## Kernel Surface Plot

A *kernel estimator* uses an explicitly defined set of
weights at each point **x** to produce the estimate
at **x**. The kernel estimator of *f* has the form

where Wis the weight function that depends on the smoothing parameter and
the diagonal matrix **V**_{x} of the squares
of the sample interquartile ranges.
The weights are derived from a single
function that is independent of the design

where *K*_{0} is a kernel function and
is the bandwidth or smoothing parameter.
The weights are nonnegative and sum to 1.
Symmetric probability density functions
commonly used as kernel functions are

You select a bandwidth for each kernel
estimator by specifying *c* in the formula

where *n* is the sample size.
Both and *c* are independent of the units of **X**.
SAS/INSIGHT software divides the range of each explanatory
variable into a number of evenly spaced intervals, then estimates
the kernel fit on this grid.
For a data point **x**_{i} that lies
between two grid points, a linear interpolation
is used to compute the predicted value.
For **x**_{i} that lies
inside a square of grid points, a pair of points that
lie on the same vertical line as **x**_{i}
and each lying between two grid points can be found.
A linear interpolation of these two points
is used to compute the predicted value.

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

**Figure 39.30:** Kernel Surface Fit Dialog

By default, SAS/INSIGHT software divides the range of
each explanatory variable into 20 evenly spaced intervals,
uses a normal weight, and uses a *c* value that minimizes
.Figure 39.31 illustrates normal
kernel estimates with *c* values of 0.5435 (the GCV value) and 1.0.
Use the slider to change the *c* value of the kernel fit.

**Figure 39.31:** Kernel Surface Plot

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