Optimum feature for high dimensional common-mean multiclass classification

Intense research efforts have contributed to feature extraction based on the discriminant information about class mean. Comparatively, the discriminant information about class covariance for the multiclass problem is not fully explored. In this study, we focus our efforts on this type of discriminant information about class covariance and limit the discussion to the multiclass common-mean case. We formulate the classification accuracy and obtain the following interesting results: (a) the classification accuracy can be expressed in terms of error functions in the univariate multiclass common-mean case. (b) For fixed largest and smallest class variances, the maximum classification accuracy occurs as all pairs of successive classes share the same ratio of class variance. An example of three classes with common class mean is given to show that the optimum feature is located in the direction in which the squared variance of class 2 is equal to the product of the variances of classes 1 and 3. The multiclass version of the Decision Boundary Feature Extraction method is applied to this example but cannot extract the optimum feature