Mandelbrot's 1/f fractional renewal models of 1963-67: The non-ergodic missing link between change points and long range dependence
aa r X i v : . [ s t a t . O T ] F e b Mandelbrot’s 1 /f fractional renewal models of1963-67: The non-ergodic missing link betweenchange points and long range dependence. Nicholas Wynn Watkins Centre for the Analysis of Time Series, London School of Economics, London, UK,
[email protected] , Faculty of Mathematics, Computing and Technology, Open University, MiltonKeynes, UK, Centre for Fusion, Space and Astrophysics, University of Warwick,UK Universit¨at Potsdam, Institut f¨ur Physik und Astronomie, Campus Golm,Potsdam-Golm, Germany Max Planck Institute for the Physics of Complex Systems, Dresden, Germany.
Abstract.
The problem of 1/f noise has been with us for about a cen-tury. Because it is so often framed in Fourier spectral language, the mostfamous solutions have tended to be the stationary long range dependent(LRD) models such as Mandelbrot’s fractional Gaussian noise. In viewof the increasing importance to physics of non-ergodic fractional renewalmodels, I present preliminary results of my research into the history ofMandelbrot’s very little known work in that area from 1963-67. I spec-ulate about how the lack of awareness of this work in the physics andstatistics communities may have affected the development of complexityscience; and I discuss the differences between the Hurst effect, 1/f noiseand LRD, concepts which are often treated as equivalent.
Keywords:
Long range dependence, Mandelbrot, change points, frac-tional renewal models, weak ergodicity breaking “The problem of 1/f noise” has been with us for about 100 years since the pio-neering work of Schottky and Johnson [10,9,30]. It is usually framed as a spectralparadox, i.e.“how can the Fourier spectral density S ′ ( f ) of a stationary processtake the form S ′ ( f ) ∼ /f and thus be singular at the origin (or equivalentlyhow can the autocorrelation function “blow up” at large lags and thus not besummable) ? ”. When a problem is seen this way, the solution will also tend tobe sought in spectral terms. The desire of for a solution to the problem with asatisfying level of generality increased in the 1950s with the recognition of ananalogous time domain effect (the Hurst phenomenon) seen in the statisticalgrowth of range in Nile minima [9]. The first stationary solution which could ex-hibit the Hurst effect and 1 /f noise was presented by Mandelbrot in 1965 using fractional Gaussian noise (fGn), the increments of fractional Brownian motion(fBm), and was subsequently developed with Van Ness and Wallis, particularlyin the hydrological context [9,19]. fGn is a stationary ergodic process, for whicha power spectrum is a natural and well-defined concept, the paradox here residesin its singular behaviour at zero.However, in the last two decades it has increasingly been realised in physicsthat another class of models, the fractional renewal processes, can also give 1 /f noise in a very different way[21]. Physical interest has come from phenomenasuch as weak ergodicity breaking (in e.g. blinking quantum dots [29,25,28]) andthe related question of how many different classes of model can share the commonproperty of the 1/f spectral shape (e.g. [26,27]). In view of this resurgence of ac-tivity, my first aim in this paper is to report (Section 2) preliminary results frommy historical research which has found, to my great surprise, that the dichotomybetween ergodic and non-ergodic origins for 1 /f spectra was not only recognisedbut also published about by Mandelbrot 50 years ago in some still remarkablylittle known work [4,15,16,17]. He carried it out in parallel with his seminal workon the ergodic, stationary fGn model. In these papers, and the bridging essayshe wrote when he revisited them late in life for his collected Selecta volumes,particularly [18,19], he developed and published a series of fractional renewalmodels. In these the periodogram, the empirical autocorrelation function (acf),and the observed waiting time distributions, all grow in extent with the lengthof time over which they are measured. He explicitly [17] drew attention to thisnon-ergodicity and its origins in