alt="{\displaystyle \mathbb {P} (C=P,{\hat {C}}=P)=\mathbb {P} (C=P)\mathbb {P} ({\hat {C}}=P)}"></span> is just the product of the unconditional probabilites since the classification and the presence of the class are <a href="/wiki/Independent_event" class="mw-redirect" title="Independent event">independent</a>. </p><p>For example the precision of a no-skill classifier is simply 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 \mathbb {P} (C=P|{\hat {C}}=P)={\frac {\mathbb {P} (C=P,{\hat {C}}=P)}{\mathbb {P} ({\hat {C}}=P)}}=\mathbb {P} (C=P),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (C=P|{\hat {C}}=P)={\frac {\mathbb {P} (C=P,{\hat {C}}=P)}{\mathbb {P} ({\hat {C}}=P)}}=\mathbb {P} (C=P),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feaeded7ac9f1fe184cdba3b88748abfe000e09e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.1ex; height:7.509ex;" alt="{\displaystyle \mathbb {P} (C=P|{\hat {C}}=P)={\frac {\mathbb {P} (C=P,{\hat {C}}=P)}{\mathbb {P} ({\hat {C}}=P)}}=\mathbb {P} (C=P),}"></span> i.e. determined by the probability/frequency with which the class P occurs. </p><p>A similar argument can be made for the recall: <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 {P} ({\hat {C}}=P|C=P)={\frac {\mathbb {P} (C=P,{\hat {C}}=P)}{\mathbb {P} (C=P)}}=\mathbb {P} ({\hat {C}}=P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ({\hat {C}}=P|C=P)={\frac {\mathbb {P} (C=P,{\hat {C}}=P)}{\mathbb {P} (C=P)}}=\mathbb {P} ({\hat {C}}=P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb908712cb0e31d1191b236420ed9a6d5cf1b087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.454ex; height:7.009ex;" alt="{\displaystyle \mathbb {P} ({\hat {C}}=P|C=P)={\frac {\mathbb {P} (C=P,{\hat {C}}=P)}{\mathbb {P} (C=P)}}=\mathbb {P} ({\hat {C}}=P)}"></span> which is the probability for a positive classification. </p> <div class="mw-heading mw-heading2"><h2 id="Imbalanced_data">Imbalanced data</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precision_and_recall&action=edit&section=6" title="Edit section: Imbalanced data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 {\text{Accuracy}}={\frac {TP+TN}{TP+TN+FP+FN}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Accuracy</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>T</mi> <mi>N</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>T</mi> <mi>N</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Accuracy}}={\frac {TP+TN}{TP+TN+FP+FN}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7588abeafe63ab4b8ae63f954978186276e54d01" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.649ex; height:5.343ex;" alt="{\displaystyle {\text{Accuracy}}={\frac {TP+TN}{TP+TN+FP+FN}}\,}"></span> </p><p>Accuracy can be a misleading metric for imbalanced data sets. Consider a sample with 95 negative and 5 positive values. Classifying all values as negative in this case gives 0.95 accuracy score. There are many metrics that don't suffer from this problem. For example, balanced accuracy<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> (bACC) normalizes true positive and true negative predictions by the number of positive and negative samples, respectively, and divides their sum by two: </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 {\text{Balanced accuracy}}={\frac {TPR+TNR}{2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Balanced accuracy</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> <mi>R</mi> <mo>+</mo> <mi>T</mi> <mi>N</mi> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Balanced accuracy}}={\frac {TPR+TNR}{2}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f36a0a23620dcc20bb32cde7335179175f3cb63b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.473ex; height:5.176ex;" alt="{\displaystyle {\text{Balanced accuracy}}={\frac {TPR+TNR}{2}}\,}"></span> </p><p>For the previous example (95 negative and 5 positive samples), classifying all as negative gives 0.5 balanced accuracy score (the maximum bACC score is one), which is equivalent to the expected value of a random guess in a balanced data set. Balanced accuracy can serve as an overall performance metric for a model, whether or not the true labels are imbalanced in the data, assuming the cost of FN is the same as FP. </p><p>The TPR and FPR are a property of a given classifier operating at a specific threshold. However, the overall number of TPs, FPs <i>etc</i> depend on the class imbalance in the data via the 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="{\textstyle r=P/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r=P/N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a39843e1befdc1e378b0d2a6387fdcc12ac287d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.119ex; height:2.843ex;" alt="{\textstyle r=P/N}"></span>. As the recall (or TPR) depends only on positive cases, it is not affected 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="{\textstyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729" 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="{\textstyle r}"></span>, but the precision is. We have that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Precision}}={\frac {TP}{TP+FP}}={\frac {P\cdot TPR}{P\cdot TPR+N\cdot FPR}}={\frac {TPR}{TPR+{\frac {1}{r}}FPR}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Precision</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo>⋅</mo> <mi>T</mi> <mi>P</mi> <mi>R</mi> </mrow> <mrow> <mi>P</mi> <mo>⋅</mo> <mi>T</mi> <mi>P</mi> <mi>R</mi> <mo>+</mo> <mi>N</mi> <mo>⋅</mo> <mi>F</mi> <mi>P</mi> <mi>R</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> <mi>R</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mi>R</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mi>F</mi> <mi>P</mi> <mi>R</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Precision}}={\frac {TP}{TP+FP}}={\frac {P\cdot TPR}{P\cdot TPR+N\cdot FPR}}={\frac {TPR}{TPR+{\frac {1}{r}}FPR}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6c7b7b356278b3c4b1ffef2aba5dc57bcb4f2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:66.681ex; height:6.509ex;" alt="{\displaystyle {\text{Precision}}={\frac {TP}{TP+FP}}={\frac {P\cdot TPR}{P\cdot TPR+N\cdot FPR}}={\frac {TPR}{TPR+{\frac {1}{r}}FPR}}.}"></span> </p><p>Thus the precision has an explicit dependence 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="{\textstyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729" 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="{\textstyle r}"></span>.<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> Starting with balanced classes at <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="{\textstyle r=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7ec5c211e5421110d2d05b930edd3dccaf3ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\textstyle r=1}"></span> and gradually decreasing <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="{\textstyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729" 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="{\textstyle r}"></span>, the corresponding precision will decrease, because the denominator increases. </p><p>Another metric is the predicted positive condition rate (PPCR), which identifies the percentage of the total population that is flagged. For example, for a search engine that returns 30 results (retrieved documents) out of 1,000,000 documents, the PPCR is 0.003%. </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 {\text{Predicted positive condition rate}}={\frac {TP+FP}{TP+FP+TN+FN}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Predicted positive condition rate</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> <mo>+</mo> <mi>T</mi> <mi>N</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Predicted positive condition rate}}={\frac {TP+FP}{TP+FP+TN+FN}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a7ffb0a3e6a146dcba376bb64fa311f9692fd3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:59.959ex; height:5.343ex;" alt="{\displaystyle {\text{Predicted positive condition rate}}={\frac {TP+FP}{TP+FP+TN+FN}}\,}"></span> </p><p>According to Saito and Rehmsmeier, precision-recall plots are more informative than ROC plots when evaluating binary classifiers on imbalanced data. In such scenarios, ROC plots may be visually deceptive with respect to conclusions about the reliability of classification performance.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Different from the above approaches, if an imbalance scaling is applied directly by weighting the confusion matrix elements, the standard metrics definitions still apply even in the case of imbalanced datasets.<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> The weighting procedure relates the confusion matrix elements to the support set of each considered class. </p> <div class="mw-heading mw-heading2"><h2 id="F-measure">F-measure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precision_and_recall&action=edit&section=7" title="Edit section: F-measure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/F1_score" class="mw-redirect" title="F1 score">F1 score</a></div> <p>A measure that combines precision and recall is the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> of precision and recall, the traditional F-measure or balanced F-score: </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 F=2\cdot {\frac {\mathrm {precision} \cdot \mathrm {recall} }{\mathrm {precision} +\mathrm {recall} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mn>2</mn> <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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=2\cdot {\frac {\mathrm {precision} \cdot \mathrm {recall} }{\mathrm {precision} +\mathrm {recall} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd577aee2dd35c5b0e349327528a5ac606c7bbbf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.724ex; height:5.843ex;" alt="{\displaystyle F=2\cdot {\frac {\mathrm {precision} \cdot \mathrm {recall} }{\mathrm {precision} +\mathrm {recall} }}}"></span> </p><p>This measure is approximately the average of the two when they are close, and is more generally the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a>, which, for the case of two numbers, coincides with the square of the <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> divided by the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a>. There are several reasons that the F-score can be criticized, in particular circumstances, due to its bias as an evaluation metric.