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For additional numerical values, see accompanying table.the vector a of parameter values is perturbed away from a(0) , then χ2 increases. Theregion within which χ2 increases by no more than a set amount ∆χ2 defines someM -dimensional confidence region around a(0). If ∆χ2 is set to be a large number,this will be a big region; if it is small, it will be small. Somewhere in between therewill be choices of ∆χ2 that cause the region to contain, variously, 68 percent, 90percent, etc. of probability distribution for a’s, as defined above.
These regions aretaken as the confidence regions for the parameters a(0) .Very frequently one is interested not in the full M -dimensional confidenceregion, but in individual confidence regions for some smaller number ν of parameters.For example, one might be interested in the confidence interval of each parametertaken separately (the bands in Figure 15.6.3), in which case ν = 1. In that case,the natural confidence regions in the ν-dimensional subspace of the M -dimensionalparameter space are the projections of the M -dimensional regions defined by fixed∆χ2 into the ν-dimensional spaces of interest.
In Figure 15.6.4, for the case M = 2,we show regions corresponding to several values of ∆χ2 . The one-dimensionalconfidence interval in a2 corresponding to the region bounded by ∆χ2 = 1 liesbetween the lines A and A0 .Notice that the projection of the higher-dimensional region on the lowerdimension space is used, not the intersection. The intersection would be the bandbetween Z and Z 0 .
It is never used. It is shown in the figure only for the purpose ofmaking this cautionary point, that it should not be confused with the projection.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).A15.6 Confidence Limits on Estimated Model Parameters695Probability Distribution of Parameters in the Normal Casewhere [α] is the curvature matrix defined in equation (15.5.8).Theorem C.
If aS(j) is drawn from the universe of simulated data sets withactual parameters a(0) , then the quantity ∆χ2 ≡ χ2 (a(j) ) − χ2 (a(0)) is distributedas a chi-square distribution with M degrees of freedom. Here the χ2 ’s are allSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).You may be wondering why we have, in this section up to now, made noconnection at all with the error estimates that come out of the χ2 fitting procedure,most notably the covariance matrix Cij . The reason is this: χ2 minimizationis a useful means for estimating parameters even if the measurement errors arenot normally distributed.
While normally distributed errors are required if the χ2parameter estimate is to be a maximum likelihood estimator (§15.1), one is oftenwilling to give up that property in return for the relative convenience of the χ2procedure. Only in extreme cases, measurement error distributions with very large“tails,” is χ2 minimization abandoned in favor of more robust techniques, as willbe discussed in §15.7.However, the formal covariance matrix that comes out of a χ2 minimization hasa clear quantitative interpretation only if (or to the extent that) the measurement errorsactually are normally distributed.
In the case of nonnormal errors, you are “allowed”• to fit for parameters by minimizing χ2• to use a contour of constant ∆χ2 as the boundary of your confidence region• to use Monte Carlo simulation or detailed analytic calculation in determining which contour ∆χ2 is the correct one for your desired confidencelevel• to give the covariance matrix Cij as the “formal covariance matrix ofthe fit.”You are not allowed• to use formulas that we now give for the case of normal errors, whichestablish quantitative relationships among ∆χ2 , Cij , and the confidencelevel.Here are the key theorems that hold when (i) the measurement errors arenormally distributed, and either (ii) the model is linear in its parameters or (iii) thesample size is large enough that the uncertainties in the fitted parameters a do notextend outside a region in which the model could be replaced by a suitable linearizedmodel.
[Note that condition (iii) does not preclude your use of a nonlinear routinelike mqrfit to find the fitted parameters.]Theorem A. χ2min is distributed as a chi-square distribution with N − Mdegrees of freedom, where N is the number of data points and M is the number offitted parameters. This is the basic theorem that lets you evaluate the goodness-of-fitof the model, as discussed above in §15.1. We list it first to remind you that unlessthe goodness-of-fit is credible, the whole estimation of parameters is suspect.Theorem B.
If aS(j) is drawn from the universe of simulated data sets withactual parameters a(0) , then the probability distribution of δa ≡ aS(j) − a(0) is themultivariate normal distribution1P (δa) da1 . . . daM = const. × exp − δa · [α] · δa da1 . . . daM2696Chapter 15.Modeling of Data∆χ2 = δa · [α] · δa(15.6.1)which follows from equation (15.5.8) applied at χ2min where βk = 0. Since δaby hypothesis minimizes χ2 in all but its first component, the second through M thcomponents of the normal equations (15.5.9) continue to hold. Therefore, thesolution of (15.5.9) is cc00 δa = [α]−1 · ...
