Mean squared prediction error
From Wikipedia, the free encyclopedia
In statistics the mean squared prediction error of a smoothing procedure is the expected sum of squared deviations of the fitted values 
from the (unobservable) function g. If the smoothing procedure has operator matrix L, then
![\operatorname{MSPE}(L)=\operatorname{E}\left[\sum_{i=1}^n\left( g(x_i)-\widehat{g}(x_i)\right)^2\right].](../../../../math/d/e/2/de2c9b097beef569ab18bdbc0d943c18.png)
The MSPE can be decomposed into two terms just like mean squared error is decomposed into bias and variance; however for MSPE one term is the sum of squared biases of the fitted values and another the sum of variances of the fitted values:
![\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/4/8/2/4828a58a2ba335bc693fbbf81f919560.png)
Note that knowledge of g is required in order to calculate MSPE exactly.
[edit] Estimation of MSPE
For the model 
where 
, one may write
![\operatorname{MSPE}(L)=g'(I-L)'(I-L)g+\sigma^2\operatorname{tr}\left[L'L\right].](../../../../math/5/b/9/5b9cbb9f82d73146029634ef06cca571.png)
The first term is equivalent to
![\sum_{i=1}^n\left(\operatorname{E}\left[\widehat{g}(x_i)\right]-g(x_i)\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)'\left(I-L\right)\right].](../../../../math/5/2/f/52f4ea59be7ea07cd76ae1265a1ce292.png)
Thus,
![\operatorname{MSPE}(L)=\operatorname{E}\left[\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2\right]-\sigma^2\left(n-2\operatorname{tr}\left[L\right]\right).](../../../../math/3/1/e/31e0faf77192a3d3bf0fd16b57047969.png)
If σ2 is known or well-estimated by 
, it becomes possible to estimate MSPE by
![\operatorname{\widehat{MSPE}}(L)=\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2-\widehat{\sigma}^2\left(n-2\operatorname{tr}\left[L\right]\right).](../../../../math/4/b/a/4bae55fddf93deded8e06ffb41b3b930.png)
Colin Mallows advocated this method in the construction of his model selection statistic Cp, which is a normalized version of the estimated MSPE:
![C_p=\frac{\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2}{\widehat{\sigma}^2}-n+2\operatorname{tr}\left[L\right].](../../../../math/d/d/f/ddfa4fd8b8fb9a0087431090cdf3974e.png)
where p comes from that fact that the number of parameters p estimated for a parametric smoother is given by ![p=\operatorname{tr}\left[L\right]](../../../../math/d/d/3/dd31cb188bd6b98def808b24f379a67a.png)
, and C is in honor of Cuthbert Daniel.
