  Under normality and homoscedasticity assumptions, Linear Discriminant
Analysis (LDA) is known to be optimal in terms of minimising the Bayes error
for binary classification. In the heteroscedastic case, LDA is not guaranteed
to minimise this error. Assuming heteroscedasticity, we derive a linear
classifier, the Gaussian Linear Discriminant (GLD), that directly minimises the
Bayes error for binary classification. In addition, we also propose a local
neighbourhood search (LNS) algorithm to obtain a more robust classifier if the
data is known to have a non-normal distribution. We evaluate the proposed
classifiers on two artificial and ten real-world datasets that cut across a
wide range of application areas including handwriting recognition, medical
diagnosis and remote sensing, and then compare our algorithm against existing
LDA approaches and other linear classifiers. The GLD is shown to outperform the
original LDA procedure in terms of the classification accuracy under
heteroscedasticity. While it compares favourably with other existing
heteroscedastic LDA approaches, the GLD requires as much as 60 times lower
training 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 150
times lower training time to achieve an equivalent classification accuracy on
some of the datasets. Thus, our algorithms can provide a cheap and reliable
option for classification in a lot of expert systems.
