Previous Page  Next Page 
Fit Analyses

Smoother Generalized Cross Validation

With the degrees of freedom of an estimate { df_{\lambda}},the mean squared error is given as

\rm{MSE}(\lambda) = \frac{1}{n- df_{\lambda}} \sum_{i=1}^n{( y_{i}- \hat{f_\lambda}( x_{i}) )^2}

Cross-validation (CV) estimates the response at each xi from the smoother that uses only the remaining n-1 observations. The resulting cross validation mean squared error is

\rm{MSE}_{CV}(\lambda) = \frac{1}n \sum_{i=1}^n{( y_{i}- \hat{f}_{\lambda (i)} ( x_{i}))^2 }
where { \hat{f}_{\lambda(i)}( x_{i}) } is the fitted value at xi computed without the ith observation.

The cross validation mean squared error can also be written as

\rm{MSE}_{CV}(\lambda) = \frac{1}n \sum_{i=1}^n ( \frac{ y_{i}- \hat{f_\lambda}( x_{i})} {1- h_{\lambda i} } )^2
where { h_{\lambda i}} is the ith diagonal element of the H_{\lambda}matrix (Hastie and Tibshirani 1990).

Generalized cross validation replaces { h_{\lambda i}} by its average value, {\frac{1}n df_{\lambda}}.The generalized cross validation mean squared error is

\rm{MSE}_{GCV}(\lambda) = \frac{1}{n (1- df_{\lambda}/n)^2 } \sum_{i=1}^n ( y_{i}- \hat{f_\lambda}( x_{i}))^2

The function { \rm{MSE}_{GCV}(\lambda)} may have multiple minima, so the value estimated by SAS/INSIGHT software may be only a local minimum, not the global minimum.

Previous Page  Next Page  Top of Page

Copyright © 2007 by SAS Institute Inc., Cary, NC, USA. All rights reserved.