Mean squared prediction error: Difference between revisions

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{{Short description|Statistics concept}}

{{Multiple issues|

In [[statistics]] the '''mean squared prediction error''' or('''MSPE'''), also known as '''mean squared error of the predictions''' , of a [[smoothing]] or, [[curve fitting]], or [[regression (statistics)|regression]] procedure is the [[expected value]] of the [[Square (algebra)|squared]] '''prediction errors''' ('''PE'''), the [[squared deviation|square difference]] between the fitted values implied by the predictive function <math>\widehat{g}</math> and the values of the (unobservable) function[[true value]] ''g''. It is an inverse measure of the '''''explanatory power''''' of <math>\widehat{g},</math> and can be used in the process of [[cross-validation (statistics)|cross-validation]] of an estimated model.

{{Unreferenced|date=December 2009}}

*Knowledge of ''g'' would be required in order to calculate the MSPE exactly; in practice, MSPE is estimated.<ref>{{cite book |firstfirst1=Robert S. |lastlast1=Pindyck |authorlink=Robert Pindyck |first2=Daniel L. |last2=Rubinfeld |authorlink2=Daniel L. Rubinfeld |title=Econometric Models & Economic Forecasts |___location=New York |publisher=McGraw-Hill |edition=3rd |year=1991 |isbn=0-07-050098-3 |chapter=Forecasting with Time-Series Models |pages=[https://archive.org/details/econometricmodel00pind/page/516 516–535] |chapter-url=https://archive.org/details/econometricmodel00pind/page/516 }}</ref>

{{Expert needed|statistics|reason=no source, and notation/definition problems regarding ''L''|date=October 2019}}

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In [[statistics]] the '''mean squared prediction error''' or '''mean squared error of the predictions''' of a [[smoothing]] or [[curve fitting]] procedure is the expected value of the squared difference between the fitted values implied by the predictive function <math>\widehat{g}</math> and the values of the (unobservable) function ''g''. It is an inverse measure of the explanatory power of <math>\widehat{g},</math> and can be used in the process of [[cross-validation (statistics)|cross-validation]] of an estimated model.

==Formulation==

If the smoothing or fitting procedure has [[projection matrix]] (i.e., hat matrix) ''L'', which maps the observed values vector <math>y</math> to [[predicted valuesvalue]]s vector <math>\hat{y}</math> via <math>\hat{y}=Ly,</math> then PE and MSPE are formulated as:

:<math>\operatorname{MSPEPE_i}(L)=\operatorname{E}\left[\left( g(x_i)-\widehat{g}(x_i)\right)^2\right].,</math>

:<math>n\cdot\operatorname{MSPE}(L)=\sum_{i=1}^n\left(\operatorname{E}\left[\widehatoperatorname{gPE}(x_i)_i^2\right]-g(x_i)\right)^2+=\sum_{i=1}^n \operatorname{varPE}\left[\widehat{g}(x_i)\right]_i^2/n.</math>

The MSPE can be decomposed into two terms: the mean of squared biases of the fitted values and the mean of variances of the fitted values:

The MSPE can be decomposed into two terms: the meansquared of[[bias squared(statistics)|bias]] biases(mean error) of the fitted values and the mean of variances[[variance]] of the fitted values:

:<math>n\cdot\operatorname{MSPE}(L)=\sum_{i=1}^n\left(\operatorname{E}\left[\widehat{g}(x_i)\right]-g(x_i)\right)^2+\sum_{i=1}^n\operatorname{var}\left[\widehat{g}(x_i)\right].</math>

:<math>\operatorname{MSPE}=\operatorname{ME}^2 + \operatorname{VAR},</math>

Knowledge of ''g'' is required in order to calculate the MSPE exactly; otherwise, it can be estimated.

:<math>\operatorname{ME}=\operatorname{E}\left[ \widehat{g}(x_i) - g(x_i)\right]</math>

:<math>\operatorname{VAR}=\operatorname{E}\left[\left(\widehat{g}(x_i) - \operatorname{E}\left[{g}(x_i)\right]\right)^2\right].</math>

The quantity {{math|SSPE{{=}}''n''MSPE}} is called '''sum squared prediction error'''.

The '''root mean squared prediction error''' is the square root of MSPE: {{math|RMSPE{{=}}{{sqrt|MSPE}}}}.

==Computation of MSPE over out-of-sample data==

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==See also==

* [[Akaike information criterion]]

* [[Bias-variance tradeoff]]

* [[Mean squared error]]

* [[Errors and residuals in statistics]]

* [[Law of total variance]]

* [[Mallows's Cp|Mallows's ''C_p'']]

* [[Model selection]]

== References ==

{{reflist}}

{{Machine learning evaluation metrics}}

== Further reading ==

*{{cite book |first=Robert S. |last=Pindyck |authorlink=Robert Pindyck |first2=Daniel L. |last2=Rubinfeld |authorlink2=Daniel L. Rubinfeld |title=Econometric Models & Economic Forecasts |___location=New York |publisher=McGraw-Hill |edition=3rd |year=1991 |isbn=0-07-050098-3 |chapter=Forecasting with Time-Series Models |pages=[https://archive.org/details/econometricmodel00pind/page/516 516–535] |url=https://archive.org/details/econometricmodel00pind/page/516 }}

{{DEFAULTSORT:Mean Squared Prediction Error}}