7 be careful and look at the solution before

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(7) Be careful and LOOK at the solution before accepting it! These nonlinear problems can produce surprising results, sometimes completely meaningless results. Don’t rely on them to be automatic or foolproof! (8) Reformulate! (?) Sometimes you can avoid all this by reformulating the problem. There are two cases: the harmless case and the not-so-harmless case. An example of the harmless case is fitting for the phase φ in the function y = cos( θ + φ ). This is definitely a nonlinear fit! But its easy to reformulate it in a linear fit using the usual trig identities to write y = A cos θ B sin θ , where B A = tan φ . Solve for ( A, B ) using linear least squares, calculate φ , and propagate the uncertainties. An example of the not-so-harmless case is in NR’s § 15.4 example: fit for ( A, B ) with equations of condition y m = Ae - Bx m . They suggest linearizing by rewriting as log( y m ) = C Bx m , solving for ( B, C ), and deriving A after-the-fact. This is not-so-harmless because you are applying a nonlinear function to the observed values y m ; thus the associated errors σ meas,m are also affected. This means you have to do weighted fitting, which is discussed in § 8 below. Suppose that A = 1, your datapoints all have σ meas,m = 0 . 05, and the observed y m ranges from 0.05 to 1. The datapoint with y m = 0 . 05 has a manageable σ meas m , but what is the corresponding value of σ meas,m for log y m = log 0 . 05? It’s ill-defined and asymmetric about the central value. Or even, God forbid, you have an observed y m that’s negative ??? Even for y m not near zero, you need to calculate new σ meas,m by error propagation; in this case, you need to reassign σ (log y ) = d log y dy σ ( y ) = σ ( y ) y . This is OK when y m is large enough so that the linear approximation is accurate, but if not the converted
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– 23 – noise becomes non-Gaussian. You should regard your datapoints as sacrosanct and never apply any nonlinear function to them. 8. CHI-SQUARE FITTING AND WEIGHTED FITTING: DISCUSSION IGNORING COVARIANCE In least-squares fitting, the derived parameters minimize the sum of squares of residuals as in equation 3.1, which we repeat here: s 2 = 1 M N M - 1 summationdisplay m =0 δy 2 m . where the m th residual δy m = ( y m y m ). Chi-square fitting is similar except for two differences. One, we divide each residual by its intrinsic measurement error σ meas,m ; and two, we define χ 2 as the sum χ 2 = M - 1 summationdisplay m =0 δy 2 m σ 2 meas,m . (8.1a) Along with χ 2 goes the reduced chi square hatwider χ 2 = χ 2 M - N hatwider χ 2 = 1 M N M - 1 summationdisplay m =0 δy 2 m σ 2 meas,m , (8.1b) which is more directly analogous to the definition of s 2 . Chi-square fitting is very much like our least-squares fitting except that we divide each data- point by its intrinsic measurement uncertainty σ meas,m . Thus, the reduced chi-square ( hatwider χ 2 ) is equal to the ratio of the variance of the datapoint residuals ( s 2 ) to the adopted intrinsic measurement variances ( σ 2 meas,m ). So it should be obvious that in chi-square fitting, you must know the mea- surement uncertainties σ meas,m of the individual datapoints beforehand. If you want to give the
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