hw2sol - Stat 315A Homework 2 Solutions November 13, 2008 1...

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Unformatted text preview: Stat 315A Homework 2 Solutions November 13, 2008 1 Problem 1 (a) The classification rule for LDA is to classify as class 2 if 2 ( x ) > 1 ( x ). This gives x T - 1 2- 1 2 2 T - 1 2 +log( N 2 N ) > x T - 1 1- 1 2 T 1 - 1 1 +log( N 1 N ) x T - 1 ( 2- 1 ) > 1 2 2 T - 1 2- 1 2 1 T - 1 1- log( N 2 N ) + log( N 1 N ) where k = g i = k x i /N k and = K k =1 g i = k ( x i- k )( x i- k ) T / ( N- K ). (b) Recall from linear regression that = ( X T X )- 1 X T Y . Note here that both X and Y are centered. (( X- x ) T ( Y- y )) j = N X i =1 ( x ij- x j ) y i y = 0 . = X g i =1 ( x ij- x j )(- N N 1 ) + X g i =2 ( x ij- x j )( N N 2 ) = N ( 2- 1 ) + N x- N x = N ( 2- 1 ) . 1 ( N- 2) + N 1 N 2 N B = X g i =1 ( x i- 1 ) x T i + N 2 N ( 1- 2 ) x T i + X g i =2 ( x i- 2 ) x T i + N 1 N ( 1- 2 ) x T i + N 1 N ( 2- 1 ) x T i = X g i =1 ( x i- x ) x T i + X g i =2 ( x i- x ) x T i = N X i =1 ( x i- x )( x i- x ) T = ( X- x )( X- x ) T where x = 1 N ( N 1 1 + N 2 2 ). (c) Notice that ( 2- 1 ) T is a scalar and we denote this by . Then from part b, ( N- 2) = N ( 2- 1 )- N 1 N 2 N B = ( 2- 1 ) N- N 1 N 2 N - 1 ( 2- 1 ) . Thus, the regression coefficient is proportional to the LDA coefficient. (d) To show that this holds for any distinct coding of Y , we must show that X T Y ( 2- 1 ). WLOG we can consider Y and X to be centered, then, assume Y is coded as- a/N 1 and a/N 2 . X T Y = N X i =1 x i y i =- a N 1 X g i =1 x i + a N 2 X g i =2 x ij = a ( 2- 1 ) . Thus, - 1 ( 2- 1 ). (e) From part d, = - 1 ( 2- 1 ) for some , and =- x T , then f = + T X =- 1 N ( N 1 1 + N 2 2 ) T - 1 ( 2- 1 ) + ( 2- 1 ) T - 1 X Thus, we classify to class 2 if f > 0, ( 2- 1 ) T - 1 X > 1 N ( N 1 1 + N 2 2 ) T - 1 ( 2- 1 ) 2 But, this is not the same as the LDA rule (below) unless N 1 = N 2 . ( 2- 1 ) t - 1 X > N 1 N h 2 T - 1 2- 1 T - 1 1 i . 2 Problem 2 (a) Note that we have an optimization problem of the form minimize + ,- L ( ) subject to X j | j | - t - + j ,- - j , for j = 1 ...p. (1) Thus, we have 2 p + 1 inequality constraints. The Lagrangian dual function for (1) is L ( ) + ( X ( + j + - j )- t )- X + j + j- X - j - j or equivalently because t does not depend on L ( ) + X ( + j + - j )- X + j + j- X - j - j (2) where ( + j- - j ) = ....
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hw2sol - Stat 315A Homework 2 Solutions November 13, 2008 1...

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