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Unformatted text preview: A Solution Manual and Notes for the Text: The Elements of Statistical Learning by Jerome Friedman, Trevor Hastie, and Robert Tibshirani John L. Weatherwax December 15, 2009 * wax@alum.mit.edu 1 Chapter 2 (Overview of Supervised Learning) Notes on the Text Statistical Decision Theory Our expected predicted error (EPE) under the squared error loss and assuming a linear model for y i.e. y = f ( x ) x T is given by EPE( ) = integraldisplay ( y x T ) 2 Pr( dx,dy ) . (1) Considering this a function of the components of i.e. i to minimize this expression with respect to i we take the i derivative, set the resulting expression equal to zero and solve for i . Taking the vector derivative with respect to the vector we obtain EPE = integraldisplay 2 ( y x T ) ( 1) x Pr( dx,dy ) = 2 integraldisplay ( y x T ) x Pr( dx,dy ) . (2) Now this expression will contain two parts. The first will have the integrand yx and the second will have the integrand x T x . This latter expression in terms of its components is given by x T x = ( x + x 1 1 + x 2 2 + + x p p ) x x 1 x 2 . . . x p = x x + x x 1 1 + x x 2 2 + ... + x x p p x 1 x + x 1 x 1 1 + x 1 x 2 2 + ... + x 1 x p p . . . x p x + x p x 1 1 + x p x 2 2 + ... + x p x p p = xx T . So with this recognition, that we can write x T x as xx T , we see that the expression EPE = 0 gives E [ yx ] E [ xx T ] = 0 . (3) Since is a constant, it can be taken out of the expectation to give = E [ xx T ] 1 E [ yx ] , (4) which gives a very simple derivation of equation 2.16 in the book. Note since y R and x R p we see that x and y commute i.e. xy = yx . Exercise Solutions Ex. 2.1 (target coding) If each of our samples from K classes is coded as a target vector t k which has a one in the k th spot. Then one way of developing a classifier is by regressing the independent variables onto the target vectors t k . Then our classification procedure would then become the following. Given the measurement vector X , predict a target vector y via linear regression and to select the class k corresponding to the component of y which has the largest value. That is k = argmax i ( y i ). Now consider the expression argmin k  y t k  , which finds the index of the target vector that is closest to the produced regression output y . By expanding the quadratic we find that argmin k  y t k  = argmin k  y t k  2 = argmin k K summationdisplay i =1 ( y i ( t k ) i ) 2 = argmin k K summationdisplay i =1 ( ( y i ) 2 2 y i ( t k ) i + ( t k ) i 2 ) = argmin k K summationdisplay i =1 ( 2 y i ( t k ) i + ( t k ) i 2 ) , since the sum K i =1 y 2 i is the same for all classes k and we have denoted ( t k ) i to be the i th component of the...
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 Winter '10
 TIBSHIRANI,R
 Statistics

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