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Statistics, Probability and Chaos Author(s): L. Mark Berliner Source: Statistical Science, Vol. 7, No. 1 (Feb., 1992), pp. 69-90 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2245991 . Accessed: 15/08/2011 11:09 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to Statistical Science. http://www.jstor.org Statistical Science 1992, Vol. 7, No. 1, 69-122 Statistics, Probabilityand Chaos L. Mark Berliner Abstract.The studyof chaotic behavior has receivedsubstantial atten- tionin manydisciplines. Although often based on deterministicmodels, chaos is associated with complex,"random" behavior and formsof unpredictability.Mathematical models and definitionsassociated with chaos are reviewed.The relationshipbetween the mathematicsof chaos and probabilisticnotions, including ergodic theoryand uncertainty modeling,are emphasized.Popular data analyticmethods appearing in the literatureare discussed.A major goal of this articleis to present some indicationsof how probabilitymodelers and statisticianscan contributeto analyses involvingchaos. Key wordsand phrases: Dynamicalsystems, ergodic theory, nonlinear time series,stationary processes, prediction. 1. INTRODUCTION tions of chaos, I offera reviewof the basic notions ofchaos withemphasis on thoseaspects of particu- unpre- Chaos is associated with complex and lar interestto statisticiansand probabilists.Many dictable behavior of phenomenaover time. Such of the referencesgiven here provideindications of in behaviorcan arise deterministicdynamical sys- the breath of interestin chaos. Jackson (1989) tems. Many examples are based on mathematical providesan introductionand an extensivebibliog- models for (discrete)time series in which, after raphy [also, see Shiraiwa (1985)]. Berge, Pomeau startingfrom some initial condition,the value of and Vidal (1984),Cooper (1989) and Rasband(1990) the series at any time is a specified,nonlinear discuss applicationsof chaos in the physical sci- functionof the previousvalue. (Continuoustime ences and engineering.Valuable sourcesfor work processesare discussed in Section 2.) These proc- on chaos in biologicaland medical scienceinclude esses are intriguingin that the realizationscorre- May (1987), Glass and Mackey (1988) and Basar spondingto different,although extremelyclose, (1990). Wegman(1988) and Chatterjeeand Yilmaz initial conditionstypically diverge. The practical (1992) present reviews of particular interestto implicationof this phenomenonis that,despite the statisticians.Finally, useful, "general audience" underlyingdeterminism, we cannot predict,with introductionsto chaos includeCrutchfield, Farmer, any reasonableprecision, the values of the process. Packard and Shaw (1986), Gleick (1987), Peterson for large time values; even the slightesterror in (1988) and Stewart(1989). specifyingthe initialcondition eventually ruins our Section 2 presents discussion of the standard attempt.Later in this article,indications that real- mathematicalsetup of nonlinear dynamical sys- izations of such dynamical systems can display tems.Definitions of chaos are reviewedand chaotic characteristicstypically associated with random- behavioris explainedmathematically, as well as by ness are presented.A majortheme of this studyis example. Next, I review relationshipsbetween thatthis connection with randomness suggests that chaos and probability.Two key pointsin this dis- statisticalreasoning may play a crucialrole in the cussionare: (i) the roleof ergodic theory and (ii) the analysis ofchaos. suggestionof uncertainty modeling and analysisby Stronginterest has recentlybeen shown in nu- probabilisticmethods. Statistical analyses related merous literaturesin the areas of nonlineardy- to chaos are discussed in Sections 3 and 4. In namical systemsand chaos. However,rather than Section3, the emphasisis on some "data analytic" attemptingto providean overviewof the applica- methodsfor analyzingchaotic data. The goals of these techniquesbasically involve attempts at un- derstandingthe structureand qualitative aspects of models and data displayingchaotic behavior. L. MarkBerliner is AssociateProfessor, Department [Specifically,the notionsof (i) estimationof dimen- of Statistics,Ohio State University,141 Cockins sion, (ii) Poincaremaps and (iii) reconstructionby Hall, 1958 Neil Avenue,Columbus, Ohio 43210. time delays are reviewed.]Although statisticians 69 70 L. M. BERLINER are now beginningto make contributionsalong specifyingthe initial conditionis in the sixth these