art%3a10.1007%2fs100510050276

Eur. Phys. J. B 2, 525{539 (1998) THE EUROPEANPHYSICAL JOURNAL Bc

EDP SciencesSpringer-Verlag 1998

Stretched exponential distributions in nature and economy:\fat tails" with characteristic scales

J. Laherrere1 and D. Sornette2;3;a

1 107 rue Louis Bleriot, 92100 Boulogne-Billancourt, France2 Laboratoire de Physique de la Matiere Condenseeb, Universite de Nice-Sophia Antipolis, BP 71, 06108 Nice Cedex 2, France3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences, UCLA, Box 951567,

Los Angeles, CA 90095-1567, USA

Received: 20 January 1998 / Received in nal form: 27 January 1998 / Accepted: 6 February 1998

Abstract. To account quantitatively for many reported \natural" fat tail distributions in Nature andEconomy, we propose the stretched exponential family as a complement to the often used power lawdistributions. It has many advantages, among which to be economical with only two adjustable parameterswith clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in termsof multiplicative processes. We show that stretched exponentials describe very well the distributions ofradio and light emissions from galaxies, of US GOM OCS oileld reserve sizes, of World, US and Frenchagglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations,of Vostok (near the south pole) temperature variations over the last 400 000 years, of the Raup-Sepkoskiskill curve and of citations of the most cited physicists in the world. We also discuss its potential forthe distribution of earthquake sizes and fault displacements. We suggest physical interpretations of theparameters and provide a short toolkit of the statistical properties of the stretched exponentials. We alsoprovide a comparison with other distributions, such as the shifted linear fractal, the log-normal and therecently introduced parabolic fractal distributions.

PACS. 02.50.+r Probability theory, stochastic processes and statistics { 89.90.+n Other areas of generalinterest to physicists { 01.75.+m Science and society

1 Introduction

Frequency or probability distribution functions (pdf) thatdecay as a power law of their argument

P (x)dx = P0x(1+)dx (1)

have acquired a special status in the last decade. Theyare sometimes called \fractal" (even if this term is moreappropriate for the description of self-similar geometri-cal objects rather than statistical distributions). A powerlaw distribution characterizes the absence of a character-istic size: independently of the value of x, the number ofrealizations larger than x is times the number ofrealizations larger than x. In contrast, an exponential forinstance or any other functional dependence does not en-joy this self-similarity, as the existence of a characteristicscale destroys this continuous scale invariance property [1].In words, a power law pdf is such that there is the sameproportion of smaller and larger events, whatever the sizeone is looking at within the power law range.

a e-mail: [email protected] CNRS UMR 6622

The asymptotic existence of power laws is a well-established fact in statistical physics and critical phenom-ena with exact solutions available for the 2D Ising model,for self-avoiding walks, for lattice animals, etc. [2], withan abundance of numerical evidence for instance for thedistribution of percolation clusters at criticality [3] and formany other models in statistical physics. There is in addi-tion the observation from numerical simulations that sim-ple \sandpile" models of spatio-temporal dynamics withstrong non-linear behavior [4] give power law distribu-tions of avalanche sizes. Furthermore, precise experimentson critical phenomena conrm the asymptotic existenceof power laws, for instance on superfluid helium at thelambda point and on binary mixtures [5]. These are the\hard" facts.

On the other hand, the relevance of power laws inNature is less clear-cut even if it has repeatedly beenclaimed to describe many natural phenomena [1,6,7]. Inaddition, power laws have also been proposed to apply toa vast set of social and economic statistics [8{14]. Powerlaws are considered as one of the most striking signaturesof complex self-organizing systems [15,16]. Empirically, apower law pdf (1) is represented by a linear dependencein a double logarithmic axis plot (a log-log plot for short)

526 The European Physical Journal B

of the frequency or cumulative number as a function ofsize. However, logarithms are notorious for contractingdata and the qualication of a power law is not as straight-forward as often believed. Claims have thus been made onthe power law dependence of many data that have or willprobably not survive a closer scrutiny. See for instance[17,18] which point out that disorder and irregular bound-aries may lead to apparent scaling over two decades. Seealso [19] which points out that the scaling range of exper-imentally declared fractality in laboratory experiments isextremely limited as most log-log plots show a straightportion typically over only 1.3 order of magnitude.

