# Discriminant Criteria for Pattern Classification

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.ch003

## Abstract

As mentioned in Chapter II, there are two kinds of LDA approaches: classification- oriented LDA and feature extraction-oriented LDA. In most chapters of this session of the book, we focus our attention on the feature extraction aspect of LDA for SSS problems. On the other hand,, with this chapter we present our studies on the pattern classification aspect of LDA for SSS problems. In this chapter, we present three novel classification-oriented linear discriminant criteria. The first one is large margin linear projection (LMLP) which makes full use of the characteristic of the SSS problems. The second one is the minimum norm minimum squared-error criterion which is a modification of the minimum squared-error discriminant criterion. The third one is the maximum scatter difference which is a modification of the Fisher discriminant criterion.
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## Introduction

### Linear Discriminant Function and Linear Classifier

Let be a set of training samples from two classes and with samples from , and let be their corresponding class labels. Here means that belongs to whereas means that belongs to . A linear discriminant function is a linear combination of the components of a feature vector which can be written as:, (1) where the vector and the scalar are called weight and bias respectively. The hyperplane is a decision surface which is used to separate samples with positive class labels from samples with negative ones.

A linear discriminant criterion is an optimization model which is used to seek the weight for a linear discriminant function. The chief goal of classification-oriented LDA is to set up an appropriated linear discriminant criterion and to calculate the optimal projection direction, i.e. the weight. Here “optimal” means that after samples are projected onto the weight, the resultant projections of samples from two distinct classes and are fully separated.

Once the weight has been derived from a certain linear discriminant criterion, the corresponding bias can be computed using:, (2) or, (3) where and are respectively the mean training sample and the mean of training samples from the class . They are defined as, (4) and

. (5)

For simplicity, we calculate the bias using the Eq. (2) throughout this chapter.

Let denote the mean of the projected training samples from the class . Thus, the binary linear classifier based on the weight and the bias is defined as follow:, (6) which assigns a class label to an unknown sample . Here, is the sign function. That is, once the weight in a linear discriminant function has been worked out the corresponding binary linear classifier is fixed.

## Complete Chapter List

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Acknowledgment
Chapter 1
Overview  (pages 1-23)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
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Chapter 2
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Chapter 3
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Chapter 4
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Chapter 5
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Chapter 6
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Chapter 7
Tensor Space  (pages 135-149)
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Chapter 8
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Chapter 9
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Chapter 10
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Chapter 11
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Chapter 12
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Chapter 13
Feature Level Fusion  (pages 273-304)
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Chapter 14
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Chapter 15
Decision Level Fusion  (pages 328-348)
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Chapter 16
Book Summary  (pages 349-358)
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