Note the excruciatingly painful difference between

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Note the excruciatingly painful difference between the intuitive equation 8.3c and the formally correct equation 8.4: on the right-hand side of the arrows, the intuitive one depends on s 2 M (the variance of the datapoint residuals ) , as you’d think it should, while the formal one depends on σ 2 meas M (the adopted intrinsic measurement variances of the data), which are chosen by the guy doing the fit. If you do an unweighted average, and derive a certain variance, and next do a weighted average in which you choose some values for σ meas that happen to be wrong, the two fits give different results for s 2 w,mean . This is crazy. To get around this difficulty, we follow the procedure in BR equations 4.20 to 4.26. This introduces an arbitrary multiplicative factor for the weights and goes through the ML calculation to derive, instead of equation 8.4, the far superior s 2 w,mean,BR = hatwider χ 2 w meas,m s 2 w M , (8.5) which is precisely the same as our intuitive guess, equation 8.3c. The difference between the formal equation 8.5 and the intuitive equations 8.3b and 8.4 is the numerator, which contains the reduced chi-square hatwider χ 2 ; for the case where all σ meas,m are identical, hatwider χ 2 = s 2 w σ 2 meas . Note that χ 2 and hatwider χ 2 are defined in equations 8.1. 8.2. The multivariate chi-square fit Here we generalize § 8.1, which dealt with the weighted average, to the multivariate case. In this case, chi-square fitting is just like least-squares fitting except for the following: 1. In the least-squares matrix X of equation 2.1a, each row m is a different measurement with a different intrinsic variance σ m . For chi-square fitting you generate a new matrix X χ , which
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– 26 – is identical to X except that each row m (which contains a particular equation of condition) is divided by σ m . This new matrix is the same as NR’s design matrix (Figure 15.4.1), which they denote by A . 2. For chi-square fitting, divide each datapoint y m in equation 2.1b by σ m . You are generating a new data vector Y χ , which is identical to Y except that each datapoint is divided by σ m . This new data vector is the same as NR’s vector b . 3. Note that the above two steps can be accomplished matrixwise by defining the M × M diagonal matrix [ σ ] in which the diagonal elements are σ m . [ σ ] = σ 0 0 . . . 0 0 σ 1 . . . 0 . . . . . . . . . 0 0 0 0 σ M - 1 (8.6) in which case we can write X χ = [ σ ] - 1 · X (8.7a) Y χ = [ σ ] - 1 · Y . (8.7b) 4. Carry through the matrix calculations in equations 8.8 below (using the matrices subscripted with χ ). You’ve divided each row, i.e. the equation of condition for each row m , by a common factor, so the solution of that particular equation of condition is unchanged. However, in the grand scheme of things—i.e. the normal equations—it receives a greater or lesser weight by a factor 1 σ 2 m .
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