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

*30 May 2018 10:43*

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.)

See also: Chaos and Dynamical Systems; Complex Networks; Self-Organized Criticality; Time Series; Tsallis Statistics

- Recommended, bigger picture:
- Michael Mitzenmacher, "A Brief History of Generative Models for
Power Law and Lognormal Distributions", Internet Mathematics
**1**(2003): 226--251 [PDF] - M. E. J. Newman, "Power laws, Pareto distributions and Zipf's law", cond-mat/0412004 [If you read one other thing on power laws, read this]
- Manfred Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise

- Recommended, more technical or more specialized:
- 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", Physica
A
**370**(2006): 49--53, physics/0603061 [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
- Robert P. Freckleton and William J. Sutherland [thanks to Nick Watkins for pointing these out]
- "Do in-hospital waiting lists show self-regulation?", Journal of the Royal Society of Medicine
**95**(2002): 164 - "Hospital waiting-lists (Communication arising): Do power laws imply self-regulation", Nature
**413**(2001): 382

- "Do in-hospital waiting lists show self-regulation?", Journal of the Royal Society of Medicine
- Alexander Gnedin, Ben Hansen, Jim Pitman, "Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws",
Probability Surveys
**4**(2007): 146--171, arxiv:math.PR/0701718 - 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.]
- Timothy Graves, Robert B. Gramacy, Christian Franzke, Nicholas Watkins, "A brief history of long memory", arxiv:1406.6018
- 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
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**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
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**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]
- Adrián López García de Lomana, Qasim K. Beg, G. de Fabritiis and Jordi Villà-Freixa, "Statistical Analysis of Global Connectivity and Activity Distributions in Cellular Networks", arxiv:1004.3138
- R. Dean Malmgren, Daniel B. Stouffer, Adilson E. Motter, Luis A.N. Amaral, "A Poissonian explanation for heavy-tails in e-mail communication",
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**105**(2008): 18153--18158, arxiv:0901.0585 - Elliott W. Montroll and Michael F. Shlesinger, "On 1/f noise and
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- 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.] - M. V. Simkin and V. P. Roychowdhury, "Re-inventing Willis", physics/0601192 [The comical, yet pathetic, history of the innumerable re-inventions of basic mechanisms which plague this area]
- Herbert Simon, "On a Class of Skew Distribution Functions",
Biometrika
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- "Multiplicative Processes and Power Laws" cond-mat/9708231
= Physical Review E
**57**(1998): 4811--4813 - "Mechanism for Powerlaws without Self-Organization" cond-mat/0110426

- "Multiplicative Processes and Power Laws" cond-mat/9708231
= Physical Review E
- Stilian A. Stoev, George Michailidis, and Murad S. Taqqu, "Estimating heavy-tail exponents through max self-similarity", math.ST/0609163
- Bruce J. West and Bill Deering, The Lure of Modern Science:
Fractal Thinking [Despite the
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- Not altogether recommended (without being actively dis-recommended either):
- 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.]

- Recommended, of a not entirely serious character:
- Mason Porter's Power Law Shop

- Modesty forbids me to recommend:
- 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]

- Pride compels me to recommend:
- Georg M. Goerg, "Lambert W random variables: A new family of generalized skewed distributions with applications to risk estimation",
Annals of Applied
Statistics
**5**(2011): 2197--2230, arxiv:0912.4554 [By my student, but entirely independent work]

