Mean squared prediction error: Difference between revisions

Revision as of 17:03, 28 May 2018

Revision as of 17:10, 28 May 2018

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In [[statistics]] the '''mean squared prediction error''' 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.

If the smoothing or fitting procedure has [[operatorprojection matrix]] (i.e., hat matrix) ''L'', which maps the observed values vector <math>y</math> to predicted values vector <math>\hat{y}</math> via <math>\hat{y}=Ly,</math> then

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

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:<math>\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>

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

==Estimation of MSPE==