Power Law Distributions, 1/f Noise, Long-Memory Time Series

Why do physicists care about power laws so much?

I'm probably not the best person to speak on behalf of our tribal obsessions (there was a long debate among the faculty at my thesis defense as to whether "this stuff is really physics"), but I'll do my best. There are two parts to this: power-law decay of correlations, and power-law size distributions. The link is tenuous, at best, but they tend to get run together in our heads, so I'll treat them both here.

The reason we care about power law correlations is that we're conditioned to think they're a sign of something interesting and complicated happening. The first step is to convince ourselves that in boring situations, we don't see power laws. This is fairly easy: there are pretty good and rather generic arguments which say that systems in thermodynamic equilibrium, i.e. boring ones, should have correlations which decay exponentially over space and time; the reciprocals of the decay rates are the correlation length and the correlation time, and say how big a typical fluctuation should be. This is roughly first-semester graduate statistical mechanics. (You can find those arguments in, say, volume one of Landau and Lifshitz's Statistical Physics.)

Second semester graduate stat. mech. is where those arguments break down --- either for systems which are far from equilibrium (e.g., turbulent flows), or in equilibrium but very close to a critical point (e.g., the transition from a solid to liquid phase, or from a non-magnetic phase to a magnetized one). Phase transitions have fluctuations which decay like power laws, and many non-equilibrium systems do too. (Again, for phase transitions, Landau and Lifshitz has a good discussion.) If you're a statistical physicist, phase transitions and non-equilibrium processes define the terms "complex" and "interesting" --- especially phase transitions, since we've spent the last forty years or so developing a very successful theory of critical phenomena. Accordingly, whenever we see power law correlations, we assume there must be something complex and interesting going on to produce them. (If this sounds like the fallacy of affirming the consequent, that's because it is.) By a kind of transitivity, this makes power laws interesting in themselves.

Since, as physicists, we're generally more comfortable working in the frequency domain than the time domain, we often transform the autocorrelation function into the Fourier spectrum. A power-law decay for the correlations as a function of time translates into a power-law decay of the spectrum as a function of frequency, so this is also called "1/f noise".

Similarly for power-law distributions. A simple use of the Einstein fluctuation formula says that thermodynamic variables will have Gaussian distributions with the equilibrium value as their mean. (The usual version of this argument is not very precise.) We're also used to seeing exponential distributions, as the probabilities of microscopic states. Other distributions weird us out. Power-law distributions weird us out even more, because they seem to say there's no typical scale or size for the variable, whereas the exponential and the Gaussian cases both have natural scale parameters. There is a connection here with fractals, which also lack typical scales, but I don't feel up to going into that, and certainly a lot of the power laws physicists get excited about have no obvious connection to any kind of (approximate) fractal geometry. And there are lots of power law distributions in all kinds of data, especially social data --- that's why they're also called Pareto distributions, after the sociologist.

Physicists have devoted quite a bit of time over the last two decades to seizing on what look like power-laws in various non-physical sets of data, and trying to explain them in terms we're familiar with, especially phase transitions. (Thus "self-organized criticality".) So badly are we infatuated that there is now a huge, rapidly growing literature devoted to "Tsallis statistics" or "non-extensive thermodynamics", which is a recipe for modifying normal statistical mechanics so that it produces power law distributions; and this, so far as I can see, is its only good feature. (I will not attempt, here, to support that sweeping negative verdict on the work of many people who have more credentials and experience than I do.) This has not been one of our more successful undertakings, though the basic motivation --- "let's see what we can do!" --- is one I'm certainly in sympathy with.

There have been two problems with the efforts to explain all power laws using the things statistical physicists know. One is that (to mangle Kipling) there turn out to be nine and sixty ways of constructing power laws, and every single one of them is right, in that it does indeed produce a power law. Power laws turn out to result from a kind of central limit theorem for multiplicative growth processes, an observation which apparently dates back to Herbert Simon, and which has been rediscovered by a number of physicists (for instance, Sornette). Reed and Hughes have established an even more deflating explanation (see below). Now, just because these simple mechanisms exist, doesn't mean they explain any particular case, but it does mean that you can't legitimately argue "My favorite mechanism produces a power law; there is a power law here; it is very unlikely there would be a power law if my mechanism were not at work; therefore, it is reasonable to believe my mechanism is at work here." (Deborah Mayo would say that finding a power law does not constitute a severe test of your hypothesis.) You need to do "differential diagnosis", by identifying other, non-power-law consequences of your mechanism, which other possible explanations don't share. This, we hardly ever do.