- To read:
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- J. A. D. Aston, "Modeling macroeconomic time series via heavy tailed distributions", math.ST/0702844
- Stefan Aulbach and Michael Falk, "Testing for a generalized Pareto
process", Electronic
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**6**(2012): 1779--1802 - Katarzyna Bartkiewicz, Adam Jakubowski, Thomas Mikosch, Olivier Wintenberger, "Infinite variance stable limits for sums of dependent random variables", arxiv:0906.2717
- Michael Batty, "Rank Clocks", Nature
**444**(2006): 592--596 - Marco Bee, Massimo Riccaboni and Stefano Schiavo, "Pareto versus lognormal: A maximum entropy test", Physical
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**84**(2011); 026104 - Jan Beran, Bikramjit Das, Dieter Schell, "On robust tail index estimation for linear long-memory processes", Journal of Time Series Analysis
**33**(2012): 406--423 - Patrice Bertail, Stéphan Clémençon, and
Jessica Tressou, "Regenerative block-bootstrap confidence intervals for tail
and extremal indexes", Electronic Journal of Statistics
**7**(2013): 1224--1248 - P. Besbeas and B. J. T. Morgan, "Improved estimation of the stable
laws", Statistics
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**18**(2008): 219--231 - Eric Beutner, Henryk Zähle, "Continuous mapping approach to the asymptotics of U- and V-statistics", arxiv:1203.1112
- Danny Bickson, Carlos Guestrin, "Linear Characteristic Graphical Models: Representation, Inference and Applications", arxiv:1008.5325 [Graphical models with heavy-tailed latent variables]
- 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.]
- Christian Brownlees, Emilien Joly, Gabor Lugosi, "Empirical risk minimization for heavy-tailed losses", arxiv:1406.2462
- 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
- Alexandra Carpentier, Arlene K.H. Kim, "Adaptive and minimax optimal estimation of the tail coefficient", arxiv:1309.2585
- C. Cattuto, V. Loreto and V. D. P. Servedio, "A Yule-Simon process with memory", cond-mat/0608672 [Memo to self: compare this to the auto-correlated Yule-Simon process in Ijiri and Simon's book.]
- 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 (?)
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Randomly: The Case of Unequal Probabilities", IEEE Transactions on
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**50**(2004): 1403--1414 - J. Danielsson, L. de Haan, L. Peng and C. G. de Vries, "Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation",
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- Anirban Dasgupta, John Hopcroft, Jon Kleinberg and Mark Sandler, "On Learning Mixtures of Heavy-Tailed Distributions"
- Nima Dehghani, Nicholas G. Hatsopoulos, Zach D. Haga, Rebecca A. Parker, Bradley Greger, Eric Halgren, Sydney S. Cash, Alain Destexhe, "Avalanche analysis from multi-electrode ensemble recordings in cat, monkey and human cerebral cortex during wakefulness and sleep", arxiv:1203.0738 [Ummm, we explain why you can't use $R^2$ that way in the paper you cite...]
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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
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**33**(2005): 2094--2126 - R. Fox and M. S. Taqqu
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**13**(1985) 428--446 - "Central Limit Theorems for Quadratic Forms in Random
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- "Noncentral Limit Thorems for Quadratic Forms in Random
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**70**(2004): 031106 = cond-mat/0506785 - "Similarity and Probability Distribution Functions in Many-body Stochastic Processes with Multiplicative Interactions", cond-mat/0508615

- "Power-law tails in nonstationary stochastic processes with
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- Xavier Gabaix, "Power Laws in Economics and Finance" [PDF preprint]
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- "The quasi-likelihood approach to statistical inference on
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**47**(2010): 1--40 - Takashi Ichinomiya, "Power-law distribution in Japanese racetrack betting", physics/0602165
- Sanja Janicevic, Lasse Laurson, Knut Jorgen Maloy, Stephane Santucci, and Mikko J. Alava, "Interevent Correlations from Avalanches Hiding Below the Detection Threshold", Physical Review Letters
**117**(2016): 230601 - Milton Jara, Tomasz Komorowski and Stefano Olla, "Limit theorems for additive functionals of a Markov chain", arxiv:0809.0177 [Convergence to alpha-stable distributions]
- Predrag R. Jelenkovic, Jian Tan, "Modulated Branching Processes, Origins of Power Laws and Queueing Duality", 0709.4297
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**83**(2011): 031123, arxiv:1103.3038 - Taisei Kaizoji, "Power laws and market crashes", physics/0603138
- Imen Kammoun, Vernoique Billat and Jean-Marc Bardet, "A new stochastic process to model Heart Rate series during exhaustive run and an estimator of its fractality parameter", arxiv:0803.3675 [Includes statistical criticism of the common, but deeply unsatisfying, "detrended fluctuation analysis" method of estimating the Hurst exponent.]
- B. Kaulakys and J. Ruseckas, "Stochastic nonlinear
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**24**(2011): 895--927, arxiv:1103.4306 - Francois M. Longin, "The Asymptotic Distribution of Extreme Stock
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**69**(1996): 383--408 [JSTOR] - Fotis Loukissas, "Precise Large Deviations for Long-Tailed Distributions", Journal of Theoretical Probability
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"Characterizing wildfire regimes in the United States", Proceedings of the
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**102**(2005): 4694--4699 - 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
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**103**(2009): 030601 - Natalia Markovich, Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice
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