Bandwidth Selection :: SAS/STAT(R) 12.1 User's Guide

Bandwidth Selection

Several different bandwidth selection methods are available in PROC KDE in the univariate case. Following the recommendations of Jones, Marron, and Sheather (1996), the default method follows a plug-in formula of Sheather and Jones.

This method solves the fixed-point equation

$h = \left[ \frac{R(\varphi )}{nR \left(\hat{f}^{}_{g(h)} \right) \left(\int x^2 \varphi (x) dx \right)^2} \right]^{1/5}$

where $R(\varphi ) = \int \varphi ^2(x) dx$ .

PROC KDE solves this equation by first evaluating it on a grid of values spaced equally on a log scale. The largest two values from this grid that bound a solution are then used as starting values for a bisection algorithm.

The simple normal reference rule works by assuming $\hat{f}$ is Gaussian in the preceding fixed-point equation. This results in

$\displaystyle h$	$\displaystyle =$	$\displaystyle {\hat\sigma } [4/(3n)]^{1/5}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 1.06 \ {\hat\sigma } n^{-1/5}$

where ${\hat\sigma }$ is the sample standard deviation.

Alternatively, the bandwidth can be computed using the interquartile range, Q:

$\displaystyle h$	$\displaystyle =$	$\displaystyle 1.06 \ {\hat\sigma } n^{-1/5}$
$\displaystyle$	$\displaystyle \approx$	$\displaystyle 1.06 \ (Q/1.34) n^{-1/5}$
$\displaystyle$	$\displaystyle \approx$	$\displaystyle 0.785 \ Q n^{-1/5}$

Silverman’s rule of thumb (Silverman, 1986, Section 3.4.2) is computed as

$h = 0.9 \min [{\hat\sigma }, Q/1.34] n^{-1/5}$

The oversmoothed bandwidth is computed as

$h = 3{\hat\sigma } [1/(70 \sqrt {\pi } n)]^{1/5}$

When you specify a WEIGHT variable, PROC KDE uses weighted versions of $Q_{3}$ , $Q_{1}$ , and ${\hat\sigma }$ in the preceding expressions. The weighted quartiles are computed as weighted order statistics, and the weighted variance takes the form

${\hat\sigma }^{2}=\frac{\sum _{i=1}^{n} W_{i}(X_{i}-\bar{X})^{2}}{\sum _{i=1}^{n} W_{i}}$

where $\bar{X} = (\sum _{i=1}^{n} W_{i}X_{i}) / (\sum _{i=1}^{n} W_{i})$ is the weighted sample mean.

For the bivariate case, Wand and Jones (1993) note that automatic bandwidth selection is both difficult and computationally expensive. Their study of various ways of specifying a bandwidth matrix also shows that using two bandwidths, one in each coordinate’s direction, is often adequate. PROC KDE enables you to adjust the two bandwidths by specifying a multiplier for the default bandwidths recommended by Bowman and Foster (1993):

$\displaystyle h_{X}$	$\displaystyle =$	$\displaystyle {\hat\sigma }_{X}n^{-1/6}$
$\displaystyle h_{Y}$	$\displaystyle =$	$\displaystyle {\hat\sigma }_{Y}n^{-1/6}$

Here ${\hat\sigma }_{X}$ and ${\hat\sigma }_{Y}$ are the sample standard deviations of X and Y, respectively. These are the optimal bandwidths for two independent normal variables that have the same variances as X and Y. They are, therefore, conservative in the sense that they tend to oversmooth the surface.

You can specify the BWM= option to adjust the aforementioned bandwidths to provide the appropriate amount of smoothing for your application.

The KDE Procedure

Bandwidth Selection