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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


Note
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.

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