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69915.7 Robust EstimationCjk =MX1Vji Vkiwi2i=1(15.6.10)Efron, B., and Tibshirani, R. 1986, Statistical Science vol. 1, pp. 54–77. [2]Avni, Y. 1976, Astrophysical Journal, vol. 210, pp. 642–646. [3]Lampton, M., Margon, M., and Bowyer, S. 1976, Astrophysical Journal, vol. 208, pp. 177–190.Brownlee, K.A. 1965, Statistical Theory and Methodology, 2nd ed. (New York: Wiley).Martin, B.R. 1971, Statistics for Physicists (New York: Academic Press).15.7 Robust EstimationThe concept of robustness has been mentioned in passing several times already.In §14.1 we noted that the median was a more robust estimator of central value thanthe mean; in §14.6 it was mentioned that rank correlation is more robust than linearcorrelation.
The concept of outlier points as exceptions to a Gaussian model forexperimental error was discussed in §15.1.The term “robust” was coined in statistics by G.E.P. Box in 1953. Variousdefinitions of greater or lesser mathematical rigor are possible for the term, but ingeneral, referring to a statistical estimator, it means “insensitive to small departuresfrom the idealized assumptions for which the estimator is optimized.” [1,2] The word“small” can have two different interpretations, both important: either fractionallysmall departures for all data points, or else fractionally large departures for a smallnumber of data points. It is the latter interpretation, leading to the notion of outlierpoints, that is generally the most stressful for statistical procedures.Statisticians have developed various sorts of robust statistical estimators.
Many,if not most, can be grouped in one of three categories.M-estimates follow from maximum-likelihood arguments very much as equations (15.1.5) and (15.1.7) followed from equation (15.1.3). M-estimates are usuallythe most relevant class for model-fitting, that is, estimation of parameters. Wetherefore consider these estimates in some detail below.L-estimates are “linear combinations of order statistics.” These are mostapplicable to estimations of central value and central tendency, though they canoccasionally be applied to some problems in estimation of parameters. Two“typical” L-estimates will give you the general idea.
They are (i) the median, and(ii) Tukey’s trimean, defined as the weighted average of the first, second, and thirdquartile points in a distribution, with weights 1/4, 1/2, and 1/4, respectively.R-estimates are estimates based on rank tests. For example, the equality orinequality of two distributions can be estimated by the Wilcoxon test of computingthe mean rank of one distribution in a combined sample of both distributions.The Kolmogorov-Smirnov statistic (equation 14.3.6) and the Spearman rank-orderSample 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).CITED REFERENCES AND FURTHER READING:Efron, B. 1982, The Jackknife, the Bootstrap, and Other Resampling Plans (Philadelphia:S.I.A.M.).
[1]700Chapter 15.Modeling of Datanarrowcentral peak(a)least squares fitrobust straight-line fit(b)Figure 15.7.1.Examples where robust statistical methods are desirable: (a) A one-dimensionaldistribution with a tail of outliers; statistical fluctuations in these outliers can preventaccurate determinationof the position of the central peak.
(b) A distribution in two dimensions fitted to a straight line; non-robusttechniques such as least-squares fitting can have undesired sensitivity to outlying points.correlation coefficient (14.6.1) are R-estimates in essence, if not always by formaldefinition.Some other kinds of robust techniques, coming from the fields of optimal controland filtering rather than from the field of mathematical statistics, are mentioned at theend of this section. Some examples where robust statistical methods are desirableare shown in Figure 15.7.1.Estimation of Parameters by Local M-EstimatesSuppose we know that our measurement errors are not normally distributed.Then, in deriving a maximum-likelihood formula for the estimated parameters a in amodel y(x; a), we would write instead of equation (15.1.3)P =NYi=1{exp [−ρ(yi , y {xi ; a})] ∆y}(15.7.1)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).tail ofoutliers70115.7 Robust Estimationwhere the function ρ is the negative logarithm of the probability density. Taking thelogarithm of (15.7.1) analogously with (15.1.4), we find that we want to minimizethe expressionNXρ(yi , y {xi ; a})(15.7.2)Very often, it is the case that the function ρ depends not independently on itstwo arguments, measured yi and predicted y(xi ), but only on their difference, at leastif scaled by some weight factors σi which we are able to assign to each point.
In thiscase the M-estimate is said to be local, and we can replace (15.7.2) by the prescriptionNXyi − y(xi ; a)ρσiminimize over a(15.7.3)i=1where the function ρ(z) is a function of a single variable z ≡ [yi − y(xi )]/σi .If we now define the derivative of ρ(z) to be a function ψ(z),ψ(z) ≡dρ(z)dz(15.7.4)then the generalization of (15.1.7) to the case of a general M-estimate is0=NX1∂y(xi ; a)yi − y(xi )ψσiσi∂akk = 1, . . . , M(15.7.5)i=1If you compare (15.7.3) to (15.1.3), and (15.7.5) to (15.1.7), you see at oncethat the specialization for normally distributed errors isρ(z) =1 2z2ψ(z) = z(normal)(15.7.6)If the errors are distributed as a double or two-sided exponential, namely yi − y(xi ) (15.7.7)Prob {yi − y(xi )} ∼ exp − σithen, by contrast,ρ(x) = |z|ψ(z) = sgn(z)(double exponential)(15.7.8)Comparing to equation (15.7.3), we see that in this case the maximum likelihoodestimator is obtained by minimizing the mean absolute deviation, rather than themean square deviation.
Here the tails of the distribution, although exponentiallydecreasing, are asymptotically much larger than any corresponding Gaussian.A distribution with even more extensive — therefore sometimes even morerealistic — tails is the Cauchy or Lorentzian distribution,Prob {yi − y(xi )} ∼11+21yi − y(xi )σi2(15.7.9)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).i=1702Chapter 15.Modeling of DataThis implies1ρ(z) = log 1 + z 22ψ(z) =z1 + 12 z 2(Lorentzian)(15.7.10)If the measurement errors happen to be normal after all, with standard deviations σi ,then it can be shown that the optimal value for the constant c is c = 2.1.Tukey’s biweight|z| < cz(1 − z 2 /c2 )2ψ(z) =(15.7.12)0|z| > cwhere the optimal value of c for normal errors is c = 6.0.Numerical Calculation of M-EstimatesTo fit a model by means of an M-estimate, you first decide which M-estimateyou want, that is, which matching pair ρ, ψ you want to use.
We rather like(15.7.8) or (15.7.10).You then have to make an unpleasant choice between two fairly difficultproblems. Either find the solution of the nonlinear set of M equations (15.7.5), orelse minimize the single function in M variables (15.7.3).Notice that the function (15.7.8) has a discontinuous ψ, and a discontinuousderivative for ρ. Such discontinuities frequently wreak havoc on both generalnonlinear equation solvers and general function minimizing routines. You mightnow think of rejecting (15.7.8) in favor of (15.7.10), which is smoother.
However,you will find that the latter choice is also bad news for many general equation solvingor minimization routines: small changes in the fitted parameters can drive ψ(z)off its peak into one or the other of its asymptotically small regimes. Therefore,different terms in the equation spring into or out of action (almost as bad as analyticdiscontinuities).Don’t despair. If your computer budget (or, for personal computers, patience)is up to it, this is an excellent application for the downhill simplex minimizationSample 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.
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