Mean squared prediction error

In statistics the mean squared prediction error of a smoothing procedure is the expected sum of squared deviations of the fitted values $\widehat{g}$ from the (unobservable) function $g$. If the smoothing procedure has operator matrix $L$, then

$\operatorname{MSPE}(L)=\operatorname{E}\left[\sum_{i=1}^n\left( g(x_i)-\widehat{g}(x_i)\right)^2\right].$

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:

$\operatorname{MSPE}(L)=\sum_{i=1}^n\left(\operatorname{E}\left[\widehat{g}(x_i)\right]-g(x_i)\right)^2+\sum_{i=1}^n\operatorname{var}\left[\widehat{g}(x_i)\right].$

Note that knowledge of $g$ is required in order to calculate MSPE exactly.

Estimation of MSPE

For the model $y_i=g(x_i)+\sigma\varepsilon_i$ where $\varepsilon_i\sim\mathcal{N}(0,1)$, one may write

$\operatorname{MSPE}(L)=g'(I-L)'(I-L)g+\sigma^2\operatorname{tr}\left[L'L\right].$

The first term is equivalent to

$\sum_{i=1}^n\left(\operatorname{E}\left[\widehat{g}(x_i)\right]-g(x_i)\right)^2 =\operatorname{E}\left[\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2\right]-\sigma^2\operatorname{tr}\left[\left(I-L\right)'\left(I-L\right)\right].$

Thus,

$\operatorname{MSPE}(L)=\operatorname{E}\left[\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2\right]-\sigma^2\left(n-2\operatorname{tr}\left[L\right]\right).$

If $\sigma^2$ is known or well-estimated by $\widehat{\sigma}^2$, it becomes possible to estimate MSPE by

$\operatorname{\widehat{MSPE}}(L)=\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2-\widehat{\sigma}^2\left(n-2\operatorname{tr}\left[L\right]\right).$

Colin Mallows advocated this method in the construction of his model selection statistic Cp, which is a normalized version of the estimated MSPE:

$C_p=\frac{\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2}{\widehat{\sigma}^2}-n+2\operatorname{tr}\left[L\right].$

where $p$ comes from that fact that the number of parameters $p$ estimated for a parametric smoother is given by $p=\operatorname{tr}\left[L\right]$, and $C$ is in honor of Cuthbert Daniel.

See also

• Mean squared error
• Errors and residuals in statistics
• Law of total variance