= [C] · ... (15.6.2)00where c is one arbitrary constant that we get to adjust to make (15.6.1) give thedesired left-hand value. Plugging (15.6.2) into (15.6.1) and using the fact that [C]and [α] are inverse matrices of one another, we getc = δa1 /C11orand∆χ2ν = (δa1 )2 /C11ppδa1 = ± ∆χ2ν C11(15.6.3)(15.6.4)At last! A relationbetween the confidence interval ±δa1 and the formal√standard error σ1 ≡ C11.
Not unreasonably, we find that the 68 percent confidenceinterval is ±σ1 , the 95 percent confidence interval is ±2σ1 , etc.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).evaluated using the fixed (actual) data set D(0) .
This theorem makes the connectionbetween particular values of ∆χ2 and the fraction of the probability distributionthat they enclose as an M -dimensional region, i.e., the confidence level of theM -dimensional confidence region.Theorem D. Suppose that aS(j) is drawn from the universe of simulated datasets (as above), that its first ν components a1 , .
. . , aν are held fixed, and that itsremaining M − ν components are varied so as to minimize χ2 . Call this minimumvalue χ2ν . Then ∆χ2ν ≡ χ2ν − χ2min is distributed as a chi-square distribution withν degrees of freedom. If you consult Figure 15.6.4, you will see that this theoremconnects the projected ∆χ2 region with a confidence level. In the figure, a point thatis held fixed in a2 and allowed to vary in a1 minimizing χ2 will seek out the ellipsewhose top or bottom edge is tangent to the line of constant a2 , and is therefore theline that projects it onto the smaller-dimensional space.As a first example, let us consider the case ν = 1, where we want to findthe confidence interval of a single parameter, say a1 .
Notice that the chi-squaredistribution with ν = 1 degree of freedom is the same distribution as that of the squareof a single normally distributed quantity. Thus ∆χ2ν < 1 occurs 68.3 percent of thetime (1-σ for the normal distribution), ∆χ2ν < 4 occurs 95.4 percent of the time (2-σfor the normal distribution), ∆χ2ν < 9 occurs 99.73 percent of the time (3-σ for thenormal distribution), etc. In this manner you find the ∆χ2ν that corresponds to yourdesired confidence level. (Additional values are given in the accompanying table.)Let δa be a change in the parameters whose first component is arbitrary, δa1 ,but the rest of whose components are chosen to minimize the ∆χ2 . Then TheoremD applies.
The value of ∆χ2 is given in general by69715.6 Confidence Limits on Estimated Model Parameters∆χ2 as a Function of Confidence Level and Degrees of Freedomνp1234561.002.714.006.639.0015.12.304.616.179.2111.818.43.536.258.0211.314.221.14.727.789.7013.316.323.55.899.2411.315.118.225.77.0410.612.816.820.127.8These considerations hold not just for the individual parameters ai , but alsofor any linear combination of them: IfMXci ai = c · a(15.6.5)then the 68 percent confidence interval on b ispδb = ± c · [C] · c(15.6.6)b≡k=1However, these simple, normal-sounding numerical relationships do not holdin the case ν > 1 [3]. In particular, ∆χ2 = 1 is not the boundary, nor does it projectonto the boundary, of a 68.3 percent confidence region when ν > 1. If you wantto calculate not confidence intervals in one parameter, but confidence ellipses intwo parameters jointly, or ellipsoids in three, or higher, then you must follow thefollowing prescription for implementing Theorems C and D above:• Let ν be the number of fitted parameters whose joint confidence region youwish to display, ν ≤M .
Call these parameters the “parameters of interest.”• Let p be the confidence limit desired, e.g., p = 0.68 or p = 0.95.• Find ∆ (i.e., ∆χ2 ) such that the probability of a chi-square variable withν degrees of freedom being less than ∆ is p. For some useful values of pand ν, ∆ is given in the table. For other values, you can use the routinegammq and a simple root-finding routine (e.g., bisection) to find ∆ suchthat gammq(ν/2, ∆/2) = 1 − p.• Take the M × M covariance matrix [C] = [α]−1 of the chi-square fit.Copy the intersection of the ν rows and columns corresponding to theparameters of interest into a ν × ν matrix denoted [Cproj].• Invert the matrix [Cproj ].
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