Chp12 - Copy

Chp12 - Copy - 12 Support Vector Machines and Flexible...

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12 Support Vector Machines and Flexible Discriminants 12.1 Introduction In this chapter we describe generalizations of linear decision boundaries for classification. Optimal separating hyperplanes are introduced in Chap- ter 4 for the case when two classes are linearly separable. Here we cover extensions to the nonseparable case, where the classes overlap. These tech- niques are then generalized to what is known as the support vector machine , which produces nonlinear boundaries by constructing a linear boundary in a large, transformed version of the feature space. The second set of methods generalize Fisher’s linear discriminant analysis (LDA). The generalizations include flexible discriminant analysis which facilitates construction of non- linear boundaries in a manner very similar to the support vector machines, penalized discriminant analysis for problems such as signal and image clas- sification where the large number of features are highly correlated, and mixture discriminant analysis for irregularly shaped classes. 12.2 The Support Vector Classifier In Chapter 4 we discussed a technique for constructing an optimal separat- ing hyperplane between two perfectly separated classes. We review this and generalize to the nonseparable case, where the classes may not be separable by a linear boundary. © Springer Science+Business Media, LLC 2009 T. Hastie et al., The Elements of Statistical Learning, Second Edition, 417 DOI: 10.1007/b94608_12,
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418 12. Flexible Discriminants margin M = 1 ± β ± M = 1 ± β ± x T β + β 0 =0 margin ξ 1 ξ 2 ξ 3 ξ 4 ξ 5 M = 1 ± β ± M = 1 ± β ± x T β + β 0 FIGURE 12.1. Support vector classifers. The leFt panel shows the separable case. The decision boundary is the solid line, while broken lines bound the shaded maximal margin oF width 2 M =2 / ± β ± . The right panel shows the nonseparable (overlap) case. The points labeled ξ j are on the wrong side oF their margin by an amount ξ j = j ; points on the correct side have ξ j . The margin is maximized subject to a total budget P ξ i constant. Hence P ξ j is the total distance oF points on the wrong side oF their margin. Our training data consists of N pairs ( x 1 ,y 1 ) , ( x 2 2 ) ,..., ( x N N ), with x i IR p and y i ∈{− 1 , 1 } . DeFne a hyperplane by { x : f ( x )= x T β + β 0 } , (12.1) where β is a unit vector: ± β ± = 1. A classiFcation rule induced by f ( x )is G ( x ) = sign[ x T β + β 0 ] . (12.2) The geometry of hyperplanes is reviewed in Section 4.5, where we show that f ( x ) in (12.1) gives the signed distance from a point x to the hyperplane f ( x x T β + β 0 = 0. Since the classes are separable, we can Fnd a function f ( x x T β + β 0 with y i f ( x i ) > 0 i . Hence we are able to Fnd the hyperplane that creates the biggest margin between the training points for class 1 and 1 (see ±igure 12.1). The optimization problem max β,β 0 , ± β ± =1 M subject to y i ( x T i β + β 0 ) M, i =1 ,...,N, (12.3) captures this concept. The band in the Fgure is
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This note was uploaded on 07/14/2010 for the course STAT 132 taught by Professor Haulk during the Spring '10 term at UBC.

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Chp12 - Copy - 12 Support Vector Machines and Flexible...

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