Page 219 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 20 1

(5.52)

(5.53)

These are the same as (5.48) and (5.49).

Effect of Design Samples

It is more difficult to discuss the effect of using afinite number of design
samples. Although we would like to keep the formula as general as possible,
in this section a specific family of discriminant functions is investigated to help
determine which approximations should be used.

Error expression: Assume that the discriminant function is a function of
two expected vectors, MI and M2, and two covariance matrices, XI and C2.
Typical examples are the quadratic and linear classifers as

1
h(X) = -(X-M1)TC;I(X-M1)
2

(5.54)
h(X) = (M*-M1)YIX + -(M:z-IM1-M;Z-lM2), 1 (5.55)
2

where c = (Cl+X2)/2. When only a finite number of design samples is avail-
able and M, and C; are estimated from them, h becomes a random variable and

Ah(X) = h(X) - h (X) = EO"!' (5.56)
m
,
I=I
where h(X) = h(X,Ml,M,,%1,&), h(X) = h(X,MI,M2,Xl,X2), and 0"" is the
kth order term of the Taylor series expansion in terms of the variations of M,
and XI. If the design samples are drawn from normal distributions, and M, and
,.
Z, are unbiased estimates (e.g., the sample mean and sample covariance),