In general, in all power laws of critical phenomena, weobserve them only asymptotically, i.e. with innite purity,innite temperature control, innitely large computer sim-ulations, waiting innitely long for equilibration, and mak-ing gravity eects innitely small by going into space. Thiswas called Asymptopia by R.A. Ferrell about thirty yearsago. In any real experiment or simulation, there are de-viations from Asymptopia and thus deviation from powerlaws. Our paper is not about these unavoidable but redu-cable observation errors, but claims that even innitely ac-curate experimental preparations and observations in ourexamples treated below give important deviations frompower laws.

Thus, in any nite critical system, it is well-known thatthe power law description must give way to another regimedominated by nite size eects [20] and the pdfs in gen-eral cross over to an exponential decay, which leads to acurvature in the log-log plots. Log-log plots of data fromnatural phenomena in Nature and Economy often exhibita limited linear regime followed by a signicant curvature.The outstanding question is whether these observed devi-ations from a power law description result simply from anite-size eect or does it invade the main body of thedistribution, thus calling for a more fundamental under-standing and also a completely dierent quantication ofthe pdfs. The rst hypothesis has often been suggested forinstance for the Gutenberg-Richter distribution of earth-quake sizes: if the power law distribution was extrapolatedto innite sizes, it would predict an innite mean rate ofenergy release, which is clearly ruled out in a nite earth.Similarly, the extrapolation of the distribution of the dis-persed habitat of oileld reserves gives an innite quantityof oil. This is clearly ruled out in a nite earth and a cross-over to another regime is called for. The fact that most ofthe natural distributions display a log-log curved plot [21],avoiding the divergence and leading to thinner tails thanpredicted by a power law, has mostly been interpreted interms of nite-size eects.

Here, we explore and test the hypothesis that the cur-vature observed in log-log plots of distributions of sev-eral data sets taken from natural and economic phenom-ena might result from a deeper departure from the powerlaw paradigm and might call for an alternative descriptionover the whole range of the distribution. The choice of agiven mathematical class of distributions corresponds toa \model" in the following sense. In its broadest sense,a model is a mathematical representation of a condition,

process, concept etc., in which the variables are dened torepresent inputs, outputs, and intrinsic states and inequal-ities are used to describe interactions of the variables andconstraints on the problem. In theoretical physics, modelstake a narrower meaning, such as in the Ising, Potts, ...,percolation models. For the description of natural and ofeconomic phenomena, the term model is usually used inthe broadest sense that we take here.

The model that we test is provided by the recentdemonstration that the tail of pdfs of products of a -nite number of random variables is generically a stretchedexponential [22], in which the exponent c is the inverse ofthe number of generations (or products) in a multiplica-tive process. We thus propose an alternative model forpdfs for natural and economic distributions in terms ofstretched exponentials:

P (x)dx = c(xc1=xc0) exp[(x=x0)c]dx; (2)

such that the cumulative distribution is

Pc(x) = exp[(x=x0)c]: (3)

Stretched exponentials are characterized by an exponent csmaller than one. The borderline c = 1 corresponds to theusual exponential distribution. For c smaller than one, thedistribution (3) presents a clear curvature in a log-log plotwhile exhibiting a relatively large apparent linear behav-ior, all the more so, the smaller c is. It can thus be usedto account both for a limited scaling regime and a cross-over to non-scaling. When using the stretched exponentialpdf, the rational is that the deviations from a power lawdescription is fundamental and not only a nite-size cor-rection.

We nd that the stretched exponential (2, 3) providesan economical description as it depends on only two mean-ingful adjustable parameters with clear physical interpre-tation (the third one being an unimportant normalizatingfactor). It accounts very well for the distribution of radioand light emissions from galaxies (Fig. 2), of US GOMOCS oileld reserve sizes (Figs. 3, 4), of World, US andFrench agglomeration sizes (Figs. 5-8), of the United Na-tion 1996 country sizes (Fig. 9), of daily Forex US-Markand Franc-Mark price variations (Figs. 10, 11), and of theRaup-Sepkoskis kill curve (Fig. 12), Even the distribu-tion of biological extinction events [23] is much better ac-counted for by a stretched exponential than by a powerlaw (linear fractal). We also show that the distribution ofthe largest 1300 earthquakes in the world from 1977 to1992 (Fig. 13) and the distribution of fault displacements(Fig. 14) can be well-described by a stretched exponential.We also study the temperature variations over the last420 000 years obtained for ice core isotope measurements(Fig. 15). Finally, we examine the distribution of citationsof the most cited physicists in the world and again nd avery t by a stretched exponential (Fig. 16).