what he called “conditional” stationarity. Heexplicitly contrasted the fractional renewal models with the stationary, ergodicfGn which is today very much better known to physicists, geoscientists and manyother time series analysts [3,9]. Mandelbrot’s work at IBM was itself in parallelwith other developments, one notable example being the work of Pierre Mertz[22,23] at RAND on modelling telephone errors, so my preliminary report is notan attempt to assign priority. I hope to return to the history of this period inmore detail in future articles.My next purpose (Section 3) is to clarify the subtle differences between 3phenomena: the empirical Hurst effect, the appearance of 1 /f noise in peri-odograms, and the concept of LRD as embodied in the stationary ergodic fGnmodel, and to set out their hierarchy with respect to each other, aided in partby this historical perspective. This paper will not deal with another possibility-multiplicative models-[18,26], though I do of course recognise that they remaina very important alternative source of 1 /f behaviour, particularly that arisingfrom turbulent cascades. I will also not be considering 1 /f -type spectra arisingfrom nonstationary self-similar walks such as fractional Brownian motion.I will (Section 4) conclude by speculating on the how the relative neglectof[4,15,16,17] at the time of their publication may have had long-term effects. fGn [3] is effectively a derivative of fractional Brownian motion Y H, ( t ): Y H, ( t ) = 1 C H, Z R dL ( s ) K H, ( t − s ) (1)which in turn extends the Wiener process to include a self-similar, memory kernel K H, ( t − s ), where K H, ( t − s ) = [( t − s ) H − / − ( − s ) H − / ] (2) thus giving a decaying, non-zero weight to all of the values in the time integralover dL .In consequence fGn shows long range dependence, and has indeed becomea very important paradigmatic model for LRD. The attention paid to its 1/fspectrum, and long-tailed acf, as diagnostics of LRD, has often led to it beingforgotten that its stationarity is an equally essential ingredient for LRD in thissense. Intuitively one can see that without stationarity there can be no LRDbecause there is no infinitely long past history over which the process can bedependent. Models like fGn, and also fractionally integrated noise (FIN) and theautoregressive fractionally integrated moving average (ARFIMA) process, whichhave been widely studied in the statistics community (e.g. [2,3]) exhibit LRD byconstruction , i.e. stationarity is assumed at the outset in defining them.While undeniably important to time series analysis and the development ofcomplexity science, we can already see from the restriction to stationary pro-cesses that the LRD concept, at least as embodied in fGn, will be insufficientto describe the whole range of either 1 /f or Hurst behaviour that observationsmay present us with. Full awareness of this limitation has been slow because ofthree widespread, deeply-ingrained, but unfortunately erroneous beliefs: i) thatan observed Fourier periodogram can always be taken to estimate a power spec-trum, ii) that the Fourier transform of an empirically obtained periodogram is always a meaningful estimator of an autocorrelation function, and iii) that theobservation of a 1/f Fourier periodogram in a time series must imply the kind oflong range dependence that is embodied in the ergodic fractional Gaussian noisemodel. The first two beliefs are routinely cautioned against in any good courseor book on time series analysis, including classics like Bendat’s [1]. The thirdbelief remains highly topical, however, because it is only relatively recently beingappreciated in the theoretical physics literature just how distinct two paradig-matic classes of 1/f noise model are, and how these differences relate not only toLRD but also to the fundamental physical question of weak ergodicity breaking(e.g. [6,21,25]).The second paradigm for 1/f noise mentioned above is the fractional renewalclass, which is a descendent of the classic random telegraph model [1], and solooks at first sight to be stationary and Markovian, but has switching timesat power law distributed intervals. A particularly well studied variant is thealternating fractal