Page 202 - Introduction to Statistical Pattern Recognition

184 Introduction to Statistical Pattern Recognition

E{i) Gf+v g(N) , (5.6)
where v = tr (a2ffaY2 K@(?))}/2 is independent of N and treated as a con-
stant determined by the underlying problem. This leads to the following pro-
cedure to estimate f.

A
Change the sample size N as N1 ,N2,. . . ,Nt. For each Ni, compute Y
.*
and subsequently f empirically. Repeat the experiment z times, and
n
approximate E { f } by the sample mean of the z experimental results.

Plot these empirical points E { i} vs. g (N). Then, find the line best fitted
to these points. The slope of this line is v and the y-cross point is the
improved estimate off. There are many possible ways of selecting a
line. The standard procedure would be the minimum mean-square error
approach.

Parametric Formulation

Moments of parameters: In the parametric approach, most of the
expressions we would like to estimate are functions of expected vectors and
covariance matrices. In this section, we will show how the general discussion
of the previous section can be applied to this particular family of parameters.
Assume that N samples are drawn from each of two n-dimensional nor-
mal distributions with expected vectors and covariance matrices given by

(5.7)

Without loss of generality, any two covariance matrices can be simultaneously
diagonalized to I and A, and a coordinate shqt can bring the expected vector of
one class to zero. Normality is assumed here for simplicity. However, the dis-
cussion can be extended to non-normal cases easily. The extension will be
presented at the end of this section. Also, N = N2 = N is assumed here. For
a
NI # NZ, similar discussion could be developed, although the results are a