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Modeling of Data15.0 IntroductionGiven a set of observations, one often wants to condense and summarize thedata by fitting it to a “model” that depends on adjustable parameters. Sometimes themodel is simply a convenient class of functions, such as polynomials or Gaussians,and the fit supplies the appropriate coefficients. Other times, the model’s parameterscome from some underlying theory that the data are supposed to satisfy; examplesare coefficients of rate equations in a complex network of chemical reactions, ororbital elements of a binary star.
Modeling can also be used as a kind of constrainedinterpolation, where you want to extend a few data points into a continuous function,but with some underlying idea of what that function should look like.The basic approach in all cases is usually the same: You choose or design afigure-of-merit function (“merit function,” for short) that measures the agreementbetween the data and the model with a particular choice of parameters. The meritfunction is conventionally arranged so that small values represent close agreement.The parameters of the model are then adjusted to achieve a minimum in the meritfunction, yielding best-fit parameters.
The adjustment process is thus a problem inminimization in many dimensions. This optimization was the subject of Chapter 10;however, there exist special, more efficient, methods that are specific to modeling,and we will discuss these in this chapter.There are important issues that go beyond the mere finding of best-fit parameters.Data are generally not exact. They are subject to measurement errors (called noisein the context of signal-processing). Thus, typical data never exactly fit the modelthat is being used, even when that model is correct. We need the means to assesswhether or not the model is appropriate, that is, we need to test the goodness-of-fitagainst some useful statistical standard.We usually also need to know the accuracy with which parameters are determined by the data set.
In other words, we need to know the likely errors ofthe best-fit parameters.Finally, it is not uncommon in fitting data to discover that the merit functionis not unimodal, with a single minimum. In some cases, we may be interested inglobal rather than local questions. Not, “how good is this fit?” but rather, “howsure am I that there is not a very much better fit in some corner of parameter space?”As we have seen in Chapter 10, especially §10.9, this kind of problem is generallyquite difficult to solve.The important message we want to deliver is that fitting of parameters is notthe end-all of parameter estimation.
To be genuinely useful, a fitting procedure656Sample 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).Chapter 15.65715.1 Least Squares as a Maximum Likelihood EstimatorCITED REFERENCES AND FURTHER READING:Bevington, P.R. 1969, Data Reduction and Error Analysis for the Physical Sciences (New York:McGraw-Hill).Brownlee, K.A. 1965, Statistical Theory and Methodology, 2nd ed. (New York: Wiley).Martin, B.R. 1971, Statistics for Physicists (New York: Academic Press).von Mises, R. 1964, Mathematical Theory of Probability and Statistics (New York: AcademicPress), Chapter X.Korn, G.A., and Korn, T.M.
1968, Mathematical Handbook for Scientists and Engineers, 2nd ed.(New York: McGraw-Hill), Chapters 18–19.15.1 Least Squares as a Maximum LikelihoodEstimatorSuppose that we are fitting N data points (xi , yi ) i = 1, . . . , N , to a model thathas M adjustable parameters aj , j = 1, . . . , M . The model predicts a functionalrelationship between the measured independent and dependent variables,y(x) = y(x; a1 . . .
aM )(15.1.1)where the dependence on the parameters is indicated explicitly on the right-hand side.What, exactly, do we want to minimize to get fitted values for the aj ’s? Thefirst thing that comes to mind is the familiar least-squares fit,minimize over a1 . . . aM :NX[yi − y(xi ; a1 . . . aM )]2(15.1.2)i=1But where does this come from? What general principles is it based on? The answerto these questions takes us into the subject of maximum likelihood estimators.Given a particular data set of xi ’s and yi ’s, we have the intuitive feeling thatsome parameter sets a1 .
. . aM are very unlikely — those for which the modelfunction y(x) looks nothing like the data — while others may be very likely — thosethat closely resemble the data. How can we quantify this intuitive feeling? How canwe select fitted parameters that are “most likely” to be correct? It is not meaningfulto ask the question, “What is the probability that a particular set of fitted parametersa1 . . . aM is correct?” The reason is that there is no statistical universe of modelsfrom which the parameters are drawn. There is just one model, the correct one, anda statistical universe of data sets that are drawn from it!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).should provide (i) parameters, (ii) error estimates on the parameters, and (iii) astatistical measure of goodness-of-fit. When the third item suggests that the modelis an unlikely match to the data, then items (i) and (ii) are probably worthless.Unfortunately, many practitioners of parameter estimation never proceed beyonditem (i). They deem a fit acceptable if a graph of data and model “looks good.” Thisapproach is known as chi-by-eye. Luckily, its practitioners get what they deserve..
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