Similarly for 1/f noise. Many different kinds of stochastic process, with no connection to critical phenomena, have power-law correlations. Econometricians and time-series analysts have studied them for quite a while, under the general heading of "long-memory" processes. You can get them from things as simple as a superposition of Gaussian autoregressive processes. (We have begun to awaken to this fact, under the heading of "fractional Brownian motion".)

The other problem with our efforts has been that a lot of the power-laws we've been trying to explain are not, in fact, power-laws. I should perhaps explain that statistical physicists are called that, not because we know a lot of statistics, but because we study the large-scaled, aggregated effects of the interactions of large numbers of particles, including, specifically, the effects which show up as fluctuations and noise. In doing this we learn, basically, nothing about drawing inferences from empirical data, beyond what we may remember about curve fitting and propagation of errors from our undergraduate lab courses. Some of us, naturally, do know a lot of statistics, and even teach it --- I might mention Josef Honerkamp's superb Stochastic Dynamical Systems. (Of course, that book is out of print and hardly ever cited...)

If I had, oh, let's say fifty dollars for every time I've seen a slide (or a preprint) where one of us physicists makes a log-log plot of their data, and then reports as the exponent of a new power law the slope they got from doing a least-squares linear fit, I'd at least not grumble. If my colleagues had gone to statistics textbooks and looked up how to estimate the parameters of a Pareto distribution, I'd be a happier man. If any of them had actually tested the hypothesis that they had a power law against alternatives like stretched exponentials, or especially log-normals, I'd think the millennium was at hand. (If you want to know how to do these things, please read this paper, whose merits are entirely due to my co-authors.) The situation for 1/f noise is not so dire, but there have been and still are plenty of abuses, starting with the fact that simply taking the fast Fourier transform of the autocovariance function does not give you a reliable estimate of the power spectrum, particularly in the tails. (On that point, see, for instance, Honerkamp.)

Michael Mitzenmacher, "A Brief History of Generative Models for Power Law and Lognormal Distributions", Internet Mathematics 1 (2003): 226--251 [PDF]
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Manfred Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise

