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Unformatted text preview: Link between LDA and OLS JeanPhilippe Vert June 9, 2011 This is the solution to exercise 4.2 of [1] which shows a link between linear discriminant analysis (LDA) and ordinary least squares (OLS) in the binary case. We have features x R p and a twoclass response, with class sizes N 1 , N 2 . The training patterns are denoted x 1 , . . . , x N R p , stored in the n p matrix X . We encode the class of each training point in the real number y i = N/N 1 for patterns x i in class 1, and y i = N/N 2 for patterns x i in class 2. (a) From equation (4.11) in [1] we know that, in the binary case, the LDA rule classifies a pattern x to class 2 if x >  1 ( 2 1 ) > 1 2 > 2  1 2 1 2 > 1  1 1 + log N 1 N log N 2 N , (1) and class 1 otherwise. (b) Let us introduce a few more notations. Let U i R n be the class indicator vector of class i , and U = U 1 + U 2 be the vector with all entries equal to 1. When we encode class 1 (resp. class 2) by the real number a 1 = N/N 1 (resp. a 2 = N/N 2), the vector of labels becomes Y = a 1 U 1 + a 2 U 2 ....
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 Winter '10
 TIBSHIRANI,R
 Statistics, Least Squares

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