15.6 Confidence Limits on Estimated Model Parameters 689 15.6 Confidence Limits on Estimated Model Parameters Several times already in this chapter we have made statements about the standard errors,or uncertainties,in a set of M estimated parameters a.We have given some formulas for computing standard deviations or variances of individual parameters (equations 15.2.9,15.4.15,15.4.19),as well as some formulas for covariances between pairs of parameters(equation 15.2.10;remark following equation 15.4.15; equation 15.4.20;equation 15.5.15). In this section,we want to be more explicit regarding the precise meaning of these quantitative uncertainties,and to give further information about how 县 quantitative confidence limits on fitted parameters can be estimated.The subject can get somewhat technical,and even somewhat confusing,so we will try to make precise statements,even when they must be offered without proof. Figure 15.6.1 shows the conceptual scheme of an experiment that"measures" a set of parameters.There is some underlying true set of parameters atre that are ⊙ known to Mother Nature but hidden from the experimenter.These true parameters 9 are statistically realized,along with random measurement errors,as a measured data set,which we will symbolize as D(o).The data set D(o)is known to the experimenter. He or she fits the data to a model by x2minimization or some other technique,and obtains measured,i.e.,fitted,values for the parameters,which we here denote a(o). Because measurement errors have a random component,D(o)is not a unique 三，09 9 realization of the true parameters atrue.Rather,there are infinitely many other realizations of the true parameters as "hypothetical data sets"each of which could OF SCIENTIFIC have been the one measured,but happened not to be.Let us symbolize these 6 by D(1),D(2),....Each one,had it been realized,would have given a slightly different set of fitted parameters,a),a(2)...,respectively.These parameter sets a()therefore occur with some probability distribution in the M-dimensional space of all possible parameter sets a.The actual measured set a (o)is one member drawn from this distribution Even more interesting than the probability distribution of a()would be the distribution of the difference a()-atrue.This distribution differs from the former one by a translation that puts Mother Nature's true value at the origin.If we knew this distribution,we would know everything that there is to know about the quantitative 高 10621 uncertainties in our experimental measurement a(o). So the name of the game is to find some way of estimating or approximating 腿 the probability distribution of a()-atrue without knowing atrue and without having North available to us an infinite universe of hypothetical data sets. Monte Carlo Simulation of Synthetic Data Sets Although the measured parameter set a(o)is not the true one,let us consider a fictitious world in which it was the true one.Since we hope that our measured parameters are not too wrong,we hope that that fictitious world is not too different from the actual world with parameters atrue.In particular,let us hope-no,let us assume-that the shape of the probability distribution a()-a(o)in the fictitious world is the same,or very nearly the same,as the shape of the probability distribution
15.6 Confidence Limits on Estimated Model Parameters 689 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). 15.6 Confidence Limits on Estimated Model Parameters Several times already in this chapter we have made statements about the standard errors, or uncertainties, in a set of M estimated parameters a. We have given some formulas for computing standard deviations or variances of individual parameters (equations 15.2.9, 15.4.15, 15.4.19), as well as some formulas for covariances between pairs of parameters (equation 15.2.10; remark following equation 15.4.15; equation 15.4.20; equation 15.5.15). In this section, we want to be more explicit regarding the precise meaning of these quantitative uncertainties, and to give further information about how quantitative confidence limits on fitted parameters can be estimated. The subject can get somewhat technical, and even somewhat confusing, so we will try to make precise statements, even when they must be offered without proof. Figure 15.6.1 shows the conceptual scheme of an experiment that “measures” a set of parameters. There is some underlying true set of parameters a true that are known to Mother Nature but hidden from the experimenter. These true parameters are statistically realized, along with random measurement errors, as a measured data set, which we will symbolize as D(0). The data set D(0) is known to the experimenter. He or she fits the data to a model by χ2 minimization or some other technique, and obtains measured, i.e., fitted, values for the parameters, which we here denote a (0). Because measurement errors have a random component, D(0) is not a unique realization of the true parameters atrue. Rather, there are infinitely many other realizations of the true parameters as “hypothetical data sets” each of which could have been the one measured, but happened not to be. Let us symbolize these by D(1), D(2),... . Each one, had it been realized, would have given a slightly different set of fitted parameters, a(1), a(2),... , respectively. These parameter sets a(i) therefore occur with some probability distribution in the M-dimensional space of all possible parameter sets a. The actual measured set a(0) is one member drawn from this distribution. Even more interesting than the probability distribution of a (i) would be the distribution of the difference a(i) − atrue. This distribution differs from the former one by a translation that puts Mother Nature’s true value at the origin. If we knew this distribution, we would know everything that there is to know about the quantitative uncertainties in our experimental measurement a(0). So the name of the game is to find some way of estimating or approximating the probability distribution of a(i) − atrue without knowing atrue and without having available to us an infinite universe of hypothetical data sets. Monte Carlo Simulation of Synthetic Data Sets Although the measured parameter set a(0) is not the true one, let us consider a fictitious world in which it was the true one. Since we hope that our measured parameters are not too wrong, we hope that that fictitious world is not too different from the actual world with parameters atrue. In particular, let us hope — no, let us assume — that the shape of the probability distribution a(i) − a(0) in the fictitious world is the same, or very nearly the same, as the shape of the probability distribution
