Mean squared prediction error

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href="/wiki/Statistics" title="Statistics">statistics</a> the <b>mean squared prediction error</b> (<b>MSPE</b>), also known as <b>mean squared error of the predictions</b>, of a <a href="/wiki/Smoothing" title="Smoothing">smoothing</a>, <a href="/wiki/Curve_fitting" title="Curve fitting">curve fitting</a>, or <a href="/wiki/Regression_(statistics)" class="mw-redirect" title="Regression (statistics)">regression</a> procedure is the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of the <a href="/wiki/Square_(algebra)" title="Square (algebra)">squared</a> <b>prediction errors</b> (<b>PE</b>), the <a href="/wiki/Squared_deviation" class="mw-redirect" title="Squared deviation">square difference</a> between the fitted values implied by the predictive function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f03ff4b434f75ad5db89ea97ae9f4336927e6d9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.232ex; height:2.509ex;" alt="{\displaystyle {\widehat {g}}}"></span> and the values of the (unobservable) <a href="/wiki/True_value" class="mw-redirect" title="True value">true value</a> <i>g</i>. It is an inverse measure of the <i><b>explanatory power</b></i> 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 {\widehat {g}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {g}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05a95ce2912aad902cf0c7babeb9250e172f8bce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.878ex; height:2.509ex;" alt="{\displaystyle {\widehat {g}},}"></span> and can be used in the process of <a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">cross-validation</a> of an estimated model. Knowledge of <i>g</i> would be required in order to calculate the MSPE exactly; in practice, MSPE is estimated.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_squared_prediction_error&action=edit&section=1" title="Edit section: Formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the smoothing or fitting procedure has <a href="/wiki/Projection_matrix" title="Projection matrix">projection matrix</a> (i.e., hat matrix) <i>L</i>, which maps the observed values vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> to <a href="/wiki/Predicted_value" class="mw-redirect" title="Predicted value">predicted values</a> vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {y}}=Ly,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>L</mi> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {y}}=Ly,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4823083c5358d6cb907015d4f989a8bda5d99c7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.786ex; height:2.509ex;" alt="{\displaystyle {\hat {y}}=Ly,}"></span> then PE and MSPE are formulated as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {PE_{i}} =g(x_{i})-{\widehat {g}}(x_{i}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">P</mi> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </msub> </mrow> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {PE_{i}} =g(x_{i})-{\widehat {g}}(x_{i}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e321e4e89fea030ffb8a2c54fbf9b10fd4c235b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.666ex; height:2.843ex;" alt="{\displaystyle \operatorname {PE_{i}} =g(x_{i})-{\widehat {g}}(x_{i}),}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {MSPE} =\operatorname {E} \left[\operatorname {PE} _{i}^{2}\right]=\sum _{i=1}^{n}\operatorname {PE} _{i}^{2}/n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>MSPE</mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msubsup> <mi>PE</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>]</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>PE</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {MSPE} =\operatorname {E} \left[\operatorname {PE} _{i}^{2}\right]=\sum _{i=1}^{n}\operatorname {PE} _{i}^{2}/n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4a5bceae972e0557d12efefe37c384d8b22e190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.081ex; height:6.843ex;" alt="{\displaystyle \operatorname {MSPE} =\operatorname {E} \left[\operatorname {PE} _{i}^{2}\right]=\sum _{i=1}^{n}\operatorname {PE} _{i}^{2}/n.}"></span></dd></dl> <p>The MSPE can be decomposed into two terms: the squared <a href="/wiki/Bias_(statistics)" title="Bias (statistics)">bias</a> (mean error) of the fitted values and the <a href="/wiki/Variance" title="Variance">variance</a> of the fitted values: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {MSPE} =\operatorname {ME} ^{2}+\operatorname {VAR} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>MSPE</mi> <mo>=</mo> <msup> <mi>ME</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>VAR</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {MSPE} =\operatorname {ME} ^{2}+\operatorname {VAR} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c761813a999e151040fbd28e3a8cc42257967d4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.882ex; height:3.009ex;" alt="{\displaystyle \operatorname {MSPE} =\operatorname {ME} ^{2}+\operatorname {VAR} ,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ME} =\operatorname {E} \left[{\widehat {g}}(x_{i})-g(x_{i})\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ME</mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ME} =\operatorname {E} \left[{\widehat {g}}(x_{i})-g(x_{i})\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54834f1e0d444f8bbda18b726e98dd157a8803ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.754ex; height:2.843ex;" alt="{\displaystyle \operatorname {ME} =\operatorname {E} \left[{\widehat {g}}(x_{i})-g(x_{i})\right]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {VAR} =\operatorname {E} \left[\left({\widehat {g}}(x_{i})-\operatorname {E} \left[{g}(x_{i})\right]\right)^{2}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>VAR</mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {VAR} =\operatorname {E} \left[\left({\widehat {g}}(x_{i})-\operatorname {E} \left[{g}(x_{i})\right]\right)^{2}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41dc0a2d2ae468dcd6e5fbc6e9bfce56a5abf4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.525ex; height:4.843ex;" alt="{\displaystyle \operatorname {VAR} =\operatorname {E} \left[\left({\widehat {g}}(x_{i})-\operatorname {E} \left[{g}(x_{i})\right]\right)^{2}\right].