Post-fit residuals Notice that we can compute the compute the covariance matrix of the post-fit residuals(a large matrix in generate Eqn 1 on previous slide gives an equation of the form V=Be; why can we not compute the actual errors with B is a singular matrix which has no unique inverse (there is in fact one inverse which would generate the true errors) Note: In this case, singularity does not mean that there is no inverse it means there are an infinite number of inverses Example Consider the case shown below When a rate of change is estimated, the slope estimate will absorb error in the last data point particularly as At increases ( Try this case yourself) Example of fitting slope to non- uniform data distribution close03/17/03 12.540 Lec 11 13 Post-fit residuals • Notice that we can compute the compute the covariance matrix of the post-fit residuals (a large matrix in generate) • v=Be; why can we not compute the actual errors with e=B-1v? • (there is in fact one inverse which would generate the true errors) • Note: In this case, singularity does not mean that there is no inverse, it means there are an infinite number of inverses. Eqn 1 on previous slide gives an equation of the form B is a singular matrix which has no unique inverse 03/17/03 12.540 Lec 11 14 Example as D 0 1 2 3 4 5 6 0.0 10.0 20.0 30.0 40.0 50.0 Data Time Dt Postfit error bar somewhat reduced • Consider the case shown below: When a rate of change is estimated, the slope estimate will absorb error in the last data point particularly t increases. (Try this case yourself) Postfit error bar very small; slope will always pass close to this data point Example of fitting slope to non-uniform data distribution 7