Page 74 - Introduction to Statistical Pattern Recognition

56 Introduction to Statistical Pattern Recognition
.. , 0 1

-p; 0 . . .

1+p: -p;

(3.14)

2
l+Pi -pi

IZ; I = (1 - p y . (3.15)

Therefore, the quadratic equation of (3.11) becomes

1-p:
><
P1
- ~xixi+l (n-1) In 7 In - (3.16)
+
,
l-p2 o? p2
where the second term shows the edge effect of terminating the observation of
random processes within a finite length, and this effect diminishes as n gets
large. If we could ignore the second and fourth terms and make
><
In ( P ,/P2) 0 (P 1= P2), the decision rule becomes (CX;X~+~)/(CX~) t ; that
=
is, the decision is made by estimating the correlation coefficient and threshold-
ing the estimate. Since pl#p2 is the only difference between o1 and o2 in this
case, this decision rule is reasonable.

Example 3: When xk's are mutually independent and exponentially
distributed,

(3.17)

where ajk is the parameter of the exponential distribution for xk and mi, and
u (.) is the step function. Then, h (X) of (3.5) becomes