690 Chapter 15.Modeling of Data x fitted actual data set parameters min experimental realization ao hypothetical data set true parameters atrue hypothetical a2 data set .com or call granted for 11-800-872 (including this one) hypothetical /Cambridge a3 users to make one paper from NUMERICAL RECIPES IN 198891992 data set (Nort : America server computer, University Press. THE Figure 15.6.1.A statistical universe of data sets from an underlying model.True parameters arue are ART realized in a data set,from which fitted (observed)parameters ao are obtained.If the experiment were repeated many times,new data sets and new values of the fitted parameters would be obtained. 9 Programs a()-atrue in the real world.Notice that we are not assuming that a(o)and atrue are equal;they are certainly not.We are only assuming that the way in which random 之h errors enter the experiment and data analysis does not vary rapidly as a function of to dir atrue,so that a(o)can serve as a reasonable surrogate. Now,often,the distribution of a()-a(o)in the fictitious world is within our OF SCIENTIFIC COMPUTING (ISBN power to calculate (see Figure 15.6.2).If we know something about the process 1988-1992 that generated our data,given an assumed set of parameters a(o),then we can usually figure out how to simulate our own sets of"synthetic"realizations of these parameters as"synthetic data sets."The procedure is to draw random numbers from 10-521 appropriate distributions(cf.$7.2-87.3)so as to mimic our best understanding of the underlying process and measurement errors in our apparatus.With such random idge.org Fuurggoglrion Numerical Recipes 43106 draws,we construct data sets with exactly the same numbers of measured points,and precisely the same values of all control (independent)variables,as our actual data set (outside Po)Let us call these simulated data setsBy construction these are Software. supposed to have exactly the same statistical relationship to a(o)as the D('s have North to atrue.(For the case where you don't know enough about what you are measuring to do a credible job of simulating it,see below.) Next,for each D perform exactly the same procedure for estimation of visit website machine parameters,e.g.,x2 minimization,as was performed on the actual data to get the parameters a(o)giving simulated measured parameters a)Each simulated measured parameter set yields a point aa)Simulate enough data sets and enough derived simulated measured parameters,and you map out the desired probability distribution in M dimensions. In fact,the ability to do Monte Carlo simulations in this fashion has revo-
690 Chapter 15. Modeling of Data Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). actual data set hypothetical data set hypothetical data set hypothetical data set a3 a2 a1 fitted parameters a0 χ2 min true parameters atrue experimental realization . . . . . . Figure 15.6.1. A statistical universe of data sets from an underlying model. True parameters atrue are realized in a data set, from which fitted (observed) parameters a0 are obtained. If the experiment were repeated many times, new data sets and new values of the fitted parameters would be obtained. a(i) − atrue in the real world. Notice that we are not assuming that a(0) and atrue are equal; they are certainly not. We are only assuming that the way in which random errors enter the experiment and data analysis does not vary rapidly as a function of atrue, so that a(0) can serve as a reasonable surrogate. Now, often, the distribution of a(i) − a(0) in the fictitious world is within our power to calculate (see Figure 15.6.2). If we know something about the process that generated our data, given an assumed set of parameters a(0), then we can usually figure out how to simulate our own sets of “synthetic” realizations of these parameters as “synthetic data sets.” The procedure is to draw random numbers from appropriate distributions (cf. §7.2–§7.3) so as to mimic our best understanding of the underlying process and measurement errors in our apparatus. With such random draws, we construct data sets with exactly the same numbers of measured points, and precisely the same values of all control (independent) variables, as our actual data set D(0). Let us call these simulated data sets DS (1), DS (2),... . By construction these are supposed to have exactly the same statistical relationship to a(0) as the D(i)’s have to atrue. (For the case where you don’t know enough about what you are measuring to do a credible job of simulating it, see below.) Next, for each DS (j), perform exactly the same procedure for estimation of parameters, e.g., χ2 minimization, as was performed on the actual data to get the parameters a(0), giving simulated measured parameters aS (1), aS (2),... . Each simulated measured parameter set yields a point aS (i) − a(0). Simulate enough data sets and enough derived simulated measured parameters, and you map out the desired probability distribution in M dimensions. In fact, the ability to do Monte Carlo simulations in this fashion has revo-
15.6 Confidence Limits on Estimated Model Parameters 691 synthetic 2 Monte Carlo parameters data set 1 min Monte Carlo realization R synthetic data set 2 actual fitted data set parameters min ao synthetic data set 3 granted for 1-800-872 synthetic data set 4 a (North. Figure 15.6.2.Monte Carlo simulation of an experiment.The fitted parameters from an actual experiment are used as surrogates for the true parameters.Computer-generated random numbers are used to simulate America computer, to make one paper University Press. THE many synthetic data sets.Each of these is analyzed to obtain its fitted parameters.The distribution of ART these fitted parameters around the (known)surrogate true parameters is thus studied Programs lutionized many fields of modern experimental science.Not only is one able to characterize the errors of parameter estimation in a very precise way;one can also try out on the computer different methods of parameter estimation,or different data reduction techniques,and seek to minimize the uncertainty of the result according 61 to any desired criteria.Offered the choice between mastery of a five-foot shelf of analytical statistics books and middling ability at performing statistical Monte Carlo OF SCIENTIFIC COMPUTING (ISBN simulations,we would surely choose to have the latter skill. 