Study on Discriminant Matrices of Commonly-used Fisher Discriminant Functions
Cheng Zheng · Acta Automatica Sinica · 2010
In linear discriminant analysis(LDA),ratio-trace,ratio-value,and trace-ratio are three Fisher discriminant functions in common use.Each has one orthogonal discriminant(OD) matrix and one uncorrelated discriminant(UD) matrix.This paper aims to systematically analyze the obtained approach and properties of these six discriminant matrices and to more clearly recognize their relations and differences.When the within-class scatter matrix is nonsingular,the research on the discriminant matrices of ratio-trace and ratio-value functions and the OD matrix of trace-ratio function have already been done.This paper only discusses the obtained approach and properties of UD matrix of trace-ratio function,and obtains such conclusions that the UD matrix of trace-ratio function is equal to that of ratio-trace and ratiovalue function,and the trace-ratio function value of UD matrix is no more than that of OD matrix.When the within-class scatter matrix is singular,it is difficult to obtain these six discriminant matrices.To overcome the difficulty,this paper introduces the limitation idea and redefines these three discriminant functions,then attains the obtained approach of the six discriminant matrices by calculating the limitations.From these obtained approaches,we can conclude that the six discriminant matrices are equal when the need discriminant vectors are all in the null space of the within-class scatter matrix.