Page 174 - Introduction to Statistical Pattern Recognition

156 Introduction to Statistical Pattern Recognition

di =ad: +p (4.106)

or

d! =g(d:). (4.107)

In a high-dimensional space, we cannot plot samples to see the distribution.
Therefore, we must rely on mathematical tools to guide us in finding a reason-
able boundary. Once samples are mapped down to a two-dimensional space as
in Fig. 4-9, we can see the distribution and use our own judgement to set up
the boundary. However, the structure of the boundary should not be too com-
plex, because the boundary must work not only for the current, existing sam-
ples but also for samples which will come in the future. We can always draw
a very complex boundary to classify the existing samples without error, but the
boundary may misclassify many of the future samples.

Stationary Processes

The quadratic classifier for stationary processes: When xi is the ith
time-sampled value of a stationary random process, x(t), the contribution of xi
in the discriminant function must be independent of i. The same is true for x’
and X;X;+~ for fixed j’s. Therefore, (4.104) may be simplified to

n n-I
+
h(X) = qo(xx?) + 2q,(~x;x;+1) . . . + 2qn4(xlxn)
i=l i=I
n
+ v(Zx;) + v, . (4.108)
i=l

This is a linear discriminant function of new variables, yo =Cx:,
.
.
.
yI =LY~X~+~, =xlxn, yn = Zx;. However, now the number of vari-
,yn-]
ables is reduced to n+l, and we need to find only n+2 coefficients,
qo, . . . r4n-I, v, and vo.
Orthonormality of the Fourier transform: Stationary processes have
another desirable property, namely that the elements of the (discrefe) Fourier
transform, F(k) and F(L), of the process are uncorrelated. In order to show this,