1988-18920 Quick-and-Dirty Monte Carlo:The Bootstrap Method 10-521 Here is a powerful technique that can often be used when you don't know uurggoglrion 43106 enough about the underlying process,or the nature of your measurement errors, Numerical Recipes to do a credible Monte Carlo simulation.Suppose that your data set consists of N independent and identically distributed (or iid)"data points."Each data point (outside probably consists of several numbers,e.g.,one or more control variables(uniformly Software. distributed,say,in the range that you have decided to measure)and one or more North associated measured values(each distributed however Mother Nature chooses)."Iid" means that the sequential order of the data points is not of consequence to the process that you are using to get the fitted parameters a.For example,a x2 sum like (15.5.5)does not care in what order the points are added.Even simpler examples are the mean value of a measured quantity,or the mean of some function of the measured quantities. Theboosrapmetduses the actual data set Dwith its N data points,to generate any number of synthetic data setsDalso with N data points. The procedure is simply to draw N data points at a time with replacement from the
15.6 Confidence Limits on Estimated Model Parameters 691 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). synthetic data set 1 synthetic data set 2 synthetic data set 3 synthetic data set 4 a2 χ2 min χ2 min (s) a1 (s) a3 (s) a4 (s) Monte Carlo parameters Monte Carlo realization fitted parameters a0 actual data set Figure 15.6.2. Monte Carlo simulation of an experiment. The fitted parameters from an actual experiment are used as surrogates for the true parameters. Computer-generated random numbers are used to simulate many synthetic data sets. Each of these is analyzed to obtain its fitted parameters. The distribution of these fitted parameters around the (known) surrogate true parameters is thus studied. lutionized many fields of modern experimental science. Not only is one able to characterize the errors of parameter estimation in a very precise way; one can also try out on the computer different methods of parameter estimation, or different data reduction techniques, and seek to minimize the uncertainty of the result according to any desired criteria. Offered the choice between mastery of a five-foot shelf of analytical statistics books and middling ability at performing statistical Monte Carlo simulations, we would surely choose to have the latter skill. Quick-and-Dirty Monte Carlo: The Bootstrap Method Here is a powerful technique that can often be used when you don’t know enough about the underlying process, or the nature of your measurement errors, to do a credible Monte Carlo simulation. Suppose that your data set consists of N independent and identically distributed (or iid) “data points.” Each data point probably consists of several numbers, e.g., one or more control variables (uniformly distributed, say, in the range that you have decided to measure) and one or more associated measured values (each distributed however Mother Nature chooses). “Iid” means that the sequential order of the data points is not of consequence to the process that you are using to get the fitted parameters a. For example, a χ2 sum like (15.5.5) does not care in what order the points are added. Even simpler examples are the mean value of a measured quantity, or the mean of some function of the measured quantities. The bootstrap method [1] uses the actual data set DS (0), with its N data points, to generate any number of synthetic data sets DS (1), DS (2),... , also with N data points. The procedure is simply to draw N data points at a time with replacement from the
692 Chapter 15.Modeling of Data set DBecause of the replacement,you do not simply get back your original data set each time.You get sets in which a random fraction of the original points, typically ~1/e37%,are replaced by duplicated original points.Now,exactly as in the previous discussion,you subject these data sets to the same estimation procedure as was performed on the actual data,giving a set of simulated measured parameters a)These will be distributed around a)in close to the same way that a(o)is distributed around atrue. Sounds like getting something for nothing,doesn't it?In fact,it has taken more than a decade for the bootstrap method to become accepted by statisticians.By now. 三 however,enough theorems have been proved to render the bootstrap reputable(see [2] for references).The basic idea behind the bootstrap is that the actual data set,viewed as a probability distribution consisting of delta functions at the measured values,is in most cases the best-or only-available estimator of the underlying probability distribution.It takes courage,but one can often simply use that distribution as the basis for Monte Carlo simulations. Watch out for cases where the bootstrap's"iid"assumption is violated.For example,if you have made measurements at evenly spaced intervals of some control ⊙】 variable,then you can usually get away with pretending that these are"iid,"uniformly distributed over the measured range.However,some estimators of a (e.g.,ones 9 involving Fourier methods)might be particularly sensitive to all the points on a grid being present.In that case,the bootstrap is going to give a wrong distribution.Also watch out for estimators that look at anything like small-scale clumpiness within the N data points,or estimators that sort the data and look at sequential differences. 9 Program Obviously the bootstrap will fail on these,too.(The theorems justifying the method are still true,but some of their technical assumptions are violated by these examples. For a large class of problems,however,the bootstrap does yield easy,very quick,Monte Carlo estimates of the errors in an estimated parameter set. 三二%uu OF SCIENTIFIC Confidence Limits Rather than present all details of the probability distribution of errors in parameter estimation,it is common practice to summarize the distribution in the form of confidence limits.The full probability distribution is a function defined on the M-dimensional space of parameters a.A confidence region (or confidence interval)is just a region of that M-dimensional space (hopefully a small region)that ridge.org Numerical Recipes 10621 43106 contains a certain (hopefully large)percentage of the total probability distribution. You point to a confidence region and say,e.g.,"there is a 99 percent chance that the (outside true parameter values fall within this region around the measured value." 腿 It is worth emphasizing that you,the experimenter,get to pick both the North confidence level(99 percent in the above example),and the shape of the confidence region.The only requirement is that your region does include the stated percentage of probability.Certain percentages are,however,customary in scientific usage:68.3 percent(the lowest confidence worthy of quoting),90 percent,95.4 percent,99 percent,and 99.73 percent.Higher confidence levels are conventionally"ninety-nine point nine...nine."As for shape,obviously you want a region that is compact and reasonably centered on your measurement a (o),since the whole purpose of a confidence limit is to inspire confidence in that measured value.In one dimension. the convention is to use a line segment centered on the measured value;in higher dimensions,ellipses or ellipsoids are most frequently used
692 Chapter 15. Modeling of Data Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). set DS (0). Because of the replacement, you do not simply get back your original data set each time. You get sets in which a random fraction of the original points, typically ∼ 1/e ≈ 37%, are replaced by duplicated original points. Now, exactly as in the previous discussion, you subject these data sets to the same estimation procedure as was performed on the actual data, giving a set of simulated measured parameters aS (1), aS (2),... . These will be distributed around a(0) in close to the same way that a(0) is distributed around atrue. Sounds like getting something for nothing, doesn’t it? In fact, it has taken more than a decade for the bootstrap method to become accepted by statisticians. By now, however, enough theorems have been proved to render the bootstrap reputable (see [2] for references). The basic idea behind the bootstrap is that the actual data set, viewed as a probability distribution consisting of delta functions at the measured values, is in most cases the best — or only — available estimator of the underlying probability distribution. It takes courage, but one can often simply use that distribution as the basis for Monte Carlo simulations. Watch out for cases where the bootstrap’s “iid” assumption is violated. For example, if you have made measurements at evenly spaced intervals of some control variable, then you can usually get away with pretending that these are “iid,” uniformly distributed over the measured range. However, some estimators of a (e.g., ones involving Fourier methods) might be particularly sensitive to all the points on a grid being present. In that case, the bootstrap is going to give a wrong distribution. Also watch out for estimators that look at anything like small-scale clumpiness within the N data points, or estimators that sort the data and look at sequential differences. Obviously the bootstrap will fail on these, too. (The theorems justifying the method are still true, but some of their technical assumptions are violated by these examples.) For a large class of problems, however, the bootstrap does yield easy, very quick, Monte Carlo estimates of the errors in an estimated parameter set. Confidence Limits Rather than present all details of the probability distribution of errors in parameter estimation, it is common practice to summarize the distribution in the form of confidence limits. The full probability distribution is a function defined on the M-dimensional space of parameters a. A confidence region (or confidence interval) is just a region of that M-dimensional space (hopefully a small region) that contains a certain (hopefully large) percentage of the total probability distribution. You point to a confidence region and say, e.g., “there is a 99 percent chance that the true parameter values fall within this region around the measured value.” It is worth emphasizing that you, the experimenter, get to pick both the confidence level (99 percent in the above example), and the shape of the confidence region. The only requirement is that your region does include the stated percentage of probability. Certain percentages are, however, customary in scientific usage: 68.3 percent (the lowest confidence worthy of quoting), 90 percent, 95.4 percent, 99 percent, and 99.73 percent. Higher confidence levels are conventionally “ninety-nine point nine ... nine.” As for shape, obviously you want a region that is compact and reasonably centered on your measurement a (0), since the whole purpose of a confidence limit is to inspire confidence in that measured value. In one dimension, the convention is to use a line segment centered on the measured value; in higher dimensions, ellipses or ellipsoids are most frequently used
15.6 Confidence Limits on Estimated Model Parameters 693 a品-aop 68%confidence interval on a 68%confidence region on a and a jointly 83 鱼 19881992 68%confidence interval on az 7 a-don Cambridge from NUMERICAL RECIPES I (Nort server bias America computer, make one paper University Press. THE ART ictly proh Programs Figure 15.6.3.Confidence intervals in I and 2 dimensions.The same fraction of measured points (here 68%)lies (i)between the two vertical lines,(ii)between the two horizontal lines,(iii)within the ellipse. to dir You might suspect,correctly,that the numbers 68.3 percent,95.4 percent, OF SCIENTIFIC COMPUTING(ISBN and 99.73 percent,and the use of ellipsoids,have some connection with a normal 1988-19920 distribution.That is true historically,but not always relevant nowadays.In general, the probability distribution of the parameters will not be normal,and the above numbers,used as levels of confidence,are purely matters of convention. 10-521 Figure 15.6.3 sketches a possible probability distribution for the case M=2. Shown are three different confidence regions which might usefully be given,all at Numerical Recipes 43198.5 the same confidence level.The two vertical lines enclose a band (horizontal interval) which represents the 68 percent confidence interval for the variable a1 without regard (outside to the value of a2.Similarly the horizontal lines enclose a 68 percent confidence North Software. interval for a2.The ellipse shows a 68 percent confidence interval for a1 and a2 jointly.Notice that to enclose the same probability as the two bands,the ellipse must necessarily extend outside of both of them(a point we will return to below). Constant Chi-Square Boundaries as Confidence Limits When the method used to estimate the parameters a (o)is chi-square minimiza- tion,as in the previous sections of this chapter,then there is a natural choice for the shape of confidence intervals,whose use is almost universal.For the observed data set D(o),the value ofx2 is a minimum at a(o).Call this minimum value x If
