Weighted Least s In standard least squares, nothing is assumed about the residuals v except that they are zero me One often sees weight-least-squares in which a weight matrix is assigned to the residuals Residuals with larger elements in W are given more weight. (V W ) △x=(AwA)Aw△y 03/1203 12540Lec10 Statistical approach to least squares If the weight matrix used in weighted least squares is the inverse of the covariance matrix of the residuals, then weighted least squares is a maximum likelihood estimator for Gaussian distributed random errors This latter form of least-squares is most statistically rigorous version ights are pirically03/12/03 12.540 Lec 10 11 mean. v( ) TWv ; Dx = (ATWA) -1 ATWDy 03/12/03 12.540 Lec 10 12 Statistical approach to least squares Weighted Least Squares • In standard least squares, nothing is assumed about the residuals v except that they are zero • One often sees weight-least-squares in which a weight matrix is assigned to the residuals. Residuals with larger elements in W are given more weight. minimize • If the weight matrix used in weighted least squares is the inverse of the covariance matrix of the residuals, then weighted least squares is a maximum likelihood estimator for Gaussian distributed random errors. • This latter form of least-squares is most statistically rigorous version. • Sometimes weights are chosen empirically 6