}"></span></dd></dl> <p>The quantity <span class="texhtml">SSPE=<i>n</i>MSPE</span> is called <b>sum squared prediction error</b>. The <b>root mean squared prediction error</b> is the square root of MSPE: <span class="texhtml">RMSPE=<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">MSPE</span></span></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Computation_of_MSPE_over_out-of-sample_data">Computation of MSPE over out-of-sample data</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_squared_prediction_error&action=edit&section=2" title="Edit section: Computation of MSPE over out-of-sample data"><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">Further information: <a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation (statistics)</a></div> <p>The mean squared prediction error can be computed exactly in two contexts. First, with a <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">data sample</a> of length <i>n</i>, the <a href="/wiki/Data_analyst" class="mw-redirect" title="Data analyst">data analyst</a> may run the <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> over only <i>q</i> of the data points (with <i>q</i> < <i>n</i>), holding back the other <i>n – q</i> data points with the specific purpose of using them to compute the estimated model’s MSPE out of sample (i.e., not using data that were used in the model estimation process). Since the regression process is tailored to the <i>q</i> in-sample points, normally the in-sample MSPE will be smaller than the out-of-sample one computed over the <i>n – q</i> held-back points. If the increase in the MSPE out of sample compared to in sample is relatively slight, that results in the model being viewed favorably. And if two models are to be compared, the one with the lower MSPE over the <i>n – q</i> out-of-sample data points is viewed more favorably, regardless of the models’ relative in-sample performances. The out-of-sample MSPE in this context is exact for the out-of-sample data points that it was computed over, but is merely an estimate of the model’s MSPE for the mostly unobserved population from which the data were drawn. </p><p>Second, as time goes on more data may become available to the data analyst, and then the MSPE can be computed over these new data. </p> <div class="mw-heading mw-heading2"><h2 id="Estimation_of_MSPE_over_the_population">Estimation of MSPE over the population</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_squared_prediction_error&action=edit&section=3" title="Edit section: Estimation of MSPE over the population"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Disputed plainlinks metadata ambox ambox-content ambox-disputed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/System-search.svg/45px-System-search.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/System-search.svg/68px-System-search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/System-search.svg/90px-System-search.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article's <b>factual accuracy is <a href="/wiki/Wikipedia:Accuracy_dispute" title="Wikipedia:Accuracy dispute">disputed</a></b>.<span class="hide-when-compact"> Relevant discussion may be found on the <a href="/wiki/Talk:Mean_squared_prediction_error#section" title="Talk:Mean squared prediction error">talk page</a>. Please help to ensure that disputed statements are <a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources">reliably sourced</a>.</span> <span class="date-container"><i>(<span class="date">May 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>When the model has been estimated over all available data with none held back, the MSPE of the model over the entire <a href="/wiki/Statistical_population" title="Statistical population">population</a> of mostly unobserved data can be estimated as follows. </p><p>For the model <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}=g(x_{i})+\sigma \varepsilon _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>σ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}=g(x_{i})+\sigma \varepsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/661a81d7fa471445f28efa805df23b51afc9a829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.145ex; height:2.843ex;" alt="{\displaystyle y_{i}=g(x_{i})+\sigma \varepsilon _{i}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}\sim {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i}\sim {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f60dc00384c2ef2802099688242782e841d17ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.425ex; height:3.009ex;" alt="{\displaystyle \varepsilon _{i}\sim {\mathcal {N}}(0,1)}"></span>, one may write </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot \operatorname {MSPE} (L)=g^{\text{T}}(I-L)^{\text{T}}(I-L)g+\sigma ^{2}\operatorname {tr} \left[L^{\text{T}}L\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <mi>MSPE</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo>−</mo> <mi>L</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo>−</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msup> <mi>L</mi> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot \operatorname {MSPE} (L)=g^{\text{T}}(I-L)^{\text{T}}(I-L)g+\sigma ^{2}\operatorname {tr} \left[L^{\text{T}}L\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa4a2de4f014dc1c2cf31bc803f5d642c68a26f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.633ex; height:3.343ex;" alt="{\displaystyle n\cdot \operatorname {MSPE} (L)=g^{\text{T}}(I-L)^{\text{T}}(I-L)g+\sigma ^{2}\operatorname {tr} \left[L^{\text{T}}L\right].}"></span></dd></dl> <p>Using in-sample data values, the first term on the right side is equivalent to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}\left(\operatorname {E} \left[g(x_{i})-{\widehat {g}}(x_{i})\right]\right)^{2}=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\operatorname {tr} \left[\left(I-L\right)^{T}\left(I-L\right)\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>−</mo> <mi>L</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>−</mo> <mi>L</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}\left(\operatorname {E} \left[g(x_{i})-{\widehat {g}}(x_{i})\right]\right)^{2}=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\operatorname {tr} \left[\left(I-L\right)^{T}\left(I-L\right)\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f60fa601829826d71a7d40c74ee73564dfab9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:73.35ex; height:7.509ex;" alt="{\displaystyle \sum _{i=1}^{n}\left(\operatorname {E} \left[g(x_{i})-{\widehat {g}}(x_{i})\right]\right)^{2}=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\operatorname {tr} \left[\left(I-L\right)^{T}\left(I-L\right)\right].