<sup id="cite_ref-Powers2011_1-3" class="reference"><a href="#cite_note-Powers2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This is also known as 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_{1}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100c7fbf174fe8b06eacc2a6b0bb2e1badd1c7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{1}}"></span> measure, because recall and precision are evenly weighted. </p><p>It is a special case of the general <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> measure (for non-negative real values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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>): </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 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> <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>+</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/49d1ff4917ee4c464f6efbee08735b4a8694e8c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.763ex; height:6.009ex;" alt="{\displaystyle F_{\beta }=(1+\beta ^{2})\cdot {\frac {\mathrm {precision} \cdot \mathrm {recall} }{\beta ^{2}\cdot \mathrm {precision} +\mathrm {recall} }}}"></span> </p><p>Two other commonly used <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> measures are 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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd17e0779153d765b40ebef91533489b87b2e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{2}}"></span> measure, which weights recall higher than precision, and 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_{0.5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0.5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0.5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9709537931fdd8af9c100ef8fc9ef1caa0efcd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.828ex; height:2.509ex;" alt="{\displaystyle F_{0.5}}"></span> measure, which puts more emphasis on precision than recall. </p><p>The F-measure was derived by van Rijsbergen (1979) 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". It is based on van Rijsbergen's effectiveness measure <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_{\alpha }=1-{\frac {1}{{\frac {\alpha }{P}}+{\frac {1-\alpha }{R}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <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> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\alpha }=1-{\frac {1}{{\frac {\alpha }{P}}+{\frac {1-\alpha }{R}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/123f907849440c9bb96f55e97548e0e1ae91295b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:19.836ex; height:7.009ex;" alt="{\displaystyle E_{\alpha }=1-{\frac {1}{{\frac {\alpha }{P}}+{\frac {1-\alpha }{R}}}}}"></span>, the second term being the weighted harmonic mean of precision and recall with weights <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 ,1-\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha ,1-\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d467f7d2541240bcbff9eb2eaf46977d1f5e9c14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.821ex; height:2.843ex;" alt="{\displaystyle (\alpha ,1-\alpha )}"></span>. 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_{\alpha }}"> <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> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }=1-E_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a714da3eb67b6d6c3fdc306b231dd00d0c41333b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.769ex; height:2.843ex;" alt="{\displaystyle F_{\beta }=1-E_{\alpha }}"></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="Limitations_as_goals">Limitations as goals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precision_and_recall&action=edit&section=8" title="Edit section: Limitations as goals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are other parameters and strategies for performance metric of information retrieval system, such as the area under the <a href="/wiki/ROC_curve" class="mw-redirect" title="ROC curve">ROC curve</a> (AUC)<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> or <a href="/wiki/Pseudo-R-squared" title="Pseudo-R-squared">pseudo-R-squared</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Multi-class_evaluation">Multi-class evaluation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Precision_and_recall&action=edit&section=9" title="Edit section: Multi-class evaluation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Precision and recall values can also be calculated for classification problems with more than two classes.<sup id="cite_ref-op24_27-0" class="reference"><a href="#cite_note-op24-27"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> To obtain the precision for a given class, we divide the number of true positives by the classifier bias towards this class (number of times that the classifier has predicted the class). To calculate the recall for a given class, we divide the number of true positives by the prevalence of this class (number of times that the class occurs in the data sample). </p><p>The class-wise precision and recall values can then be combined into an overall multi-class evaluation score, e.g., using the <a href="/wiki/F-score#Macro_F1" title="F-score">macro F1 metric</a>.