Heteroscedastic Gaussian based Correction term for Fisher Discriminant Analysis and Its Kernel Extension

Fisher discriminant analysis (FDA) is a very famous analysis method for classification. However, FDA does give an optimal projection only for Gaussian distributions with equal covariance matrices. In other words, FDA is not optimal in the case of heteroscedastic Gaussian distributions. In this paper, we propose a novel criterion for FDA including a correction term based on the Bhattacharyya distance which is closely related to classification rate. Furthermore, the Chernoff distance based criterion and its kernelized version are proposed as its extension. These proposed criteria have three strong points. The first one is that the correction term based on the Bhattacharyya distance can deal with heteroscedastic Gaussian distributions. The second one is that the correction term based on the Chernoff distance can handle easily the difference in the number of class samples. The third point is that their kernel extensions are easily implemented to be applied to non-Gaussian distributions. As a result, the proposed method is applicable in a wide variety of classification problems. Experimental results using toy simulations and nine kinds of UCI real-world datasets show marked advantages of the proposed method over the conventional FDA.