Covariance matrix of parameter estimates Propagation of covariance can be applied to the weighted least squares problem x=(AVYA)"A'VYy XX >=(A VA)AV<yy >Va(aVa Vx=(AVa) Notice that the covariance matrix of parameter estimates is a natural output of the estimator if av-1A is inverted ( does not need to be 03/1703 12540Lec11 Covariance matrix of estimated parameters Notice that for the rigorous estimation, the inverse of the data covariance is needed ( time consuming if non-diagonal To compute to parameter estimate covariance, only the covariance matrix of the data is needed (not the inverse) In some cases, a non-rigorous inverse can be done with say a diagonal covariance matrix, but the parameter covariance matrix is rigorously computed using the full covariance matrix. This is a non-MLE but the covariance matrix of the parameters should be orrect ust not the best estimates that can found) This techniques could be used if storage of the full covariance matrix is possible, but inversion of the matrix is not because it would take too long or inverse can not be performed in place 03/703 12540Lec11† 03/17/03 12.540 Lec 11 9 • Propagation of covariance can be applied to the • Notice that the covariance matrix of parameter estimates is a natural output of the estimator if ATV-1A is inverted (does not need to be) x ˆ = (ATVyy -1 A)-1 ATVyy -1 y < xˆxˆ T >= (ATVyy -1 A)-1 ATVyy -1 < yyT > Vyy -1 TVyy -1 A)-1 Vxˆx ˆ = (ATVyy -1 A)-1 Covariance matrix of parameter estimates weighted least squares problem: A(A 03/17/03 12.540 Lec 11 10 Covariance matrix of estimated • Notice that for the rigorous estimation, the inverse of the data covariance is needed (time consuming if non-diagonal) • To compute to parameter estimate covariance, only the covariance matrix of the data is needed (not the inverse) • In some cases, a non-rigorous inverse can be done with say a diagonal covariance matrix, but the parameter covariance matrix is rigorously computed using the full covariance matrix. This is a correct (just not the best estimates that can found). • This techniques could be used if storage of the full covariance parameters non-MLE but the covariance matrix of the parameters should be matrix is possible, but inversion of the matrix is not because it would take too long or inverse can not be performed in place. 5