<sup id="cite_ref-op24_27-1" class="reference"><a href="#cite_note-op24-27"><span class="cite-bracket">[</span>21<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=Precision_and_recall&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Uncertainty_coefficient" title="Uncertainty coefficient">Uncertainty coefficient</a>, also called <i>proficiency</i></li> <li><a href="/wiki/Sensitivity_and_specificity" title="Sensitivity and specificity">Sensitivity and specificity</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li> <li><a href="/wiki/Scoring_rule" title="Scoring rule">Scoring rule</a></li> <li><a href="/wiki/Base_rate_fallacy" title="Base rate fallacy">Base rate fallacy</a></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=Precision_and_recall&action=edit&section=11" 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-Powers2011-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Powers2011_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Powers2011_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Powers2011_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Powers2011_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="CITEREFPowers2011" class="citation journal cs1">Powers, David M W (2011). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191114213255/https://www.flinders.edu.au/science_engineering/fms/School-CSEM/publications/tech_reps-research_artfcts/TRRA_2007.pdf">"Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Machine Learning Technologies</i>. <b>2</b> (1): <span class="nowrap">37–</span>63. 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What you wanted to know about AUC. <a rel="nofollow" class="external free" href="http://fastml.com/what-you-wanted-to-know-about-auc/">http://fastml.com/what-you-wanted-to-know-about-auc/</a></span> </li> <li id="cite_note-op24-27"><span class="mw-cite-backlink">^ <a href="#cite_ref-op24_27-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-op24_27-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOpitz2024" class="citation journal cs1">Opitz, Juri (2024). <a rel="nofollow" class="external text" href="https://doi.org/10.1162/tacl_a_00675">"A Closer Look at Classification Evaluation Metrics and a Critical Reflection of Common Evaluation Practice"</a>. <i>Transactions of the Association for Computational Linguistics</i>. <b>12</b>: <span class="nowrap">820–</span>836. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2404.16958">2404.16958</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2Ftacl_a_00675">10.1162/tacl_a_00675</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Association+for+Computational+Linguistics&rft.atitle=A+Closer+Look+at+Classification+Evaluation+Metrics+and+a+Critical+Reflection+of+Common+Evaluation+Practice&rft.volume=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E820-%3C%2Fspan%3E836&rft.date=2024&rft_id=info%3Aarxiv%2F2404.16958&rft_id=info%3Adoi%2F10.1162%2Ftacl_a_00675&rft.aulast=Opitz&rft.aufirst=Juri&rft_id=https%3A%2F%2Fdoi.org%2F10.1162%2Ftacl_a_00675&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrecision+and+recall" class="Z3988"></span></span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns 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>Baeza-Yates, Ricardo; Ribeiro-Neto, Berthier (1999). <i>Modern Information Retrieval</i>. New York, NY: ACM Press, Addison-Wesley, Seiten 75 ff. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-39829-X" title="Special:BookSources/0-201-39829-X">0-201-39829-X</a></li> <li>Hjørland, Birger (2010); <i>The foundation of the concept of relevance</i>, Journal of the American Society for Information Science and Technology, 61(2), 217-237</li> <li><a href="/wiki/John_Makhoul" title="John Makhoul">Makhoul, John</a>; Kubala, Francis; Schwartz, Richard; and Weischedel, Ralph (1999); <a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.4637"><i>Performance measures for information extraction</i></a>, in <i>Proceedings of DARPA Broadcast News Workshop, Herndon, VA, February 1999</i></li> <li>van Rijsbergen, Cornelis Joost "Keith" (1979); <i>Information Retrieval</i>, London, GB; Boston, MA: Butterworth, 2nd Edition, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-408-70929-4" title="Special:BookSources/0-408-70929-4">0-408-70929-4</a></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=Precision_and_recall&action=edit&section=12" 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="http://www.dcs.gla.ac.uk/Keith/Preface.html">Information Retrieval – C. J. van Rijsbergen 1979</a></li> <li><a rel="nofollow" class="external text" href="http://www.text-analytics101.com/2014/10/computing-precision-and-recall-for.html">Computing Precision and Recall for a Multi-class Classification Problem</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 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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_metrics329" 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 href="/wiki/F-score" title="F-score">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 class="mw-selflink selflink">Precision</a></li> <li><a class="mw-selflink selflink">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 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