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Sol8 - SOLUTION TO ASSIGNMENT 8 MATH 251 WINTER 2007 In...

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SOLUTION TO ASSIGNMENT 8 - MATH 251, WINTER 2007 In this assignment F = R or C . 1. Consider results of an experiment given by a series of points: ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) , where x 1 < x 2 < · · · < x n and the x i , y i are real numbers. We assume that the actual law governing this data is linear. Namely, that there is an equation of the form f A,B ( x ) = Ax + B that fits the data up to experimental errors. Therefore, we look for such an equation Ax + B that fits the data best. Our measure for that is “the method of list squares”. Namely, given a line Ax + B , let d i = | y i - ( Ax i + B ) | (the distance between the theoretical y and the observed y ). Then we seek to minimize d 2 1 + d 2 2 + · · · + d 2 n . Let T : R 2 R n , be the map T ( A, B ) = ( f A,B ( x 1 ) , . . . , f A,B ( x n )) . Prove that T is a linear map: We can represent T by a matrix ( A, B ) 7→ x 1 1 x 2 1 . . . . . . x n 1 A B . So clearly it is linear. and that the problem we seek to solve is to minimize k T ( A, B ) - ( y 1 , . . . , y n ) k 2 . Let us calculate: k T ( A, B ) - ( y 1 , . . . , y n ) k 2 = k ( Ax 1 + B, . . . , Ax n + B ) - ( y 1 , . . . , y n ) k 2 = n i =1 ( Ax i + B - y i ) 2 = n i =1 d 2 i . Let W be the subspace of R n which is the image of T . Prove that W is two dimensional and that { s 1 , s 2 } is a basis for W , where s 1 = (1 , 1 , . . . , 1) , s 2 = ( x 1 , x 2 , . . . , x n ). W is spanned by T (0 , 1) and T (1 , 0) , which are just s 1 and s 2 . W is two dimensional, unless x 1 = x 2 = · · · = x n , which is not the case. Assume for simplicity that n i =1 x i = 0 (this can always be achieved by shifting the data). Find an orthonormal basis for W and use it to find the vector in W closest to ( y 1 , . . . , y n ). 1
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2 SOLUTION TO ASSIGNMENT 8 - MATH 251, WINTER 2007 The vector closest to ( y 1 , . . . , y n ) is its orthogonal projection on W . Let v 1 = n - 1 / 2 (1 , 1 , . . . , 1) , v 2 = ( n X i =1 x 2 i ) - 1 / 2 ( x 1 , . . . , x n ) . Then { v 1 , v 2 } are an orthonormal basis for W . The projection is the function v 7→ h v, v 1 i v 1 + h v, v 2 i v 2 . Taking v = ( y 1 , . . . , y n ) we get that the projection is n i =1 y i n (1 , . . . , 1) + h x ,y i k x k 2 x , where we have let x = ( x 1 , . . . , x n ) , y = ( y 1 , . . . , y n ) . Put now everything together to get explicit formulas for A, B such that f A,B ( x ) is the best linear approximation to the data ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) .
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