We do not claim that all power law distributions haveto be replaced but that observable curvatures in log-logplots that are often present may signal that another sta-tistical representation, such as a stretched exponential, isbetter suited. This in turn may inspire the identication

J. Laherrere and D. Sornette: Stretched exponential distribution 527

10

100

1000

10000

1 10 100 1000 10000rank

lognorm s=0.75, m=70

shifted linear a=0.42, A=12

PF a=0.01, b=0.1

str exp a=40, b=71, c=0.6,xo=29, x95=184

log SEPF

SL

Fig. 1. Comparison between parabolic fractal, shifted linear fractal, lognormal and stretched exponential tted up to rank 500.

of a relevant physical mechanism. Among the possible can-didates, stretched exponentials are particularly appealingbecause, not only do they provide a simple and economi-cal description but, there is a generic mechanism in termsof multiplicative processes. Multiplicative processes oftenconstitute zeroth-order descriptions of a large variety ofphysical systems, exhibiting anomalous pdfs and relax-ation behaviors.

Stretched exponential laws are familiar in the contextof anomalous relaxations in glasses [24] and can be de-rived from a multiplicative process [22]. The Ising fer-romagnet in two dimensions was also predicted to relaxby a stretched-exponential law [25]. This was conrmednumerically [26] and by improved arguments [27]. It hasbeen claimed that the eect also exists in three dimen-sions [28] but is not conrmed numerically (D. Stauer,private communication). The theory is based on the ex-istence of very rare large droplets (heterophase fluctua-tions). Ising models deal with interacting spins and thusdependent random variables. This may be transformedinto independent variables by considering suitable groupsof spins (the droplets), as can be done for variables withlong range correlation [29], and then the theory of the ex-treme deviations for the product of independent randomvariables can apply [22]. This might suggest a deep linkbetween the anomalous relaxation in the Ising model andthe data sets that we analyze here. Let us also mentionthat, up to the space dimension 3+1 for which numericalsimulations have been carried out [30], the distribution ofheights in the Kardar-Parisi-Zhang equation of non-linearstochastic interface growth is also found to be a stretchedexponential.

We present the dierent data sets in the next sectionsand provide in the appendix a short toolkit calculus for thestatistical analysis of stretched exponentials. The quali-cation of stretched exponentials is particularly importantwith regards to extrapolations to large events that havenot yet been observed.

2 Evidence of stretched exponentialsand comparison with other laws

In our analysis, we use the rank-ordering technique[9,31,32] that amounts to order the variables by descend-ing values Y1 > Y2 > > YN , and plot Yn as a func-tion of the rank n. Rank-ordering statistics and cumula-tive plots are equivalent except that the former providesa perspective on the rare, largest elements of a popula-tion, whereas the statistics of cumulative distributions aredominated by the more numerous small events. As a conse-quence, statistical fluctuations describe the uncertaintiesin the values of Yn for a given rank in the rank-orderingmethod while they describe the variations of the numberof events of a given size in the cumulative representation.This shows that the former statistics is better suited forthe analysis of the tails of pdfs characterized by relativelyfew events.

Within the rank-ordering plot, a stretched exponen-tial is qualied by a straight line when plotting Y cn as afunction of log n, as seen from equation (3). A t of thestraight line gives

Y cn = a lnn+ b (4)


p

y = -0,357x + 2,8952R2 = 0,9971

y = -0,1036x + 1,5842R2 = 0,9962

1

1,2

1,4

1,6

1,8

2

2,2

2,4

2,6

2,8

3

0 1 2 3 4 5 6log rank

radio c=0.11, xo=4E-8, x95=1E-3

light c=0.04, xo=2E-34, x95=2E-22

Fig. 2. Galaxys radio and light intensities: \stretched expo-nential".

corresponding to

x0 = a1=c: (5)

Indeed, the denition (3) of the cumulative distribu-tion Pc(x) = exp[(x=x0)c] for the stretched exponen-tial means that lnPc(x) = (x=x0)c. Since Pc(x) = n=N ,where n is the number of events larger or equal to x,hence n is the rank, we obtain (4) immediately. The anal-ysis of [22] predicts that multiplicative processes lead to astretched exponential of the form

Pc(x) = exp[m(x=)1=m=]; (6)

where the index c stands for the cumulative distribution,dened by expression (3). Here,

m = 1=c (7)

is the number of levels in the multiplicative cascade. isthe unit of the variable x, such that x= is dimensionless. is a typical multiplicative factor dened by the fact thatx= is the product of m random dimensionless variables oftypical size . We see that a t to a data set by a stretchedexponential gives access only to the product 1=m by therelation x0 = c

c.In order to interpret correctly the meaning of x0, the

appendix shows that the mean hxi is given by

hxi = x0 (1=c)=c; (8)

where (x) is the gamma function (equal to (x 1)! for xinteger). When c is small, hxi will be much larger than x0.x0 is thus not the average scale of x, but a reference scalefrom which all moments can be determined. Another char-acteristic scale x95% can be obtained such that the proba-bility to exceed this value is less than 5% (correspondingto a 95% level condence). Using (6), this yields

x95% = 31=cx0: (9)

Notice that the gures and the ts are done using thedecimal logarithm. We thus convert the values of the tsto the natural logarithm to estimate these numbers.

