Page 146 - Introduction to Statistical Pattern Recognition

128 Introduction to Statistical Pattern Recognition

Now the decision rule has the geometrical interpretation of comparing the
Euclidean distances from X to MI, and M2 according to a threshold. When
PI = P2 = 0.5, the decision boundary is the perpendicular bisector of the line
joining MI and M2, as shown in Fig. 4-4.

Nonwhite Observation Noise

In the more general case when XI = C2 + I, the observation noise is
correlated and is often called colored noise. The Bayes classifier of (4.2)
should be used in this case instead of (4.3). However, it is still useful to view
the decision rule of (4.2) as a correlation classifier or a distance classifier. To
see this, we introduce the "whitening" transformation, Y = ATX, where
A~XA =I. (4.1 1)

It is important to note that as long as C is positive definite, A exists and is non-
singular. Thus, the whitening transformation is reversible, and the observation
Y can be classified as effectively as X.
The expected vector of Y is

Di =E(Ylo;} =ATMi (i = 1,2) (4.12)
for class ai, and the covariance of Y is I for both classes. Hence, all of the
discussion of the preceding section applies to Y if we replace M; [or rni(t)]
with Di [or d,(t)].
In the continuous time case, the transformation becomes an integral as
7
Y=ATX + y(t)=ta(r,T)x(T)dT. (4.13)
The kernel, a (r, T), can be viewed as the impulse response of a whitening filter.
A possible structure for this classifier is shown in Fig. 4-5. We see that we
have the correlation classifier of Fig. 4-1 modified by the addition of whitening
filters.

Example 1: Figure 4-6 shows a two-dimensional example in which a
whitening transformation is effective. Although the two distributions of Fig.