Flexible Discriminant Analysis Penalized Discriminant Analysis Mixture Discriminant Analysi.pdf

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Flexible Discriminant Analysis Penalized Discriminant Analysis Mixture Discriminant Analysi.pdf

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Flexible Discriminant Analysis Penalized Discriminant Analysis Mixture Discriminant Analysi

Support Vector Machines and Flexible Discriminants Chapter 12: Hastie et al. (2001) Madhusudana Shashanka Department of Cognitive and Neural Systems Boston University CN700 - SVMs/Flexible Discriminants March 23, 2004 – p.1/28 Overview Part I – SVMs Support Vector Classifier Support Vector Machines Part II – Flexible Discriminants Flexible Discriminant Analysis Penalized Discriminant Analysis Mixture Discriminant Analysis CN700 - SVMs/Flexible Discriminants March 23, 2004 – p.2/28 Introduction Generalizations of linear decision boundaries for classification. Optimal separating hyperplanes for linearly separable classes. Extensions to the non-separable case generalize to support vector machines. SVM - produce nonlinear decision boundaries by constructing a linear boundary in a large, transformed space. CN700 - SVMs/Flexible Discriminants March 23, 2004 – p.3/28 Separating Hyperplanes - Recap Consider classes separable by a linear boundary. Data of the form (xi, yi) with xi ∈ Rp, yi ∈ {?1, 1} ?i = 1, 2, . . . , N . Define a hyperplane {x : f(x) = xT β + β0 = 0}. Find the hyperplane that creates the biggest margin between the training points for classes -1 and 1. minβ,β0 1 2 ||β||2 subject to yi(xTi β + β0) ≥ 1, i = 1, 2, . . . , N . Margin is C = 1/||β||. Classification rule G(x) = sign[xT β + β0]. Convex optimization problem: quadratic criterion and linear inequality constraints. CN700 - SVMs/Flexible Discriminants March 23, 2004 – p.4/28 Non-separable classes margin A A F ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ca d b e A A F margin Allow some points on the wrong side. Define slack variables ξ = (ξ1, ξ2, . . . , ξn). Modify the constraint as: yi(xTi β + β0) ≥ C(1 ? ξi), ?i, ξi ≥ 0, ∑N i=1 ξi ≤ constant. Misclassifications occur when ξi 1. CN700 - SVMs/Flexible Discriminants March 23, 2004 – p.5/28 Support Vector Classifier minβ,β0 1 2 ||β||2 + γ ∑N i=1 ξi subject to ξi ≥ 0, yi(xTi βxi + b) ≥ 1 ? ξi?i. LP = 1 2 ||β||2 + γ