1412 the covariance matrix and errors of the derived

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14.12. The Covariance Matrix (and errors) of the Derived Parameters The matrix α , which is defined above in equation 14.21b, is the conventionally-defined cur- vature matrix and its inverse α - 1 is the conventionally-defined covariance matrix. Because we have formulated this problem as a chi-squared one, the elements of this matrix give the covariance directly. Thus, the errors in the derived parameters can be taken as the square-root of the diagonal elements of α - 1 . One usually wants to calculate the chi-squared of the solution. This involves not only the best-fit parameters a * but also the best fit datapoints x * . To do this, use equation 14.8 using Δx g in place of δ x . This is the same as Jefferys’ equation (43). The expectation value of χ 2 is the number of degrees of freedom. In one sense this is just like the usual least-squares solution: it’s equal to the number of datapoints minus the number of derived parameters. Here the number of datapoints is not the number of experiments M ; rather, it’s JM . So the number of degrees of freedom is JM N . 15. NOTATION COMPARISON WITH NUMERICAL RECIPES I learned least squares from Appendix A of Chauvenet (1863). He didn’t use χ 2 and didn’t use matrix techniques, but § 0 and 1 follows his development quite closely. I wrote the first version of this document before knowing of NR’s treatment, which explains my orientation towards least squares instead of chi-square. I’m fortunate in this approach because it made me realize the pitfalls one can get into with chi-square, as I discuss in § 8. On the other hand, NR describe the least-squares approach with some disdain in the discussion of equation (15.1.6) and warn that it is “dangerous” because you aren’t comparing the residuals to the intrinsic inaccuracies of the data. In astronomy, though, more often than not you don’t have an independent assessment of σ m . But you might know the relative weights, and this is a plus for chi-square fitting. In any case, heed our warnings about chi-square fitting in § 8.
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– 65 – In this writeup I have revised my old notation to agree, partially, with NR’s. This effort wasn’t completely successful because I didn’t read NR very carefully before starting. To make it easier to cross-reference this document with NR, I provide the following table of correspondences (left of the double arrow is ours, to the right is theirs): X ←→ A (15.1a) Y ←→ b (15.1b) X T · X = XX = [ α ] ←→ A T · A = [ α ] (15.1c) XX - 1 = XXI = [ α ] - 1 ←→ [ α ] - 1 = [ C ] = C (15.1d) I use M for the number of measurements and N for the number of unknown coefficients; NR uses the opposite, so we have N ←→ M (15.1e) M ←→ N (15.1f) Confusing, hey what? It is a great pleasure to thank Tim Robishaw for his considerable effort in providing detailed comments on several aspects of this paper. These comments led to significant revisions and im- provements. He also fixed bad formatting and manuscript glitches. Also, I am deeply appreciative to Berkeley undergraduate students (Spring 2006) Ashley Bacon and Tiffany Ussery for their per-
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