CH11.Classification.pdf - Chapter 11 Discrimination and...

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Chapter 11. Discrimination and Classification 1. Introduction Discrimination and classification are multivariate techniques concerned with sep- arating distinct sets of objects (or observations) and allocating new objects to previously defined groups. Discrimination is rather exploratory while classification procedures are less ex- ploratory. In theses days, people use the term classification more often than discrimination. Classification is also called supervised learning. Example Group: { good credit, bad credit } Input variables: { age, education level, income , . . . } . The objective is to make a rule - a function of input variables, which discriminate the two groups, based on the given data, and, classify a new customer, who has only input variables, based on the rules constructed. 1
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2. Classification with Two Population (A) Fisher’s idea Two populations with common variance : π 1 : ( µ 1 , Σ ) , π 2 : ( µ 2 , Σ ) Objective : Observe X either from π 1 or π 2 , and specify a classification rule ( R 1 , R 2 ) or R 2 where X is from π 1 ” if X R 1 X is from π 2 ” if X R 2 with R 1 R 2 = R p , R 1 R 2 = ϕ . Idea of “maximally separating projection” (p.590) : arg max a ( E ( a X | π 2 ) E ( a X | π 1 )) 2 Var( a X | π ) = Σ 1 ( µ 2 µ 1 ) Note on “the scaling invariance”. (p.589) Classification with linear discriminant function (p.591) : R LD 2 : | a X a µ 2 | < | a X a µ 1 | for a = Σ 1 ( µ 2 µ 1 ) , i.e., ( µ 2 µ 1 ) Σ 1 X > 1 2 ( µ 2 µ 1 ) Σ 1 ( µ 2 + µ 1 ) . 2
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Unknown parameter case (p.591) : X 11 , · · · , X 1 n 1 : r.s. from ( µ 1 , Σ ) X 21 , · · · , X 2 n 2 : r.s. from ( µ 2 , Σ ), indep. of X 1 j ( j = 1 , · · · , n 1 ) X : new observation from either π 1 or π 2 R LD 2 : ˆ a X > 1 2 ˆ a µ 2 + ˆ µ 1 ) ˆ a = (ˆ µ 2 ˆ µ 1 ) ˆ Σ 1 , ˆ µ i = ¯ X i ( i = 1 , 2) ˆ Σ = S pooled - maximum separation through t -statistic (p.590) : arg max a | a ¯ X 2 a ¯ X 1 | p n 1 1 + n 1 2 p a S pooled a = S 1 ( ¯ X 2 ¯ X 1 ) Example 11.4 (p.591) 3
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(B) Bayes and Minimax classification Two populations with known pdf’s : π 1 : pdf f 1 , π 2 : pdf f 2 Assume f 1 and f 2 are of continuous type. Costs (“losses” : terminology in decision theory) (p.581) : Classification (“action”) π 1 π 2 True π 1 0 c (2 | 1) π 2 c (1 | 2) 0 Expected costs (“risks”) ( c (2 | 1) P ( X R 2 | π 1 ) , c (1 | 2) P ( X R 1 | π 2 )) with R 1 = R c 2 . - misclassification probabilities for 0-1 loss. Minimax criterion : - minimize the maximum risk : minimize max { c (2 | 1) P ( X R 2 | π 1 ) , c (1 | 2) P ( X R 1 | π 2 ) } with respect to R 2 = R c 1 Bayes criterion (p.581, p.583) : Given “prior” probabilities p 1 = P ( π 1 ) & p 2 = P ( π 2 ), minimize the average (w.r.t. prior) risk (Bayes risk) c (2 | 1) P ( X R 2 | π 1 ) p 1 + c (1 | 2) P ( X R 1 | π 2 ) p 2 with respect to R 2 = R c 1 - “ECM on p.581” “TPM on p.583 for 0-1 loss” 4
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Bayes classification rule (p.581) R B 2 : ( p 2 f 2 ( x )) / ( p 1 f 1 ( x )) > c (2 | 1) /c (1 | 2) where the l.h.s. (r.h.s.) is called the posterior odds (cost, respectively) ratio (p.584).
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