Page 306 - Introduction to Statistical Pattern Recognition

288 Introduction to Statistical Pattern Recognition

E{p(X)&(X)) = h;c; . (6.126)

When we terminate the expansion of (6.122) for i = m, the squared error
is given by

(6.127)

Thus, hi IC; l2 represents the error due to the elimination of the ith term in the
expansion. This means that, if we can find a set of basis functions such that
hi I ci I decreases quickly as i increases, the set of basis functions forms an
economical representation of the density function.
There is no known procedure for choosing a set of basis functions in the
general multivariate case. Therefore, we will only consider special cases where
the basis functions are well defined.
Both the Fourier series and the Fourier transform are examples of
expanding a function in a set of basis functions. The characteristic function of
a density function is a Fourier transform and is thus one kind of expansion of a
density function. Here we seek a simpler kind of expansion.

One-dimensional case: When a density function is one-dimensional, we
may try many well-known basis functions, such as Fourier series, Legendre,
Gegenbauer, Jacohi, Hermite, and Leguerre polynomials, etc. [22]. Most of
them have been developed for approximating a waveform, but obviously we
can look at a one-dimensional density function as a waveform.
As a typical example of the expansion, let us study the Hermite polyno-
mial which is used to approximate a density function distorted from a normal
distribution. That is,