renewal process (AFRP, e.g. [13,14]), which is also closely connected to the renewal reward process in mathematics. When studied in thetelecommunications context, however, the AFRP has often had a cutoff appliedto its switching time distribution for large times to allow analytical tractability.The use of an upper cutoff unfortunately masks some of its most physicallyinteresting behaviour, because when the cutoffs are not used the periodogram,the empirical acf, and observed waiting time distributions, all grow with thelength of time over which they are measured, rendering the process both non-ergodic and non-stationary in an important sense (Mandelbrot preferred hisown term “conditionally stationary”). In particular, Mandelbrot stressed thatthe process no longer obeys the necessary conditions on the Wiener-Khinchinetheorem for its empirical periodogram to be interpreted as an estimate of thepower spectrum. This property of weak ergodicity breaking (named by Bouchaudin the early 1990s [6]) is now attracting much interest in physics, see e.g. Niemannet al [25], on the resolution of the low frequency cutoff paradox, and subsequentdevelopments [12,7,26,27].The existence of this alternative, nonstationary, nonergodic fractional re-newal model makes it clear that there is a difference between the observation ofan empirical 1/f noise alone, and the presence of the type of LRD that is em-bodied in the stationary ergodic fGn model. We will develop this point furtherin section 3, but will first go back to the 1960s and Mandelbrot’s twin tracks to1 /f . What seems to have gone almost completely unnoticed, is the remarkable factthat Mandelbrot was not only aware of the distinction between fGn and frac-tional renewal models [18,19], but also published a nonstationary model of theAFRP type in 1965 [15,16] and had explicitly discussed the time dependence ofits power spectrum as a symptom on non-ergodicity by 1967 [17].There are 4 key papers in Mandelbrot’s consideration of fractional renewalmodels. The first, cowritten with physicist Jay Berger [4], appeared in IBM Jour-nal of Research and Development. It dealt with errors in telephone circuits, andits key point point was the power law distribution of times between errors, whichwere themselves assumed to take discrete values. Switching models were alreadybeing looked at, and the authors acknowledged that Pierre Mertz of RAND hadalready studied a power law switching model [22], but Mandelbrot’s early expo-sure to the extended central limit theorem, and the fact that he was studyingheavy tailed models in economics and neuroscience among other applications,evidently helped him to see their broader significance.The second, [15] was in the IEEE Transactions on Communication Technol-ogy, and essentially also used the model of Berger and Mandelbrot. The abstractmakes it clear that it describes: ... a model of certain random perturbations that appear to come in clus-ters, or bursts. This will be achieved by introducing the concept of “self-similar stochastic point process in continuous time.” The resulting mech- anism presents fascinating peculiarities from the mathematical viewpoint.In order to make them more palatable as well as to help in the searchfor further developments, the basic concept of “conditional stationarity”will be discussed in greater detail than would be strictly necessary fromthe viewpoint of the immediate engineering problem of errors of trans-mission.
It is clear that by 1965 Mandelbrot had come to appreciate that the appli-cation of the Fourier periodogram to the fractional renewal process would giveambiguous results, saying in [15] that:
The now classical technique of spectral analysis is inapplicable to theprocesses examined in this paper but it is sometimes unavoidable thatotherwise excellent spectral estimates be applied in this context. Anotherpublication of the author[that paper’s Ref 18] is devoted to an examina-tion of the expected outcomes of such operations. This will lead to freshconcepts that appear most promising indeed in the context of a statisticalstudy of turbulence, excess noise, and other phenomena when interestingevents are intermittent and bunched together (see also [that paper’s Ref19]).
