Page 142 - Introduction to Statistical Pattern Recognition

Chapter 4

PARAMETRIC CLASSIFIERS

The Bayes likelihood ratio test has been shown to be optimal in the
sense that it minimizes the cost or the probability of error. However, in order
to construct the likelihood ratio, we must have the conditional probability den-
sity function for each class. In most applications, we must estimate these den-
sity functions using a finite number of sample observation vectors. Estimation
procedures are available, and will be discussed in Chapters 6 and 7. However,
they may be very complex or require a large number of samples to give accu-
rate results.
Even if we can obtain the densities, the likelihood ratio test may be
difficult to implement; time and storage requirements for the classification pro-
cess may be excessive. Therefore, we are often led to consider a simpler pro-
cedure for designing a pattern classifier. In particular, we may specify the
mathematical form of the classifier, leaving a finite set of parameters to be
determined. The most common choices are linear, quadratic, or piecewise
classifiers which we will discuss in this chapter.
First, we will consider under what conditions the Bayes classifier
becomes quadratic, linear, or piecewise. We will then develop alternative
methods for deriving "good" parametric classifiers even when these conditions
are not met.
The reader should be reminded, however, that the Bayes classifier is the

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