}"></span></dd></dl> <p>Thus, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot \operatorname {MSPE} (L)=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\left(n-\operatorname {tr} \left[L\right]\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <mi>MSPE</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mi>L</mi> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot \operatorname {MSPE} (L)=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\left(n-\operatorname {tr} \left[L\right]\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6da6d150e8f3f19e291b2e7fd536f366690eff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.385ex; height:7.509ex;" alt="{\displaystyle n\cdot \operatorname {MSPE} (L)=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\left(n-\operatorname {tr} \left[L\right]\right).}"></span></dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span> is known or well-estimated 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 {\widehat {\sigma }}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\sigma }}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afe69bbdc03070ef019f43fe451b847d2acf1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle {\widehat {\sigma }}^{2}}"></span>, it becomes possible to estimate MSPE by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot \operatorname {\widehat {MSPE}} (L)=\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}-{\widehat {\sigma }}^{2}\left(n-\operatorname {tr} \left[L\right]\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">E</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mi>L</mi> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot \operatorname {\widehat {MSPE}} (L)=\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}-{\widehat {\sigma }}^{2}\left(n-\operatorname {tr} \left[L\right]\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4890e1a2ff444ff87af3a9a80dfd00ee733f1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.091ex; height:6.843ex;" alt="{\displaystyle n\cdot \operatorname {\widehat {MSPE}} (L)=\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}-{\widehat {\sigma }}^{2}\left(n-\operatorname {tr} \left[L\right]\right).}"></span></dd></dl> <p><a href="/wiki/Colin_Mallows" class="mw-redirect" title="Colin Mallows">Colin Mallows</a> advocated this method in the construction of his model selection statistic <a href="/wiki/Mallows%27s_Cp" title="Mallows's Cp"><i>C<sub>p</sub></i></a>, which is a normalized version of the estimated MSPE: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}={\frac {\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}}{{\widehat {\sigma }}^{2}}}-n+2p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{p}={\frac {\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}}{{\widehat {\sigma }}^{2}}}-n+2p.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5f99724aeefc7da275f0d324ea2a4b295e9adf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.264ex; height:6.843ex;" alt="{\displaystyle C_{p}={\frac {\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}}{{\widehat {\sigma }}^{2}}}-n+2p.}"></span></dd></dl> <p>where <i>p</i> the number of estimated parameters <i>p</i> 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 {\widehat {\sigma }}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\sigma }}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afe69bbdc03070ef019f43fe451b847d2acf1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle {\widehat {\sigma }}^{2}}"></span> is computed from the version of the model that includes all possible regressors. That concludes this proof. </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=Mean_squared_prediction_error&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">Akaike information criterion</a></li> <li><a href="/wiki/Bias-variance_tradeoff" class="mw-redirect" title="Bias-variance tradeoff">Bias-variance tradeoff</a></li> <li><a href="/wiki/Mean_squared_error" title="Mean squared error">Mean squared error</a></li> <li><a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">Errors and residuals in statistics</a></li> <li><a href="/wiki/Law_of_total_variance" title="Law of total variance">Law of total variance</a></li> <li><a href="/wiki/Mallows%27s_Cp" title="Mallows's Cp">Mallows's <i>C<sub>p</sub></i></a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</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=Mean_squared_prediction_error&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFPindyckRubinfeld1991" class="citation book cs1"><a href="/wiki/Robert_Pindyck" title="Robert Pindyck">Pindyck, Robert S.</a>; <a href="/wiki/Daniel_L._Rubinfeld" title="Daniel L. Rubinfeld">Rubinfeld, Daniel L.</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/econometricmodel00pind/page/516">"Forecasting with Time-Series Models"</a>. <i>Econometric Models & Economic Forecasts</i> (3rd ed.). New York: McGraw-Hill. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/econometricmodel00pind/page/516">516–535</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-050098-3" title="Special:BookSources/0-07-050098-3"><bdi>0-07-050098-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Forecasting+with+Time-Series+Models&rft.btitle=Econometric+Models+%26+Economic+Forecasts&rft.place=New+York&rft.pages=516-535&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1991&rft.isbn=0-07-050098-3&rft.aulast=Pindyck&rft.aufirst=Robert+S.&rft.au=Rubinfeld%2C+Daniel+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feconometricmodel00pind%2Fpage%2F516&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+squared+prediction+error" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><style 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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 class="mw-selflink selflink">MSPE</a></li> <li><a href="/wiki/Root_mean_square" title="Root mean square">RMS</a></li> <li><a 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Mean squared prediction error - Wikipedia