The third key paper, the “other publication ... Ref 18”, resulted from an IEEEconference talk in 1965. It [16] is now available but in the post hoc edited form inwhich all his papers appeared in his Selecta[18,19]. “Reference 19”, meanwhile,seems originally to have been intended to be a paper in the physics literature,the fate of which is not clear to me but whose role was effectively taken overby the fourth key paper [17]. With the proviso that the Selecta version of [16]may not fully reflect the original’s content, one can nonetheless see that by mid-1965 Mandelbrot was already focusing on the implications for ergodicity of theconditional stationarity idea. He remarked that:
In other words, the existence of f θ − noises challenges the mathemati-cian to reinterpret spectral measurements otherwise than in “Wiener-Khinchin” terms. [...] operations meant to measure the Wiener-Khinchinspectrum may unvoluntarily measure something else, to be referred to asthe “conditional spectrum” of a “conditionally covariance stationary”random function. [17]Taking the two papers [16,17] together we can see that Mandelbrot expandedon this vision by discussing several fractional renewal models, including in [16] athree state, explicitly nonstationary model with waiting times whose probabilitydensity function decayed as a power law p ( t ) ∼ t − (1+ θ ) . This stochastic processwas intended as a “cartoon” to model intermittency, in which “off” periods ofno activity were interrupted by jumps to a negative (or positive) “on” activestate. His key finding, confirmed in [17] for a model with an arbitrary number ofdiscrete levels, was that the traditional Wiener-Khinchine spectral diagnosticswould return a 1 /f periodogram and thus a spectral “infrared catastrophe”when viewed with traditional methods, but building on the notion of conditional stationarity proposed in [15], that a conditional power spectrum S ( f, T ) couldbe decomposed into a stationary part in which no catastrophe was seen, and onedepending on the time series’ length T , multiplying a slowly varying function L ( f ). He found S ( f, T ) ∼ f θ − L ( f ) Q ( T ) (3)where Q ( T ) T − θ was slowly varying, and that the conditional spectral density S ′ ( f, T ) obeys S ′ ( f, T ) = ddf S ( f, T ) ∼ f θ − T θ − L ( f ) (4)Rather than representing a true singularity in power at the lowest frequencies,in the Selecta [18] he described the apparent infrared catastrophe in the powerspectral density in the fractional renewal models as a “mirage” resulting fromthe fact that the moments of the model varied in time in a step-like fashion, aproperty he called “conditional covariance stationarity”.In [17] Mandelbrot noted a clear contrast between his conditionally station-ary, non-Gaussian fractional renewal 1 /f model and his stationary Gaussian fGnmodel (the 1968 paper concerning which, with Van Ness, was then in press atSIAM Review): Section VI [... of this paper... ] showed that some f θ − L ( f ) noises havea very erratic sampling behavior. Some other f θ − noises are Gaussianand, therefore, perfectly “well-behaved;” an example is provided by the“fractional white noise” [i.e. fGn] which is the formal derivative of theprocess of Mandelbrot and Van Ness [i.e. fBm] He identified the origin of this erratic sampling behaviour in the non-ergodicityof the fractional renewal processes. Niemann et al [25] have recently given a veryprecise analysis of the behaviour of the random prefactor S ( T ) , obtaining itsMittag-Leffler distribution and checking it by simulations. Informed in part by the above historical investigations, the purpose of this sec-tion is now to distinguish conceptually between 3 phenomena which are stillfrequently elided.To recap, the phenomena are: – The Hurst effect: the observation of “anomalous” growth of range in a timeseries using a diagnostic such as Hurst and Mandelbrot’s
R/S or detrendedfluctuation analysis (DFA)(e.g. [9,3]). – /f noise: the observation of singular low frequency behaviour in the empir-ical periodogram of a time series. – Long range dependence (LRD): a property of a stationary model by con-struction . This can only be inferred to be a property of an empirical timeseries if certain additional conditions are known to be met, including theimportant one of stationarity
The reason why it is necessary to unpick the relationship between these ideasis that there are three commonly held misperceptions about them.
The first is that observation of the Hurst effect in a time series necessarily impliesstationary LRD.
This is “well known” to be erroneous, see e.g. the work of [5]who showed the Hurst effect arising from an imposed trend rather than fromstationary LRD, but is nonetheless in practice still not very widely appreciated.
The second is that observation of the Hurst effect in a time series necessarilyimplies a /f periodogram. Although less “well known”, [8], for example, haveshown an example where the Hurst effect arose in the Lorenz model which hasan exponential power spectrum rather than 1 /f . The third is the idea that observation of a /f periodogram necessarily impliesstationary LRD. As noted above, this is a more subtle issue, and although littleappreciated since the pioneering work of [15,16,17] it has now become central tothe investigation of weak ergodicity breaking in physics.
