Page 128 - Introduction to Statistical Pattern Recognition

110 Introduction to Statistical Pattern Recognition

3.5 Sequential Hypothesis Testing

In the problems considered so far, all of the information about the sam-
ple to be classified is presented at one instant. The classifier uses the single
observation vector to make a decision via Bayes rule since no further observa-
tions will be made, and, as a result, we essentially have no control over the
error, unless we can modify the observation process.
In many practical problems, however, the observations are sequential in
nature, and more and more information becomes available as time procedes.
For example, the vibration of a machine is observed to determine whether the
machine is in good or bad condition. In this case, a sequence of observed
waveforms should belong to the same category: either "good" or "bad' condi-
tion. Another popular example is a radar detection problem. Again the
sequence of return pulses over a certain period of time should be from the
same class: either existence or nonexistence of a target. A basic approach to
problems of this type is the averaging of the sequence of observation vectors.
This has the effect of filtering the noise and reducing the observed vectors
down to the expected vector. Thus, it is possible, at least theoretically, to
achieve zero error, provided that the expected vectors of the two classes are not
the same. However, since obtaining an infinite number of observation vectors
is obviously not feasible, it is necessary to have a condition, or rule, which
helps us decide when to terminate the observations. The sequential hypothesis
test, the subject of this section, is a mathematical tool for this type of problem.

The Sequential Test

Let XI,. . . ,X, be the random vectors observed in sequence. These are
assumed to be drawn from the same distribution and thus to be independent
and identically distributed. Using the joint density functions of these m vec-
tors, pj(Xj,. . . .Xnl) (i = 1,2), the minus-log likelihood ratio becomes