Page 132 - Introduction to Statistical Pattern Recognition

114 Introduction to Statistical Pattern Recognition

When pi= 1, Var(sIoi} =mo?+2o?[(rn-l)+ ...+ l]=rn20?. There-
fore, the error of the sequential classifier is the same as the one of a classifier
with a single observation, regardless of the value of in.
Multi-sensor fusion: The multi-sensor fusion problem may be handled
in a similar way as the sequential test. Suppose that in different sensors (such
as radar, infrared, and so on) are used to gather data, and the ith sensor gen-
erates a vector Xi with ki measurements. Then, we form a vector with
(k, + . . . + k,) component‘s, concatinating XI, . . . ,X,. However, often in
practice, X, , . . . ,X, are mutually independent. Therefore, the Bayes classifier
becomes

o1 P,
>< ln-, (3.183)
03 p2

where - In px, (Xi I o1 )/px, (Xi I 02) is the minus-log likelihood ratio for the irh
sensor outputs. The Bayes classifier for the multi-sensor system can be
designed by computing the minus-log likelihood ratio for each individual sen-
sor outputs, adding these ratios, and thresholding the summation. Note that
(3.183) is similar to (3.175). However, there is a difference between the
multi-sensor and sequential classifiers in that each likelihood function is dif-
ferent for the multi-sensor classifier, while it is the same for the sequential
classifier. When the outputs of different sensors are correlated, we need to
treat the problem in the (k I + . . . +k,)-dimensional space.

The Wald Sequential Test

Wald sequential test: Instead of fixing in, we may terminate the obser-
vations when s of (3.175) reaches a certain threshold value. That is

s,, 2 a +X’s E 01 ,
a < s,, < h + take the (rn+l)th sample , (3.184)

h I s,, + X’s E w2 ,