Mean squared prediction error
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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 
. If the smoothing procedure has operator matrix 
, then
![\operatorname{MSPE}(L)=\operatorname{E}\left[\sum_{i=1}^n\left( g(x_i)-\widehat{g}(x_i)\right)^2\right].](http://images.wikia.com/psychology/images/math/c/6/b/c6b8d687a595405051fa82bccc8d1790.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].](http://images.wikia.com/psychology/images/math/b/5/e/b5e7051b6fa0ecf2b715ea782b398685.png)
Note that knowledge of is required in order to calculate MSPE exactly.
Estimation of MSPE
Edit
For the model 
where 
, one may write
![\operatorname{MSPE}(L)=g'(I-L)'(I-L)g+\sigma^2\operatorname{tr}\left[L'L\right].](http://images.wikia.com/psychology/images/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].](http://images.wikia.com/psychology/images/math/8/3/5/8355505f8fc474e973dc83e30ecf8fed.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).](http://images.wikia.com/psychology/images/math/3/4/a/34a612d9c780b20a598ee0378989ed08.png)
If 
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).](http://images.wikia.com/psychology/images/math/1/7/6/176f6bc63477d6a9a26973140df0e988.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].](http://images.wikia.com/psychology/images/math/1/5/7/157103c715ef26ff3ac5b4020118387c.png)
where 
comes from that fact that the number of parameters estimated for a parametric smoother is given by ![p=\operatorname{tr}\left[L\right]](http://images.wikia.com/psychology/images/math/d/d/3/dd31cb188bd6b98def808b24f379a67a.png)
, and 
is in honor of Cuthbert Daniel.
See alsoEdit
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