15.6 Confidence Limits on Estimated Model Parameters 693 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). 68% confidence interval on a2 68% confidence interval on a1 68% confidence region on a1 and a2 jointly bias a(i)1 − a(0)1 (s) a(i)2 − a(0)2 (s) Figure 15.6.3. Confidence intervals in 1 and 2 dimensions. The same fraction of measured points (here 68%) lies (i) between the two vertical lines, (ii) between the two horizontal lines, (iii) within the ellipse. You might suspect, correctly, that the numbers 68.3 percent, 95.4 percent, and 99.73 percent, and the use of ellipsoids, have some connection with a normal distribution. That is true historically, but not always relevant nowadays. In general, the probability distribution of the parameters will not be normal, and the above numbers, used as levels of confidence, are purely matters of convention. Figure 15.6.3 sketches a possible probability distribution for the case M = 2. Shown are three different confidence regions which might usefully be given, all at the same confidence level. The two vertical lines enclose a band (horizontal interval) which represents the 68 percent confidence interval for the variable a 1 without regard to the value of a2. Similarly the horizontal lines enclose a 68 percent confidence interval for a2. The ellipse shows a 68 percent confidence interval for a 1 and a2 jointly. Notice that to enclose the same probability as the two bands, the ellipse must necessarily extend outside of both of them (a point we will return to below). Constant Chi-Square Boundaries as Confidence Limits When the method used to estimate the parameters a(0) is chi-square minimization, as in the previous sections of this chapter, then there is a natural choice for the shape of confidence intervals, whose use is almost universal. For the observed data set D(0), the value of χ2 is a minimum at a(0). Call this minimum value χ2 min. If
694 Chapter 15.Modeling of Data △x2=6.63 B △x2=2.71 △x2=1.00 W今 鱼 1.800 42=2.30 B server n NUMERICAL RECIPES IN C: C (Nort America computer, THE ART Figure 15.6.4.Confidence region ellipses corresponding to values of chi-square larger than the fitted minimum.The solid curves,with Ax2=1.00,2.71,6.63 project onto one-dimensional intervals AA, BB',CC.These intervals-not the ellipses themselves-contain 68.3%,90%,and 99%of normally Progra distributed data.The ellipse that contains 68.3%of normally distributed data is shown dashed,and has Ax2=2.30.For additional numerical values,see accompanying table. to dir the vector a of parameter values is perturbed away from a )thenx2 increases.The region within which x2 increases by no more than a set amount Ax2 defines some OF SCIENTIFIC COMPUTING (ISBN M-dimensional confidence region around a(o).If Ax2 is set to be a large number, 188812920 this will be a big region;if it is small,it will be small.Somewhere in between there will be choices of Ax2 that cause the region to contain,variously,68 percent,90 percent,etc.of probability distribution for a's,as defined above.These regions are taken as the confidence regions for the parameters a(o). uurggoglrion Numerical Recipes 10-521 4310 Very frequently one is interested not in the full M-dimensional confidence region,but in individual confidence regions for some smaller numbervofparameters. (outside g For example,one might be interested in the confidence interval of each parameter taken separately (the bands in Figure 15.6.3).in which case v=1.In that case North Software. the natural confidence regions in the v-dimensional subspace of the M-dimensional 6 parameter space are the projections of the M-dimensional regions defined by fixed Ay into the v-dimensional spaces of interest.In Figure 15.6.4.for the case M =2. B visit website we show regions corresponding to several values of Ax2.The one-dimensional machine confidence interval in a2 corresponding to the region bounded by Ax2=1 lies between the lines A and A'. Notice that the projection of the higher-dimensional region on the lower- dimension space is used,not the intersection.The intersection would be the band between Z and Z'.It is never used.It is shown in the figure only for the purpose of making this cautionary point,that it should not be confused with the projection
694 Chapter 15. Modeling of Data Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). C B A Z ′ Z C′ ∆χ2 = 6.63 ∆χ2 = 2.71 ∆χ2 = 1.00 ∆χ2 = 2.30 A′ B′ Figure 15.6.4. Confidence region ellipses corresponding to values of chi-square larger than the fitted minimum. The solid curves, with ∆χ2 = 1.00, 2.71, 6.63 project onto one-dimensional intervals AA , BB , CC . These intervals — not the ellipses themselves — contain 68.3%, 90%, and 99% of normally distributed data. The ellipse that contains 68.3% of normally distributed data is shown dashed, and has ∆χ2 = 2.30. For additional numerical values, see accompanying table. the vector a of parameter values is perturbed away from a (0), then χ2 increases. The region within which χ2 increases by no more than a set amount ∆χ2 defines some M-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 there will be choices of ∆χ2 that cause the region to contain, variously, 68 percent, 90 percent, etc. of probability distribution for a’s, as defined above. These regions are taken as the confidence regions for the parameters a(0). Very frequently one is interested not in the full M-dimensional confidence region, but in individual confidence regions for some smaller number ν of parameters. For example, one might be interested in the confidence interval of each parameter taken 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-dimensional parameter 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-dimensional confidence interval in a2 corresponding to the region bounded by ∆χ2 = 1 lies between the lines A and A . Notice that the projection of the higher-dimensional region on the lowerdimension space is used, not the intersection. The intersection would be the band between Z and Z . It is never used. It is shown in the figure only for the purpose of making this cautionary point, that it should not be confused with the projection
