Stretched exponential distributions in nature and economy: “fat tails” with characteristic scales

Abstract

To account quantitatively for many reported « natural » fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok temperature variations, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also briefly discuss its potential for the distribution of earthquake sizes and fault displacements and earth temperature variations over the last 400 000 years. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions. 1-IntroductionFrequency or probability distribution functions (pdf) that decay as a power law of their argument P(x) dx = P 0 x -(1+µ) dx (1) have acquired a special status in the last decade. They are sometimes called ``fractal'' (even if this term is more appropriate for the description of self-similar geometrical objects rather than statistical distributions). A power law distribution characterizes the absence of a characteristic size : independently of the value of x, the number of realizations larger dans λx is λ -µ times the number of realizations larger than x. In contrast, an exponential for instance or any other functional dependence does not enjoy this self-similarity, as the existence of a characteristic scale destroys this continuous scale invariance property [1]. In words, a power law pdf is such that there is the same proportion of smaller and larger events, whatever the size one is looking at within the power law range.The asymptotic existence of power laws is a well-established fact in statistical physics and critical phenomena with exact solutions available for the 2D Ising model, for selfavoiding walks, for lattice animals, etc.[2], with an abondance of numerical evidence for instance for the distribution of percolation clusters at criticality [3] and for many other models in statistical physics. There is in addition the observation from numerical simulations that simple ``sandpile'' models of spatio-temporal dynamics with strong non-linear behavior [4] give power law distributions of avalanche sizes. Furthermore, precise experiments on critical phenomena confirm the asymptotic existence of power laws, for instance on superfluid helium at the lambda point and on binary mixtures [5]. These are th...

Workload Modeling for Computer Systems Performance Evaluation

“…Finally, the Weibull distribution has characteristics that are similar to those of the lognormal. Laherrére and Sornette present evidence that this distribution provides excellent fits for many phenomena in nature and society [422]. However, it has so far not been used much in the context of workload modeling, perhaps because fitting the distribution's parameters is more complicated than for other candidate distributions.…”

“…Similar models can also lead to a Weibull distribution [269,422]. Moreover, the differences between the models may be small in practical situations.…”

Tagging Scientific Publications Using Wikipedia and Natural Language Processing Tools

“…It turns out that the observed rank-frequency dependencies are much better approximated by one of the alternatives to the power-law (2), namely the stretched exponential distribution. This type of distribution is used to describe large variety of phenomena from physics to finance [17]. It was observed, e.g., for rank distributions of radio/light emission intensities from galaxies, French and US agglomeration sizes, daily Forex US-Mark price variation, etc.…”

Power-law distributions for the citation index of scientific publications and scientists

“…ponential distribution as done by Laherrere and Sornette [31]. However, the stretched exponential distribution is not suitable for the citation index of Brazilian physicists.…”

“…Laherrere and Sornette [31] presented numerical evidence, on the basis of data for the 1120 most cited physicists from 1981 to June 1997, that the citation distribution of individual authors is associated with a stretched exponential form, N(x) ∼ exp − (x/x 0 ) β , with β 0.3. Using the technique of the Zipf plots, Redner [32] has recently shown that the distribution of citations of the most cited scientific papers is described by a power law, N(x) ∼ x −α , with α 3.0.…”