lines, the methods describedin Section 3 decimalplace. This sortof behavior, known as sen- have been developedprimarily by mathematicians sitivity to initial conditions, is one of the key and physicists.In Section4, I discusssome possible componentsof chaos. To amplifyon this phe- strategiesfor methods of chaotic data analysisbased nomenon,the firstframe of Figure 2 presentsdot on main streamtechniques for statistical modeling plots, at selected time values, of the dynamical and inference.Finally, Section 5 is devotedto gen- system correspondingto 18 initial conditions eral remarksconcerning statistics and chaos. equally spaced in the interval [0.2340, 0.2357]. There are two messages in this plot. First, note that the images of these 18 points are quickly 2. MATHEMATICS,PROBABILITY AND CHAOS attractedto the unit interval.Second, the initial 2.1 The Complexityof Nonlinear Dynamical conditionsappear to get "mixed" up in an almost noncontinuousmanner. (However,for the logistic Systems map, xt is, of course,a continuousfunction of xo A simpledeterministic dynamical system may be forall t). The secondframe of Figure 2 is a scatter- definedas follows.For a discretetime index set, plot of the values of the logisticmap after2000 T = {0, 1, 2, ... }, considera timeseries { x,; te T}. iterates against the correspondinginitial condi- Assume that xo is an initial conditionand that tions for4000 initials equally spaced in the inter- xt+1= f(xt), forsome functionf that maps a do- val [0.10005, 0.3]. There is clearly essentiallyno main D into D. (D is typicallya compactsubset of meaningfulstatements about the relationshipbe- a metricspace). Chaoticbehavior may arise when f tween x2000and xo, even though x2000is a well- is a nonlinearfunction. definedpolynomial function (of admittedlyhigh To begin,some numerical examples for one ofthe order) of xo. (Note that presentingthis graph more popular examples of dynamicalsystems, the for 2000 iterates is a bit of "overkill." Corre- logisticmap, are given. The dynamicalsystem is spondingscatterplots after even a 100 or so iterates obtainedby iteratingthe function f( x) = ax(l - x), wouldlook quite the same.) where a is a fixedparameter in the interval[0,4]. Let xo be an initialpoint in the interval[0,1]; note 2.1.1 Some mathematicsfor nonlinear dynamical that then all futurevalues ofthe systemalso lie in systems [0,1].To get a bit ofthe flavorof this map,example This discussionis intendedto providesome flavor computationsare presentedfor an importantvalue of the mathematicsconcerning the appearance of of a: namely,a = 4.0. Figure 1 presentstime series complexor chaoticbehavior in nonlineardynami- plots of the first500 iterates of the logisticmap cal systems.The presentationis a bit quick, and correspondingto the initial values 0.31, 0.310001 untilSection 2.1.3, considersone-dimensional maps and 0.32. The firstthing to notice about these only.More completedetails maybe foundin Collet series is that their appearance is "complex." In- and Eckmann(1980), Rasband (1990) and Devaney deed, one mightbe temptedto suggestthese series (1989). We begin by consideringthe long-run are "random." Also, despitethe similarityin the behavior of a dynamicalsystem generated by a initial conditions,visual inspectionof the series nonlinearfunction f. The studybegins with the indicatesthat they are not quantitativelysimilar. considerationof fixedpoints of f; namely,those To make the point,I have includedscatterplots of pointsthat are solutionsto f(x) = x. The key re- these series,matched by time. The first25 iterates sult in this contextis the followingproposition. ofthe maps in these plots are indicatedby a differ- Using conventionalnotation, let ffn(.) denotethe ent symbol fromthe rest. Points falling on the 450 n-foldcomposition of f. line in these plotssuggest time values at whichthe correspondingvalues ofthe systemsare quite close. PROPOSITION 2.1. Let p be a fixed point of f. If We see that quite early in time, the three series I f'(p) I < 1, then thereexists an open intervalU "predict"each otherreasonably well. However,the about p such that, forall x in U, limn-+o fn(x) = p. similarityin the series diminishesrapidly as time increases. (The rate of this "separation" is in fact Under the conditionsof this proposition,p is an exponentialin time.) Note' that, except for very attractingfixed point and the set U is a stable set. early times, the series correspondingto xo = It is also true that if if'( p) i > 1, p is a
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