Robert J. Adler, Raise E. Feldman and Murad S. Taqqu (eds.), A Practical Guide to Heavy Tails [Presumes that you already know something about statistics and stochastic processes, so not suitable for beginners.]
Barry C. Arnold, Pareto Distributions [Fine guide to the statistical literature, as it was in 1983; still valuable, though many things which were nasty computations then are easy now.]
Arijit Chakrabarty, "Effect of truncation on large deviations for heavy-tailed random vectors", arxiv:1107.2476
Aaron Clauset, Maxwell Young, and Kristian Skrede Gleditsch, "Scale Invariance in the Severity of Terrorism", physics/0606007 [Surprising, but well-supported]
F. Clementi, T. Di Matteo, M. Gallegati, "The Power-law Tail Exponent of Income Distributions", physics/0603061 = Physica A 370 (2006): 49--53 [An interesting way to improve the accuracy of Hill-type (tail-conditional maximum likelihood) estimates of the scaling parameter. Written with few concessions to those who are neither statisticians nor econometricians. Not directly suitable for determining the range of the scaling region. Income distribution is used only as an example.]
Andrew M. Edwards, Richard A. Phillips, Nicholas W. Watkins, Mervyn P. Freeman, Eugene J. Murphy, Vsevolod Afanasyev, Sergey V. Buldyrev, M. G. E. da Luz, E. P. Raposo, H. Eugene Stanley and Gandhimohan M. Viswanathan, "Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer", Nature 449 (2007): 1044--1048
Paul Embrechts and Makoto Maejima, Selfsimilar Processes
Michel L. Goldstein, Steven A. Morris and Gary G. Yen, "Fitting to the Power-Law Distribution", cond-mat/0402322 [Pedestrian, but accurate, exposition in terms physicists and engineers are likely to understand. Insufficiently sourced to the statistical literature; e.g., their calculation of the maximum likelihood estimator was first published in 1952.]
Josef Honerkamp, Stochastic Dynamical Systems: Concepts, Numerical Methods, Data Analysis
Yuji Ijiri and Herbert Simon, Skew Distributions and the Sizes of Business Firms [Collects Simon and co.'s pioneering papers on power laws and related distributions --- including "On a Class of Skew Distribution Functions", below --- as well as considering the limitations, alternatives, modifications to match data, statistical issues, the connection to Bose-Einstein statistics, the importance of going beyond just staring at distributional plots if you want to learn about mechanisms, etc., etc. This was all published in 1977...]
A. James and M. J. Plank, "On fitting power laws to ecological data", arxiv:0712.0613
Raya Khanin and Ernst Wit, "How Scale-Free Are Biological Networks?", Journal of Computational Biology 13 (2006): 810--818 [Ans.: not very scale-free at all.]
Joel Keizer, Statistical Thermodynamics of Nonequilibrium Processes [Has a good discussion of critical fluctuations in chapter 8. Review: Molecular Fluctuations for Fun and Profit]
Paul Krugman, The Self-Organizing Economy [Has a nice discussion of power-law size distributions in economics. Review]
Michael LaBarbera, "Analyzing Body Size as a Factor in Ecology and Evolution", Annual Review of Ecology and Systematics 20 (1989): 91--117 [Statistical problems in many studies of power-law scaling in biology, their effects on the conclusions of those studies (ranging from "wrong, but correctable" to "meaningless"), and how to do it right. JSTOR]
J. Laherrère and D. Sornette, "Stretched exponential distributions in nature and economy: 'fat tails' with characteristic scales", The European Physical Journal B 2 (1998): 525--539
L. D. Landau and E. M. Lifshitz, Statistical Physics [For the theory of fluctuations in statistical mechanics, and for critical phenomena in equilibrium]
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Elliott W. Montroll and Michael F. Shlesinger, "On 1/f noise and other distributions with long tails", Proceedings of the National Academy of Sciences (USA) 79 (1982): 3380--3383
V. F. Pisarenko and D. Sornette, "New statistic for financial return distributions: power-law or exponential?", physics/0403075 [Actually, two new statistics: one converges to a constant if the distribution you're sampling from is an exponential, independent of the exponent, and the other converges to a constant if the distribution is a power law, independent of the power. They even have some indications of the sampling distributions, so you can at least gauge the statistical signifcance, i.e., the probability of deviations from the ideal value, even though the distribution really is of the appropriate type. I don't recall anything about the power of these statistics, however (i.e., the probability that a power law will look like an exponential, or vice-versa).]
William J. Reed and Barry D. Hughes, "From Gene Families and Genera to Incomes and Internet File Sizes: Why Power Laws are so Common in Nature", Physical Review E 66 (2002): 067103 [This is, as I said, perhaps the most deflating possible explanation for power law size distributions. Imagine you have some set of piles, each of which grows, multiplicatively, at a constant rate. New piles are started at random times, with a constant probability per unit time. (This is a good model of my office.) Then, at any time, the age of the piles is exponentially distributed, and their size is an exponential function of their age; the two exponentials cancel and give you a power-law size distribution. The basic combination of exponential growth and random observation times turns out to work even if it's only the mean size of piles which grows exponentially.]
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Didier Sornette
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Damian H. Zanette, "Zipf's law and the creation of musical context", cs.CL/0406015 [This sounds bizarre, and I'd not have bothered to even note it if I didn't know Zanette's work in other areas, which shows him to be a good and careful scientist. And this is actually an interesting and meaningful little paper, which has something non-trivial to say about music. It's worth noting, perhaps, that the distribution he actually ends up fitting isn't a pure power law, but a modification inspired by Simon's paper. Thanks to John Burke for prodding me to actually read it.]

R. Alexander Bentley, Paul Ormerod, Michael Batty, "An evolutionary model of long tailed distributions in the social sciences", arxiv:0903.2533 [This is a minor modification of the classical Yule/Simon mechanism for random growth, with the main advantage being that (with the right parameter tweaking) it allows for more turn-over of which values are most common. Unsurprisingly, this is done by adding extra parameters, and so the family of distributions is more flexible. But they use bad statistical procedures, and the finding that the estimated power law exponent grows as the amount of data held in the tail shrinks is simply explained: the tails aren't power laws.]

