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By William S. Meisel
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Extra info for Computer-Oriented Approaches to Pattern Recognition
The boundary between classes i a n d j is clearly the locus of points where pi(x) = pj(x); hence, the boundary functions of Eq. 9) are given by In particular, in the two-class problem, we may work efficiently with Eq. 8) or use two discriminant functions, in which case the two approaches are related by A third alternative is to convert an N-class problem into a series of twoclass problems by successive dichotomy, by successively splitting the classes remaining into two groups. If we began with eight classes, we could solve the following three problems in the order given: (1) Find a decision boundary separating classes 1 through 4 from classes 5 through 8.
Several authors discuss the theoretical validity of this approach from a probabilistic viewpoint [5,6,311. We have viewed nearest neighbor pattern classification in terms of the boundaries between classes. It could as well be viewed in terms of discriminant functions. For, if we define = &(X) -d(x, y p ) for y:) the closest sample point of class i to x , where d(x, y) is a distance function, the nearest neighbor rule is equivalent to placing a point z in class i if p,(z) > pj(z) for j # i. 15 illustrates such a discriminant function for a one-dimensional example.
And Mickey, M. , Estimation of Error Rates in Discriminant Analysis, Technomefrics 10, 71 5-725 (1968). 22. , and Green, D. , On the Effectiveness of Receptors in Pattern Recognition Systems, I€€€ Trans. Informutioil Theory 9, 11-17 (1963). 23. Martin, L. , and Meisel, W. , Successive Dichotomies in Pattern Recognition (to be published). 24. Mason, C. J. , Pattern Recognition Bibliography, IEEE Sys. Science and Cybernetics Group Newsletfer, February, October, and December 1970. 25. Meisel. W. , Potential Functions in Mathematical Pattern Recognition, I€€€ Trans.
Computer-Oriented Approaches to Pattern Recognition by William S. Meisel