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Webb and Lowe (1990) extend the result to MLPs with non-linear Choosing optimum weights to minimize p chooses R such th9t _the transformation from the data to the hidden layer outputs maximizes trace(C,C:) in the space spanned by the outputs of the hidden layer. However, we still need to know how the non-linear transformations of the MLP work in the process of producing a decision boundary.
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Note that the derivation of the linear discriminant in Equations (3.2) to (3.9) does not depend on any assumptions about the distributions of the classes. If we assume a Gaussian distribution then the LDA can be derived via a maximum likelihood argument. Just considering two classes, with distributions N ( p 1 , C,.) and N ( p 2 . maximizing (3.2) reduces to calculating (p1 - pz)7C;1. We write (p1 - p2)7C;1x = g(z), the discriminant scores. The likelihood of being in class 1 versus class 2 is
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where and rt are the densities and priors for classes rearranging we have
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c say. The classification rule is then:
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It would appear that a common practice in the literature (see Brown et al., 2000, for an example) is to form (bl - b 2 ) 7 ( 2 ) - *and estimate c by evaluating the classification criterion at points along the linear discriminant If we do not wish to assume Gaussian distributions with equal covariance matrices we have a number of options. We can use quadratic discriminant analysis 01 some other more general method which makes no distributional assumptions. Another option is to continue to use LDA. Despite the fact that the underlying assumptions are not met, LDA may still perform well due to the small number of parameters that need to be estimated, as compared to quadratic discriminant analysis. Yet another option is to modify LDA to give the best linear classifier in the case where the covariance matrices are not equal. Consider two classes, 1 and 2, for which discriminant scores have been calculated and a c value selected. Figure 3.3 shows the probability of a member of class 1 being misclassified as a member of class 2 for the given c value, and Figure 3.4 shows the probability of a member of class 2 being correctly classified. The curve formed by the locus of points {Pc(211),Pc(212)} is called the receiver operating characteristic (ROC) curve and is shown in Figure 3.5. A better classifier will result in a ROC with smaller P(211) for the same P(Z(2). The measure of classification accuracy that we have generally used (Pc(211) + Pc(112)) is minimized by the point on the
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Figure 3.3 For two classes, class 1 and class 2, the probability of a member of class 1 being misclassified a member of class 2 is shown for a given c value.
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Figure 3.4 For two classes, class 1 and class 2, the probability of a member of class 2 being classified a member of class 2 is shown for a given c value.
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Figure The ROC curve for the two classes shown in Figure 3.4. The curve is formed by the locus of points {Pc(211), Pc(212)}.
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Figure The ROC curve for t.he two classes shown in Figure 3.2, with generalized LDA and QDA.
ROC curve such that length of the sum of the line segments marked P ( 2 ) l ) and 112) in Figure 3.5 is minimized. If we can assume Gaussian distributions with proportional covariance matrices then LDA, that is (p1 - p ~ ) ~ ( C 1 will give the optimum discriminant (Su and Liu, 1993). If we can assume Gaussian distributions with unequal covariance matrices then Anderson and Bahadur (1962) and Clunies-Ross and Riffenburgh ~ 1 (1960) show that the optimum discriminant is given by ( p - p ~ ) ~ ( t l C tzCz)- . Anderson and Bahadur show that no linear procedure is superior to all others across the range of the ROC curve and give iterative procedures for determining t z and c in the following special cases:
minimizing one probability of misclassification for a specified probability of the other;