Mason Porter's Power Law Shop

Aaron Clauset, CRS and M. E. J. Newman, "Power-law distributions in empirical data", SIAM Review 51 (2009): 661--703 = arxiv:0706.1062 [with commentary by Aaron and myself]

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Thierry Bochud and Damien Challet, "Optimal approximations of power-laws with exponentials", physics/0605149 ["We propose an explicit recursive method to approximate a power-law with a finite sum of weighted exponentials. Applications to moving averages with long memory are discussed in relationship with stochastic volatility models." The last part sounds like a rediscovery of Granger.]
Laurent E. Calvet and Adlai J. Fisher, Multifractal Volatility: Theory, Forecasting, and Pricing [Thanks to Prof. Calvet for bringing this to my attention]
Anna Carbone and Giuliano Castelli, "Scaling Properties of Long-Range Correlated Noisy Signals," cond-mat/0303465
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Arijit Chakrabarty, "Central Limit Theorem and Large Deviations for truncated heavy-tailed random vectors", arxiv:1003.2159
Arijit Chakrabarty, Gennady Samorodnitsky, "Understanding heavy tails in a bounded world or, is a truncated heavy tail heavy or not?", arxiv:1001.3218
Anirban Chakraborti, Marco Patriarca, "A Variational Principle for Pareto's power law", cond-mat/0605325
Ali Chaouche and Jean-Noel Bacro, "Statistical Inference for the Generalized Pareto Distribution: Maximum Likelihood Revisited", Communications in Statistics: Theory and Methods 35 (2006): 785--802
F. Clementi, M. Gallegati, "Pareto's Law of Income Distribution: Evidence for Germany, the United Kingdom, and the United States", physics/0504217
Cline, heavy-tailed noise, 1983 (?)
B. Conrad and M. Mitzenmacher, "Power Laws for Monkeys Typing Randomly: The Case of Unequal Probabilities", IEEE Transactions on Information Theory 50 (2004): 1403--1414
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Anirban Dasgupta, John Hopcroft, Jon Kleinberg and Mark Sandler, "On Learning Mixtures of Heavy-Tailed Distributions"
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Paul Doukhan, George Oppenheim and Murad S. Taqqu (eds.), Theory and Applications of Long-Range Dependence
Rick Durrett and Jason Schweinsberg, "Power laws for family sizes in a duplication model", math.PR/0406216 = Annals of Probability 33 (2005): 2094--2126
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Akihiro Fujihara, Toshiya Ohtsuki and Hiroshi Yamamoto
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Alexander Gnedin, Ben Hansen, Jim Pitman, "Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws", math.PR/0701718
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Takashi Ichinomiya, "Power-law distribution in Japanese racetrack betting", physics/0602165
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Y. Malevergne, V.F. Pisarenko, D. Sornette, "Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?", physics/0305089
Alon Manor and Nadav M. Shnerb, "Multiplicative Noise and Second Order Phase Transitions", Physical Review Letters 103 (2009): 030601
Natalia Markovich, Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice
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Joseph L. McCauley, Gemunu H. Gunaratne, Kevin E. Bassler, "Hurst Exponents, Markov Processes, and Fractional Brownian motion", cond-mat/0609671
Richard Metzler, "Comment on 'Power-law correlations in the southern-oscillation-index fluctuations characterizing El Nino'", Physical Review E 67 (2003): 018201
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Holger Rootzen, M. Ross Leadbetter and Laurens de Haan, "On the distribution of tail array sums for strongly mixing stationary sequences", Annals of Applied Probability 8 (1998): 868--885
Gennady Samorodnitsky and Murad S. Taqqu, Stable Non-Gaussian Random Processes
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Attilio L. Stella, Fulvio Baldovin, "Anomalous scaling due to correlations: Limit theorems and self-similar processes", arxiv:0909.0906
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Rafal Weron
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Notebooks

Power Law Distributions, 1/f Noise, Long-Memory Time Series