Orthogonal Discriminant Analysis Methods

Orthogonal Discriminant Analysis Methods

David Zhang (Hong Kong Polytechnic University, Hong Kong), Fengxi Song (New Star Research Institute Of Applied Technology, China), Yong Xu (Harbin Institute of Technology, China) and Zhizhen Liang (Shanghai Jiao Tong University, China)
DOI: 10.4018/978-1-60566-200-8.ch004

Abstract

In this chapter, we first give a brief introduction to Fisher linear discriminant, Foley- Sammon discriminant, orthogonal component discriminant, and application strategies for solving the SSS problems. We then present two novel orthogonal discriminant analysis methods, orthogonalized Fisher discriminant and Fisher discriminant with Schur decomposition. At last, we compare the performance of several main orthogonal discriminant analysis methods under various SSS strategies.
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Introduction

Fisher Linear Discriminant

Fisher linear discriminant (FLD) (Duda, Hart, & Stork, 2001) operates by learning a discriminant matrix which maps a d-dimensional input space into an r-dimensional feature space by maximizing the multiple Fisher discriminant criterion.Specifically, a Fisher discriminant matrix is an optimal solution of the following optimization model:

978-1-60566-200-8.ch004.m01. (1)

Here 978-1-60566-200-8.ch004.m02 is an arbitrary matrix, and 978-1-60566-200-8.ch004.m03 and 978-1-60566-200-8.ch004.m04 are the between- and within- class scatter matrices, and 978-1-60566-200-8.ch004.m05 is the determinant of a square matrix978-1-60566-200-8.ch004.m06.

The between-class scatter matrix SB and the within-class scatter matrix SW are defined as follows,978-1-60566-200-8.ch004.m07, (2) and

978-1-60566-200-8.ch004.m08. (3)

Here Ni and 978-1-60566-200-8.ch004.m09 are respectively the number and the mean of samples from the ith class 978-1-60566-200-8.ch004.m10, 978-1-60566-200-8.ch004.m11 the mean of samples from all classes, and l the number of classes.

It has been proved that if 978-1-60566-200-8.ch004.m12 is nonsingular, the matrix composed of unit eigenvectors of the matrix 978-1-60566-200-8.ch004.m13 corresponding to the first r largest eigenvalues is an optimal solution of the optimization model defined in Eq. (1) (Wilks, 1962). The matrix 978-1-60566-200-8.ch004.m14 is the Fisher discriminant matrix commonly used in Fisher linear discriminant.

Since the matrix 978-1-60566-200-8.ch004.m15 is usually asymmetric, Fisher discriminant vectors, i.e. column vectors of the Fisher discriminant matrix are unnecessary orthogonal to each other.

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