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Suppose that your data set consists ofN independent and identically distributed (or iid) “data points.” Each data pointprobably consists of several numbers, e.g., one or more control variables (uniformlydistributed, say, in the range that you have decided to measure) and one or moreassociated measured values (each distributed however Mother Nature chooses).“Iid” means that the sequential order of the data points is not of consequence tothe process that you are using to get the fitted parameters a.
For example, a χ2sum like (15.5.5) does not care in what order the points are added. Even simplerexamples are the mean value of a measured quantity, or the mean of some functionof the measured quantities.SThe bootstrap method [1] uses the actual data set D(0), with its N data points, toSSgenerate any number of synthetic data sets D(1), D(2), . . . , also with N data points.The procedure is simply to draw N data points at a time with replacement from theSample 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).onteCarlorealization15.6 Confidence Limits on Estimated Model Parameters692Chapter 15.Modeling of DataConfidence LimitsRather than present all details of the probability distribution of errors inparameter estimation, it is common practice to summarize the distribution in theform of confidence limits.
The full probability distribution is a function definedon the M -dimensional space of parameters a. A confidence region (or confidenceinterval) is just a region of that M -dimensional space (hopefully a small region) thatcontains 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 thetrue parameter values fall within this region around the measured value.”It is worth emphasizing that you, the experimenter, get to pick both theconfidence level (99 percent in the above example), and the shape of the confidenceregion.
The only requirement is that your region does include the stated percentageof probability. Certain percentages are, however, customary in scientific usage:68.3 percent (the lowest confidence worthy of quoting), 90 percent, 95.4 percent, 99percent, and 99.73 percent. Higher confidence levels are conventionally “ninety-ninepoint nine . . . nine.” As for shape, obviously you want a region that is compactand reasonably centered on your measurement a(0) , since the whole purpose of aconfidence 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 higherdimensions, ellipses or ellipsoids are most frequently used.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).Sset D(0). Because of the replacement, you do not simply get back your originaldata 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, exactlyas in the previous discussion, you subject these data sets to the same estimationprocedure as was performed on the actual data, giving a set of simulated measuredparameters aS(1) , aS(2) , . . . . These will be distributed around a(0) in close to the sameway that a(0) is distributed around atrue .Sounds like getting something for nothing, doesn’t it? In fact, it has taken morethan 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, viewedas a probability distribution consisting of delta functions at the measured values, isin most cases the best — or only — available estimator of the underlying probabilitydistribution.
It takes courage, but one can often simply use that distribution as thebasis for Monte Carlo simulations.Watch out for cases where the bootstrap’s “iid” assumption is violated. Forexample, if you have made measurements at evenly spaced intervals of some controlvariable, then you can usually get away with pretending that these are “iid,” uniformlydistributed over the measured range. However, some estimators of a (e.g., onesinvolving Fourier methods) might be particularly sensitive to all the points on a gridbeing present.
In that case, the bootstrap is going to give a wrong distribution. Alsowatch out for estimators that look at anything like small-scale clumpiness within theN 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 methodare 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, veryquick, Monte Carlo estimates of the errors in an estimated parameter set.69315.6 Confidence Limits on Estimated Model Parameters(s)a (i)2− a(0)268% confidence regionon a1 and a2 jointly68% confidence interval on a2(s)− a(0)1a (i)1biasFigure 15.6.3. Confidence intervals in 1 and 2 dimensions. The same fraction of measured points (here68%) 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 normaldistribution.
That is true historically, but not always relevant nowadays. In general,the probability distribution of the parameters will not be normal, and the abovenumbers, 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 thesame confidence level. The two vertical lines enclose a band (horizontal inverval)which represents the 68 percent confidence interval for the variable a1 without regardto the value of a2 . Similarly the horizontal lines enclose a 68 percent confidenceinterval for a2 .
The ellipse shows a 68 percent confidence interval for a1 and a2jointly. Notice that to enclose the same probability as the two bands, the ellipse mustnecessarily extend outside of both of them (a point we will return to below).Constant Chi-Square Boundaries as Confidence LimitsWhen 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 theshape of confidence intervals, whose use is almost universal.
For the observed dataset D(0) , the value of χ2 is a minimum at a(0) . Call this minimum value χ2min . IfSample 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).68% confidenceinterval on a1694Chapter 15.Modeling of Data∆χ 2 = 6.63C∆χ 2 = 2.71B∆χ 2 = 1.00ZZ′A′∆χ 2 = 2.30B′C′Figure 15.6.4.
Confidence region ellipses corresponding to values of chi-square larger than the fittedminimum. The solid curves, with ∆χ2 = 1.00, 2.71, 6.63 project onto one-dimensional intervals AA0 ,BB 0, CC 0 . These intervals — not the ellipses themselves — contain 68.3%, 90%, and 99% of normallydistributed data. The ellipse that contains 68.3% of normally distributed data is shown dashed, and has∆χ2 = 2.30.
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