15.6 Confidence Limits on Estimated Model Parameters 695 Probability Distribution of Parameters in the Normal Case You may be wondering why we have,in this section up to now,made no connection at all with the error estimates that come out of the x2fitting procedure, most notably the covariance matrix Cij.The reason is this:x2 minimization is a useful means for estimating parameters even if the measurement errors are not normally distributed.While normally distributed errors are required if the x2 parameter estimate is to be a maximum likelihood estimator(815.1),one is often willing to give up that property in return for the relative convenience of the x2 procedure.Only in extreme cases,measurement error distributions with very large 81 "tails,"is x2 minimization abandoned in favor of more robust techniques,as will be discussed in $15.7. 虽2 However,the formal covariance matrix that comes out of a x2 minimization has a clear quantitative interpretation only if(or to the extent that)the measurement errors 的 actually are normally distributed.In the case of nonnormal errors,you are"allowed" .to fit for parameters by minimizing x2 to use a contour of constant Ax2 as the boundary of your confidence region to use Monte Carlo simulation or detailed analytic calculation in deter- mining which contour Ax2 is the correct one for your desired confidence 9 level to give the covariance matrix Cii as the "formal covariance matrix of the fit.” You are not allowed to use formulas that we now give for the case of normal errors,which 、a是号分 establish quantitative relationships among Ax2,Cii,and the confidence level. Here are the key theorems that hold when (i)the measurement errors are normally distributed,and either(ii)the model is linear in its parameters or (iii)the sample size is large enough that the uncertainties in the fitted parameters a do not extend outside a region in which the model could be replaced by a suitable linearized model.[Note that condition(iii)does not preclude your use of a nonlinear routine like mqrfit to find the fitted parameters.] Theorem A.Xi is distributed as a chi-square distribution with N-M 、 Numerica 10621 degrees of freedom,where N is the number of data points and M is the number of fitted parameters.This is the basic theorem that lets you evaluate the goodness-of-fit of the model,as discussed above in 815.1.We list it first to remind you that unless the goodness-of-fit is credible,the whole estimation of parameters is suspect. Theorem B.If a is drawn from the universe of simulated data sets with actual parameters a(),then the probability distribution of aa-a(o)is the multivariate normal distribution P(6a)da1...dam const.x exp da...dam where [a]is the curvature matrix defined in equation(15.5.8). Theorem C.If a)is drawn from the universe of simulated data sets with actual parameters a(o),then the quantity x=x2(a())-x2(a())is distributed as a chi-square distribution with M degrees of freedom.Here the x2's are all evaluated
15.6 Confidence Limits on Estimated Model Parameters 695 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Probability Distribution of Parameters in the Normal Case You may be wondering why we have, in this section up to now, made no connection 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 minimization is a useful means for estimating parameters even if the measurement errors are not normally distributed. While normally distributed errors are required if the χ 2 parameter estimate is to be a maximum likelihood estimator (§15.1), one is often willing to give up that property in return for the relative convenience of the χ 2 procedure. Only in extreme cases, measurement error distributions with very large “tails,” is χ2 minimization abandoned in favor of more robust techniques, as will be discussed in §15.7. However, the formal covariance matrix that comes out of a χ2 minimization has a clear quantitative interpretation only if (or to the extent that) the measurement errors actually 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 confidence level • to give the covariance matrix Cij as the “formal covariance matrix of the fit.” You are not allowed • to use formulas that we now give for the case of normal errors, which establish quantitative relationships among ∆χ2, Cij , and the confidence level. Here are the key theorems that hold when (i) the measurement errors are normally distributed, and either (ii) the model is linear in its parameters or (iii) the sample size is large enough that the uncertainties in the fitted parameters a do not extend outside a region in which the model could be replaced by a suitable linearized model. [Note that condition (iii) does not preclude your use of a nonlinear routine like mqrfit to find the fitted parameters.] Theorem A. χ2 min is distributed as a chi-square distribution with N − M degrees of freedom, where N is the number of data points and M is the number of fitted parameters. This is the basic theorem that lets you evaluate the goodness-of-fit of the model, as discussed above in §15.1. We list it first to remind you that unless the 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 with actual parameters a(0), then the probability distribution of δa ≡ aS (j) − a(0) is the multivariate normal distribution P(δa) da1 . . . daM = const. × exp −1 2 δa · [α] · δa da1 . . . daM where [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 with actual parameters a(0), then the quantity ∆χ2 ≡ χ2(a(j))−χ2(a(0)) is distributed as a chi-square distribution with M degrees of freedom. Here the χ2’s are all evaluated