The Hurst effect was originally observed as the growth of range in a time series, atfirst the Nile. The original diagnostic for this effect was R/S. Using the notation J (not H ) for the Joseph (i.e. Hurst) exponent that Mandelbrot latterly advocated[19], the Hurst effect is seen when the rescaled range[3,9] grows with time as RS ∼ τ J (5)in the case that J = 1 /
2. During the period between Feller’s proof that aniid stationary process had J = 1 /
2, and Mandelbrot’s papers of 1965-68 onlong range dependence in fGn [9], there was a controversy about whether theHurst effect was a consequence of nonstationarity and/or a pre-asymptotic effect.This controversy has never fully subsided [9] because Occam’s Razor frequentlyfavours at least the possibility of change points in an empirically determined timeseries (e.g. [24]), and because of the (at first sight surprising) non-Markovianproperty of fGn.A key point to appreciate is that it is easier to generate the Hurst effect overa finite scaling range, as measured for example by
R/S , than a true wideband1/f spectrum. [8] for example shows how a Hurst effect can appear over a finiterange even when the power spectrum is known a priori not be 1 /f , e.g. in theLorenz attractor case where the low frequency spectrum is exponential. The term 1 /f spectrum is usually used to denote periodograms where the spec-tral density S ′ ( f ) has an inverse power law form, e.g. the definition used in[16,17] S ′ ( f ) ∼ f θ − (6) where θ runs between 0 and 2.One needs to distinguish here between bounded and unbounded processes.Brownian, and fractional Brownian, motion are unbounded, nonstationary ran-dom walks and one can view their 1 /f J spectral density as a direct conse-quence of nonstationarity, as Mandelbrot did (see pp 78-79 of [18]). In manyphysical contexts however, such as the on-off blinking quantum dot process[25]or the river Nile minima studied by Hurst[9] the signal amplitude is alwaysbounded and does not grow in time, requiring a different explanation that iseither stationary or “conditionally stationary”.Mandelbrot’s best known model for 1 /f noise remains the stationary, er-godic, fractional Gaussian noise (fGn) that he advocated so energetically in the1960s. But, evidently himself aware that this had had received a disproportion-ate amount of attention, he was at pains late in his life (e.g. Selecta Volume N[18] p.207, introducing the reprinted [16,17]) to stress that: Self-affinity and an 1/f spectrum can reveal themselves in several quitedistinct fashions ... forms of 1/f behaviour that are predominantly due tothe fact that a process does not vary in “clock time” but in an “intrinsictime” that is fractal. Those 1/f noises are called “sporadic” or “absolutelyintermittent”, and can also be said to be “dustborne” and “acting infractal time”.
He clearly distinguishes the LRD stationary Gaussian models like fGn fromfrom his “conditionally stationary” fractal time process, noting also that:
There is a sharp contrast between a highly anomalous (“non-white”)noise that proceeds in ordinary clock time and a noise whose principalanomaly is that it is restricted to fractal time.
In practise the main importance of this is to caution that, used on its own,even a very sophisticated approach to the periodogram like the GPH method [3]cannot tell the difference between a time series being stationary LRD of the fGntype and “just” a “1/f” noise, unless independent information about stationarityis also available.One route to reducing the ambiguity in future studies of 1 /f is to developnon-stationary extensions to the Wiener-Khinchine theorem. An important step[12] has been to distinguish between one which relates the spectrum and theensemble average correlation function, and a second relating the spectrum to thetime average correlation function. The importance of this distinction can be seenby considering the Fourier inversion of the power spectrum-does the inversionyield the time or the ensemble average? [E. Barkai, personal communication]. My readers will, I hope, now be able to see why I believe that the commonly usedspectral definition of LRD has caused misunderstandings. The problem has been that on its own a “1/f” behaviour is necessary but not sufficient, and stationarityis also essential for LRD in the sense so widely studied in statistics community(e.g. in [2] and [3]). One may in fact argue that the more crucial aspect of LRD isthus the “loose” one embodied in its name, rather than the formal one embodiedin the spectral definition, because a /f spectrum can only be synonymous withLRD when there is an infinitely long past . The fact that fGn exhibits LRD byconstruction because the stationarity property is assumed, and also shows 1/fnoise, and the Hurst effect has led to the widespread misconception that theconverse is true, and that observing “1/f” spectra and/or the Hurst effect must imply LRD.
