Smoother Degrees of Freedom

Fit Analyses

Smoother Degrees of Freedom

For a nonparametric smoother with a parameter $\lambda$ ,the fitted values can be written as

$\hat{y} = H_{\lambda} y$

where y is the n×1 vector of observed responses y_i, $\hat{y}$ is the n×1 vector of fitted values $\hat{ y_{i}} =\hat{f_\lambda}( x_{i})$ ,and the smoother matrix $H_{\lambda}$ is an n×n matrix that depends on the value of $\lambda$ .

The degrees of freedom, or the effective number of parameters, of a smoother can be used to compare different smoothers and to describe the flexibility of the smoother. SAS/INSIGHT software defines the degrees of freedom of a smoother as

$df_{\lambda} = \rm{trace}( H_{\lambda})$

which is the sum of the diagonal elements of $H_{\lambda}$ .

Note Two other popular definitions of degrees of freedom for a smoother are ${\rm{trace}( H_{\lambda}{ H_{\lambda}'}) }$ and ${\rm{trace}(2 H_{\lambda}- H_{\lambda}{ H_{\lambda}'}) }$ (Hastie and Tibshirani 1990).

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