696 Chapter 15.Modeling of Data using the fixed (actual)data set D(o).This theorem makes the connection between particular values of Ax2 and the fraction of the probability distribution that they enclose as an M-dimensional region,i.e.,the confidence level of the M-dimensional confidence region. Theorem D.Suppose that a is drawn from the universe of simulated data sets (as above),that its first v components a1,...,a are held fixed,and that its remaining M-vcomponents are varied so as to minimize x2.Call this minimum value x2.Then Axx-x is distributed as a chi-square distribution with vdegrees of freedom.If you consult Figure 15.6.4,you will see that this theorem connects the projected Ax2region with a confidence level.In the figure,a point that 81 is held fixed in a2 and allowed to vary in a minimizing x2 will seek out the ellipse whose top or bottom edge is tangent to the line of constant a2,and is therefore the line that projects it onto the smaller-dimensional space. As a first example,let us consider the case v=1,where we want to find 人 the confidence interval of a single parameter,say a1.Notice that the chi-square distribution withv=1 degree of freedom is the same distribution as that of the square of a single normally distributed quantity.Thus Ax<1 occurs 68.3 percent of the time(1-o for the normal distribution),x<4 occurs 95.4 percent of the time(2- 公 2 for the normal distribution),Ax<9 occurs 99.73 percent of the time (3-0 for the normal distribution),etc.In this manner you find the Ax that corresponds to your desired confidence level.(Additional values are given in the accompanying table.) Press. Let a be a change in the parameters whose first component is arbitrary,oa1, but the rest of whose components are chosen to minimize the Ax2.Then Theorem D applies.The value of Ax2 is given in general by △x2=6a.[a.6a (15.6.1) 6 which follows from equation(15.5.8)applied at x where =0.Since 6a by hypothesis minimizes x2 in all but its first component,the second through Mth components of the normal equations (15.5.9)continue to hold.Therefore,the solution of (15.5.9)is 0 Numerica 10.621 6a [a] (15.6.2) E喜 43106 0 0 (outside where c is one arbitrary constant that we get to adjust to make(15.6.1)give the desired left-hand value.Plugging(15.6.2)into (15.6.1)and using the fact that [C] North and [a]are inverse matrices of one another,we get c=0a1/Cu and △X2=(6a1)2/C1 (15.6.3) or 6a1=±V△x2VCi (15.6.4) At last!A relation between the confidence interval toa and the formal standard error1=VC11.Not unreasonably,we find that the 68 percent confidence interval is t,the 95 percent confidence interval is +201,etc
696 Chapter 15. Modeling of Data Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). using the fixed (actual) data set D(0). This theorem makes the connection between particular values of ∆χ2 and the fraction of the probability distribution that they enclose as an M-dimensional region, i.e., the confidence level of the M-dimensional confidence region. Theorem D. Suppose that aS (j) is drawn from the universe of simulated data sets (as above), that its first ν components a1,...,aν are held fixed, and that its remaining M − ν components are varied so as to minimize χ2. Call this minimum value χ2 ν . Then ∆χ2 ν ≡ χ2 ν − χ2 min is distributed as a chi-square distribution with ν degrees of freedom. If you consult Figure 15.6.4, you will see that this theorem connects the projected ∆χ2 region with a confidence level. In the figure, a point that is held fixed in a2 and allowed to vary in a1 minimizing χ2 will seek out the ellipse whose top or bottom edge is tangent to the line of constant a 2, and is therefore the line that projects it onto the smaller-dimensional space. As a first example, let us consider the case ν = 1, where we want to find the confidence interval of a single parameter, say a 1. Notice that the chi-square distribution with ν = 1 degree of freedom is the same distribution as that of the square of a single normally distributed quantity. Thus ∆χ2 ν < 1 occurs 68.3 percent of the time (1-σ for the normal distribution), ∆χ2 ν < 4 occurs 95.4 percent of the time (2-σ for the normal distribution), ∆χ2 ν < 9 occurs 99.73 percent of the time (3-σ for the normal distribution), etc. In this manner you find the ∆χ2 ν that corresponds to your desired confidence level. (Additional values are given in the accompanying table.) Let δa be a change in the parameters whose first component is arbitrary, δa 1, but the rest of whose components are chosen to minimize the ∆χ2. Then Theorem D applies. The value of ∆χ2 is given in general by ∆χ2 = δa · [α] · δa (15.6.1) which follows from equation (15.5.8) applied at χ2 min where βk = 0. Since δa by hypothesis minimizes χ2 in all but its first component, the second through Mth components of the normal equations (15.5.9) continue to hold. Therefore, the solution of (15.5.9) is δa = [α] −1 · c 0 . . . 0 = [C] · c 0 . . . 0 (15.6.2) where c is one arbitrary constant that we get to adjust to make (15.6.1) give the desired 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 get c = δa1/C11 and ∆χ2 ν = (δa1) 2/C11 (15.6.3) or δa1 = ± ∆χ2 ν C11 (15.6.4) At last! A relation between the confidence interval ±δa1 and the formal standard error σ1 ≡ √C11. Not unreasonably, we find that the 68 percent confidence interval is ±σ1, the 95 percent confidence interval is ±2σ1, etc
15.6 Confidence Limits on Estimated Model Parameters 697 Ax2 as a Function of Confidence Level and Degrees of Freedom p 1 2 3 6 68.3% 1.00 2.30 3.53 4.72 5.89 7.04 90% 2.71 4.61 6.25 7.78 9.24 10.6 95.4% 4.00 6.17 8.02 9.70 11.3 12.8 99% 6.63 9.21 11.3 13.3 15.1 16.8 99.73% 9.00 11.8 14.2 16.3 18.2 20.1 99.99% 15.1 18.4 21.1 23.5 25.7 27.8 These considerations hold not just for the individual parameters ai,but also for any linear combination of them:If M h Ciai=c·a (15.6.5) 9 k=1 then the 68 percent confidence interval on b is b=±Vc.[C·c (15.6.6) However,these simple,normal-sounding numerical relationships do not hold in 2、《》 the case >1 [3].In particular,Ax2=1 is not the boundary,nor does it project onto the boundary,of a 68.3 percent confidence region whenv>1.If you want to calculate not confidence intervals in one parameter,but confidence ellipses in two parameters jointly,or ellipsoids in three,or higher,then you must follow the following prescription for implementing Theorems C and D above: Let v be the number of fitted parameters whose joint confidence region you wish to display,<M.Call these parameters the"parameters of interest." Let p be the confidence limit desired,e.g.,p=0.68 or p =0.95. Numerica 10621 Find A(i.e.,Ax2)such that the probability of a chi-square variable with 431 vdegrees of freedom being less than A is p.For some useful values of p and v,A is given in the table.For other values,you can use the routine Recipes gammq and a simple root-finding routine (e.g.,bisection)to find A such (outside that gammq(v/2,A/2)=1-p. North Take the Mx M covariance matrix [C]=[a]-1 of the chi-square fit. Copy the intersection of the v rows and columns corresponding to the parameters of interest into a vxvmatrix denoted [Cprojl. .Invert the matrix [Cproj].(In the one-dimensional case this was just taking the reciprocal of the element C1.) The equation for the elliptical boundary of your desired confidence region in the v-dimensional subspace of interest is △=da'·[Cprojl-1.ia (15.6.7 where oa'is the v-dimensional vector of parameters of interest