Unfortunately [17] received far less contemporary attention than did Mandel-brot’s papers on heavy tails in finance in the early 1960s or the series with vanNess and Wallis in 1968-69 on stationary fractional Gaussian models for LRD,gaining only about 20 citations in its first 20 years. This has been rectified sincebut I believe the consequences have been lasting. Perhaps it was because hiswork on the AFRP was communicated primarily in the (IEEE) journals andconferences of telecommunications and computer science, that it was largely in-visible to the contemporary audience that encountered fGn and fBm in SIAMReview and Water Resources Research. In fact, so invisible was it that one hismost articulate critics, hydrologist, Vit Klemeˇs [11] used an AFRP model as aparadigm for the absence of the type of LRD seen in the stationary fGn model,clearly unaware of Mandelbrot’s work. Klemeˇs’ paper remains very worthwhilereading even today. It showed how at least two other classes of model couldexhibit the Hurst effect, the AFRP class and also integrated processes, such asAR(1) with a φ parameter close to 1. Fascinatingly, despite the fact that Man-delbrot had colleagues such as the hydrologist Wallis who published in WaterResources Research, and thus may well have seen the paper, if he did he chosenot to enlighten Klemeˇs about his earlier work. Sadly Klemeˇs and Mandelbrotseem also not to have subsequently debated nonstationary approaches on anequal footing with fGn, as with the advantage of historical distance one can seethe importance of both as non-ergodic and ergodic solutions to the 1 /f paradox.Although he revisited the 1963-67 fractional renewal papers with new com-mentaries in the volume of his Selecta [18] that dealt with multifractals and“1 /f ” noise, Mandelbrot himself neglected to mention them explicitly in hispopular historical account of the genesis of LRD in [20]. That he saw them asa representing a different strand of his work to fractional Brownian motion isclear from the way that fBm and fGn and the Gaussian paths to 1 /f were eachallocated a separate Selecta volume [19]. Despite the Selecta, the relatively lowvisibility has remained to the present day. Mandelbrot’s fractional renewal pa-pers are for example not cited or discussed even in encyclopedic books on LRDsuch as Beran et al’s [3]. The long term consequence of this in the physics and statistics literaturesmay have been to emphasise ergodic solutions to the 1 /f problem at the expenseof non-ergodic ones. This seems to me to be important, because, for example,Per Bak’s paradigm of Self-Organised Criticality, in which stationary spectra andcorrelation functions play an essential role, could not surely have been positionedas the unique solution to the 1 /f problem [30] if it had been widely recognisedhow different Mandelbrot’s two existing routes to 1/f already were. Acknowledgements.
I would like to thank Rebecca Killick for inviting me to talkat ITISE 2016, and helpful comments on the manuscript from Eli Barkai. I alsogratefully acknowledge many valuable discussions about the history of LRD andweak ergodicity breaking with Nick Moloney, Christian Franzke, Ralf Metzler,Holger Kantz, Igor Sokolov, Rainer Klages, Tim Graves, Bobby Gramacy, An-drey Cherstvy, Aljaz Godec, Sandra Chapman, Thordis Thorarinsdottir, Kristof-fer Rypdal,Martin Rypdal, Bogdan Hnat, Daniela Froemberg, and Igor Goychukamong many others. I acknowledge travel support from KLIMAFORSK projectnumber 229754 and the London Mathematical Laboratory, a senior visiting fel-lowship from the Max Planck Society in Dresden, and Office of Naval ResearchNICOP grant NICOP - N62909-15-1-N143 at Warwick and Potsdam.
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