Page 274 - Introduction to Statistical Pattern Recognition

256 Introduction to Statistical Pattern Recognition

Fig. 6-1 Parzen kernel density estimate.

ne1 function is very limited to either a normal or uniform kernel. In this book,
we will use the following kernel which includes both normal and uniform ker-
nels as special cases:

where r(.) is the gamma function, and m is a parameter determining the shape
of the kernel. It may be verified that, for any value of m, the covariance matrix
of the kernel density (6.3) is r2A. The parameter rn determines the rate at
which the kernel function drops off. For m = 1, (6.3) reduces to a simple nor-
mal kernel. As m becomes large, (6.3) approaches a uniform (hyperelliptical)
kernel, always with a smooth roll-off. The matrix A determines the shape of
the hyperellipsoid, and I' controls the size or volume of the kernel. Other
coefficients are selected to satisfy the two conditions mentioned previously: