摘要:
在线性判别分析(Linear discriminant analysis, LDA)中, 比迹函数、比值函数和迹比函数是三种常用的Fisher判别函数, 每一个判别函数都可得到一个正交判别(Orthogonal discriminant, OD)矩阵和一个不相关判别(Uncorrelated discriminant, UD)矩阵. 本文的主要目的是对这6种判别矩阵的获取方法及其性质进行系统分析, 拟期更清楚地认识它们的联系与区别. 当类内协方差阵非奇异时, 比迹、比值函数的判别矩阵和迹比函数的OD矩阵的获取方法及性质已有研究, 本文对迹比函数的UD矩阵的获取方法及性质进行了补充研究, 得到了迹比函数的UD矩阵与比迹、比值函数的UD矩阵是同一矩阵以及迹比函数的UD矩阵的判别函数值不超过它的OD矩阵的结论. 当类内协方差阵奇异时, 6种判别矩阵的获取方法遇到了困难, 为克服这一困难, 本文首先用极限的思想重新定义了这三种判别函数, 然后采用求极限的方法得到了6种判别矩阵的获取方法. 从所得的获取方法可以看出, 当所需的判别向量均在类内协方差阵的零空间中时, 6个判别矩阵是同一矩阵.
Abstract:
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 ratio-value 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.
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