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Under normality and homoscedasticity assumptions, Linear DiscriminantAnalysis (LDA) is known to be optimal in terms of minimising the Bayes errorfor binary classification. In the heteroscedastic case, LDA is not guaranteedto minimise this error. Assuming heteroscedasticity, we derive a linearclassifier, the Gaussian Linear Discriminant (GLD), that directly minimises theBayes error for binary classification. In addition, we also propose a localneighbourhood search (LNS) algorithm to obtain a more robust classifier if thedata is known to have a non-normal distribution. We evaluate the proposedclassifiers on two artificial and ten real-world datasets that cut across awide range of application areas including handwriting recognition, medicaldiagnosis and remote sensing, and then compare our algorithm against existingLDA approaches and other linear classifiers. The GLD is shown to outperform theoriginal LDA procedure in terms of the classification accuracy underheteroscedasticity. While it compares favourably with other existingheteroscedastic LDA approaches, the GLD requires as much as 60 times lowertraining time on some datasets. Our comparison with the support vector machine(SVM) also shows that, the GLD, together with the LNS, requires as much as 150times lower training time to achieve an equivalent classification accuracy onsome of the datasets. Thus, our algorithms can provide a cheap and reliableoption for classification in a lot of expert systems.

Linear classifier design under heteroscedasticity in Linear Discriminant Analysis