12.4 Weighted least squares. In least squares, the objective (to be minimized) is || Ar ...
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12.4 Weighted least squares. In least squares, the objective (to be minimized) is || Ar – b|| = (a?x – bi)°, i=1 where ã are the rows of A, and the n-vector x is to chosen. In the weighted least squares problem, we minimize the objective i=1 where w; are given positive weights. The weights allow us to assign different weights to the different components of the residual vector. (The objective of the weighted least squares problem is the square of the weighted norm, ||Ax – b||, as defined in exercise 3.28.) (a) Show that the weighted least squares objective can be expressed as ||D(Ax – b)||² for an appropriate diagonal matrix D. This allows us to solve the weighted least squares problem as a standard least squares problem, by minimizing ||Bx - d||2, where B = DA and d = Db. (b) Show that when A has linearly independent columns, so does the matrix B. (c) The least squares approximate solution is given by = (A" A)-AT b. Give a similar formula for the solution of the weighted least squares problem. You might want to use the matrix W = diag(w) in your formula. 12.4 Weighted least squares. In least squares, the objective (to be minimized) is || Ar – b|| = (a?x – bi)°, i=1 where ã are the rows of A, and the n-vector x is to chosen. In the weighted least squares problem, we minimize the objective i=1 where w; are given positive weights. The weights allow us to assign different weights to the different components of the residual vector. (The objective of the weighted least squares problem is the square of the weighted norm, ||Ax – b||, as defined in exercise 3.28.) (a) Show that the weighted least squares objective can be expressed as ||D(Ax – b)||² for an appropriate diagonal matrix D. This allows us to solve the weighted least squares problem as a standard least squares problem, by minimizing ||Bx - d||2, where B = DA and d = Db. (b) Show that when A has linearly independent columns, so does the matrix B. (c) The least squares approximate solution is given by = (A" A)-AT b. Give a similar formula for the solution of the weighted least squares problem. You might want to use the matrix W = diag(w) in your formula.
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Related Book For
Statistics for Business and Economics
ISBN: 978-0321826237
12th edition
Authors: James T. McClave, P. George Benson, Terry T Sincich
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