Covariance matrix of post-fit residuals Post-fit residuals are the differences between the observations and the values computed from the estimated parameters Because some of the noise in the data are absorbed into the parameter estimates, in general, the post-fit residuals are not the same as the errors in the data In some cases, they can be considerably smaller The covariance matrix of the post-fit residuals can be computed using propagation of covariances 03/1703 12540Lec11 Covariance matrix of post-fit residuals This can be computed using propagation on covariances: e is the vector of true errors and v is vector of residuals y=Ax+e X=(A VA)A V y-Ai=I-A(AVYA) Amount error reduced V,=<VV>=V-A(AVAA 12540Lec11† 03/17/03 12.540 Lec 11 11 Covariance matrix of post-fit residuals • Post-fit residuals are the differences between the observations and the values computed from the estimated parameters • Because some of the noise in the data are absorbed into the parameter estimates, in general, the post-fit residuals are not the same as the errors in the data. • In some cases, they can be considerably smaller. • The covariance matrix of the post-fit residuals can be computed using propagation of covariances. 6 03/17/03 12.540 Lec 11 12 Covariance matrix of post-fit residuals covariances: e v y = Ax + e x ˆ = (A T Vyy -1 A)-1 A T Vyy -1 y v = y - Ax ˆ = I - A(A T Vyy -1 A)-1 A T Vyy -1 Amount error reduced 1 2 3 È Î Í Í ˘ ˚ ˙ ˙ e Eqn 1 Vvv =< vvT >= Vyy - A(A T Vyy -1 A)-1 A T • This can be computed using propagation on is the vector of true errors, and is vector of residuals 444 444