First, dispersion (divergence)

Deviation is one of the early statistical separability measures used in pattern recognition (Swain and Davis, 1987). Deviation is related to the likelihood ratio of class I and class J:

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As shown in Figure 6- 1, for the classification of two kinds of problems, the greater a is, the greater the possibility that the classifier will correctly classify X0 into Class I, and the size of A can be described by the likelihood ratio of X0 points.

In order to facilitate mathematical processing, a logarithmic likelihood ratio can be defined, which is also related to the size of a:

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Figure 6- 1 Point Likelihood Ratio Definition

At this point, the deviation Dij between classes I and J can be defined by log-likelihood ratio:

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These include:

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When the measurement space is multidimensional, the integrals in the above formula all become volume fractions.

From the above analysis, it can be seen that the deviation can be regarded as the sum of the mean of the likelihood ratio of class I mode and the mean of the likelihood ratio of class J mode.

The difficulty in calculating dispersion lies in the volume fraction in Equation (6-4). If the probability density function of each class is assumed to be normal distribution, that is:

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The deviation can be represented by the average vector and covariance matrix:

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Where tr [a] represents the trace of matrix a.

The first term in Equation (6-6) represents the contribution only caused by the difference of covariance matrix, while the second term is the standardized distance between the mean values. Dij is equal to zero only when the mean vector and covariance matrix of the two classes are exactly the same.

The deviation defined above is the deviation between two categories. When the category is greater than 2, there is no formula to express the deviation between multiple categories. The common method is to define the average deviation, that is, to calculate the average deviation of all class pairs. The average deviation is defined as:

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It is the weighted average of the prior probabilities of categories of deviations between pairs of categories. Where m is the number of categories.