Mean squared prediction error: Difference between revisions

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{{Short description|Statistics concept}}

{{Multiple issues|

In [[statistics]] the '''mean squared prediction error''' ('''MSPE'''), also known as '''mean squared error of the predictions''', of a [[smoothing]], [[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) [[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 |first1=Robert S. |last1=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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==Formulation==

In [[statistics]] the '''mean squared prediction error''' ('''MSPE'''), also known as '''mean squared error of the predictions''', of a [[smoothing]], [[curve fitting]], or [[regression (statistics)|regression]] procedure is the expected value of the [[squared deviation|square difference]] between the fitted values implied by the predictive function <math>\widehat{g}</math> and the values of the (unobservable) [[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.

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 value]]s vector <math>\hat{y}=Ly,</math> then PE and MSPE are formulated as:

:<math>\operatorname{PE_i}=g(x_i)-\widehat{g}(x_i),</math>

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 value]]s vector <math>\hat{y}=Ly,</math> then

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

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

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

:<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''' ('''SSPE''').

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

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

==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'']]

{{Machine learning evaluation metrics}}

* [[Model selection]]

== Further readingReferences ==

{{reflist}}

*{{cite book |first1=Robert S. |last1=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 }}

{{Machine learning evaluation metrics}}

{{DEFAULTSORT:Mean Squared Prediction Error}}