## Smoothing Spline Surface Plot

Two criteria can be used to select an estimator
for the function *f*:
- goodness of fit to the data
- smoothness of the fit

A standard measure of goodness of fit is
the mean residual sum of squares

A measure of the smoothness of a fit is
an integrated squared second derivative

A single criterion that combines
the two criteria is then given by

where belongs to the set of
all continuously differentiable functions with square integrable
second derivatives, and is a positive constant.
The estimator that results from minimizing *S*()is called a *thin-plate smoothing spline estimator*.
Wahba and Wendelberger (1980) derived a closed form expression
for the thin-plate smoothing spline estimator.

Note | The computations for a thin-plate smoothing spline are time intensive,
especially for large data sets. |

The smoothing parameter controls the amount of
smoothing; that is, it controls the trade-off between the
goodness of fit to the data and the smoothness of the fit.
You select a smoothing parameter by
specifying a constant *c* in the formula

The values of the explanatory variables are scaled by their
corresponding interquartile ranges before the computations.
This makes the computations independent of the units of
*X*_{1} and *X*_{2}.

After choosing **Graphs:Surface Plot:Spline** from the menu,
you specify a smoothing parameter selection method in
the **Spline Fit** dialog.

**Figure 39.28:** Spline Surface Fit Dialog

The default **Method:GCV** uses a *c* value that
minimizes the generalized cross validation mean
squared error .Figure 39.29 displays smoothing spline estimates
with *c* values of 0.0000831 (the GCV value) and
0.4127 (DF=6).
Use the slider in the table to change
the *c* value of the spline fit.

**Figure 39.29:** Smoothing Spline Surface Plot

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.