15.6 Confidence Limits on Estimated Model Parameters 697 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). ∆χ2 as a Function of Confidence Level and Degrees of Freedom ν p 123456 68.3% 1.00 2.30 3.53 4.72 5.89 7.04 90% 2.71 4.61 6.25 7.78 9.24 10.6 95.4% 4.00 6.17 8.02 9.70 11.3 12.8 99% 6.63 9.21 11.3 13.3 15.1 16.8 99.73% 9.00 11.8 14.2 16.3 18.2 20.1 99.99% 15.1 18.4 21.1 23.5 25.7 27.8 These considerations hold not just for the individual parameters a i, but also for any linear combination of them: If b ≡ M k=1 ciai = c · a (15.6.5) then the 68 percent confidence interval on b is δb = ± c · [C] · c (15.6.6) However, these simple, normal-sounding numerical relationships do not hold in the case ν > 1 [3]. In particular, ∆χ2 = 1 is not the boundary, nor does it project onto the boundary, of a 68.3 percent confidence region when ν > 1. If you want to calculate not confidence intervals in one parameter, but confidence ellipses in two parameters jointly, or ellipsoids in three, or higher, then you must follow the following prescription for implementing Theorems C and D above: • Let ν be the number of fitted parameters whose joint confidence region you wish 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 with ν degrees of freedom being less than ∆ is p. For some useful values of p and ν, ∆ is given in the table. For other values, you can use the routine gammq and a simple root-finding routine (e.g., bisection) to find ∆ such that 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 the parameters of interest into a ν × ν matrix denoted [Cproj]. • Invert the matrix [Cproj]. (In the one-dimensional case this was just taking the reciprocal of the element C11.) • The equation for the elliptical boundary of your desired confidence region in the ν-dimensional subspace of interest is ∆ = δa · [Cproj] −1 · δa (15.6.7) where δa is the ν-dimensional vector of parameters of interest
698 Chapter 15.Modeling of Data △x2=1 length (1) ve) http://www.nr. Per readable files length W2 -800472 co (including this one) granted fori 18819920的e 19 a from NUMERICAL RECIPES IN C: Figure 15.6.5.Relation of the confidence region ellipse x=1 to quantities computed by singular value decomposition.The vectors V are unit vectors along the principal axes of the confidence region (North server The semi-axes have lengths equal to the reciprocal of the singular values.If the axes are all scaled by some constant factor a.Ax2 is scaled by the factor a2 computer, Americ tusers to make one paper UnN电.t THE ART If you are confused at this point,you may find it helpful to compare Figure 15.6.4 and the accompanying table,considering the case M =2 with v=1 and 是 ictly proh Programs v=2.You should be able to verify the following statements:(i)The horizontal band between C and C'contains 99 percent of the probability distribution,so it is a confidence limit on a2 alone at this level of confidence.(ii)Ditto the band between B and B'at the 90 percent confidence level.(iii)The dashed ellipse. labeled by Ax2=2.30,contains 68.3 percent of the probability distribution,so it is to dir a confidence region for a and a2 jointly,at this level of confidence. OF SCIENTIFIC COMPUTING(ISBN Confidence Limits from Singular Value Decomposition 198918920 When you have obtained your x2 fit by singular value decomposition(15.4),the .Further reproduction, Numerical Recipes 10-6211 information about the fit's formal errors comes packaged in a somewhat different,but generally more convenient,form.The columns of the matrix V are an orthonormal 43108 set of M vectors that are the principal axes of the Ax2=constant ellipsoids. We denote the columns as V()...V(M).The lengths of those axes are inversely (outside proportional to the corresponding singular values w1...wM;see Figure 15.6.5.The boundaries of the ellipsoids are thus given by North Software. △x2=w(V(a·6a)2+…+wr(Nan·ia)2 (15.6.8) which is the justification for writing equation(15.4.18)above.Keep in mind that visit website it is much easier to plot an ellipsoid given a list of its vector principal axes,than machine given its matrix quadratic form! The formula for the covariance matrix [C]in terms of the columns V()is M lC]=2Vo⑧Va (15.6.9 or,in components
698 Chapter 15. Modeling of Data Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). 1 w2 V(2) V(1) ∆χ2 = 1 a2 a1 length length 1 w1 Figure 15.6.5. Relation of the confidence region ellipse ∆χ2 = 1 to quantities computed by singular value decomposition. The vectors V(i) are unit vectors along the principal axes of the confidence region. The semi-axes have lengths equal to the reciprocal of the singular values wi. If the axes are all scaled by some constant factor α, ∆χ2 is scaled by the factor α2. If you are confused at this point, you may find it helpful to compare Figure 15.6.4 and the accompanying table, considering the case M = 2 with ν = 1 and ν = 2. You should be able to verify the following statements: (i) The horizontal band between C and C contains 99 percent of the probability distribution, so it is a confidence limit on a2 alone at this level of confidence. (ii) Ditto the band between B and B at the 90 percent confidence level. (iii) The dashed ellipse, labeled by ∆χ2 = 2.30, contains 68.3 percent of the probability distribution, so it is a confidence region for a1 and a2 jointly, at this level of confidence. Confidence Limits from Singular Value Decomposition When you have obtained your χ2 fit by singular value decomposition (§15.4), the information about the fit’s formal errors comes packaged in a somewhat different, but generally more convenient, form. The columns of the matrix V are an orthonormal set of M vectors that are the principal axes of the ∆χ2 = constant ellipsoids. We denote the columns as V(1) ... V(M). The lengths of those axes are inversely proportional to the corresponding singular values w1 ...wM ; see Figure 15.6.5. The boundaries of the ellipsoids are thus given by ∆χ2 = w2 1(V(1) · δa) 2 + ··· + w2 M (V(M) · δa) 2 (15.6.8) which is the justification for writing equation (15.4.18) above. Keep in mind that it is much easier to plot an ellipsoid given a list of its vector principal axes, than given its matrix quadratic form! The formula for the covariance matrix [C] in terms of the columns V (i) is [C] = M i=1 1 w2 i V(i) ⊗ V(i) (15.6.9) or, in components