Abstract-Discriminants are often used in pattern recognition to separate clusters of points in some multidimensional "feature" space. This paper provides two fast and simple techniques for improving on the classification performance provided by Fisher's linear discriminant for two classes. Both of these methods are also extended to nonlinear decision surfaces through the use of Mercer kernels.

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The current study provides a simple algorithm for finding the optimal ROC curve for a linear discriminant between two point distributions, given only information about the classes' means and covariances. The method makes no assumptions concerning the exact type of distribution and is shown to provide the best possible discrimination for any physically reasonable measure of the classification error. This very general solution is shown to specialise to results obtained in other papers which assumed multi-dimensional Gaussian distributed classes, or minimised the maximum classification error. Some numerical examples are provided which show the improvement in classification of this method over previously used methods.

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