The intuitive interpretation of the three parameters a,b and c of the stretched exponential rank ordering t (4)is the following: b1=c is the size of the event of rank 1 (i.e.the largest event in the population), x0 = a

1=c accordingto equation (5) is a characteristic scale from which one candeduce the value of the mean and of various moments asseen from equations (8, 9). Finally, the exponent c quan-ties the fatness of the tail of the stretched exponential,the smaller the exponent, the fatter the tail. Within themultiplicative model [22], its inverse is proportional to thenumber of generations.

Figure 1 compares the stretched exponential modelwith three other models also exhibiting a curvature inthe fractal display (log-log size-rank). These three modelsare the following. The rst one corresponds to a simpleparabola in the log-log plot, and is called the parabolicfractal [21],

logSn = log S1 a logn b(logn)2: (10)

In expression (10), Sn is the size of rank n. The caseb = 0 recovers the usual linear log-log plot qualifyinga pure power law distribution. The introduction of thequadratic term b (log n)2 is a natural parametric addi-tion to account for the existence of curvature in the log-log plot. The three parameters log S1, a and b of the twith equation (10) have simple interpretations: log S1 isthe (decimal) logarithm the size of the event of rank 1(i.e. the largest event in the population), a is the slope(inverse of the power law exponent ) for the largest val-ues (smallest ranks) and b quanties the curvature while1=(a+b logn) is the apparent power law exponent. Solvingthis quadratic equation for n as a function of Sn, we ndthe corresponding cumulative distribution function (cdf)giving the number of events n larger than Sn:

n = n0 exph(1=pb) flog(Smax=Sn)g

1=2i; (11)

where

Smax = S1 exp[a2=4b]; (12)

and

n0 = ea=2b: (13)

The most remarkable property of this parabolic fractaldistribution is the existence of a maximum Smax beyondwhich the distribution does not exist. In other words, thepdf and cdf of the parabolic fractal have nite compactsupport. This is dierent from a nite-size eect leadingto a maximum size in a nite system as here the maximumsize remains nite even in the innite asymptotic case.This property contrasts the parabolic fractal from theother distributions that are dened for arbitrarily largearguments.

The second model is the curved shifted linear fractal

logSn = logS0 a log(n+A); (14)


0,1

1

10

100

1000

1 10 100 1000 10000

rank

parabola S1=530,a=0.035, b=0.26up to 1959

up to 1969

up to 1979

up to 1989

up to 1993

Fig. 3. US Gulf of Mexico OCS: parabolic fractal.

and the last model we consider is the lognormal distribu-tion (with standard deviation s and most probable valuem). We have chosen the parameters of these four models insuch a way that they approach each other the most closelyin the interval of ranks 5{500, as shown in Figure 1.

All these models need three parameters, one for nor-malization and two that characterize the shape. As for theweights of small events in this numerical simulation whichtakes as element of comparison the overlap from rank 5to 500, the smallest is the lognormal, then the stretchedexponential, then the parabolic fractal and last the shiftedlinear fractal (which is completely linear beyond rank100) in this numerical example. For the tail towards largeevents, the thinner distribution is the parabolic fractal(since it is limited), then the stretched exponential, thenthe lognormal and last the shifted linear fractal.

2.1 Radio and light emissions from galaxies

There is an undergoing controversy on the nature of thespatial distribution of galaxies and galaxy clusters inthe universe with recent suggestions that scale invariancecould apply [33{35]. Motivated by this question and therelationship between galaxy luminosities and their spa-tial patterns [36], we investigate the pdf of radio andlight intensities radiated by galaxies. The data is obtainedfrom [37]. The problem is that it is dicult to obtain acomplete distribution of the global universe as the obser-vation are done by sectors and the universe is not homoge-neous. It has been shown [21] that the rank-ordering plotin log-log coordinates is not strictly linear for these datasets and that a noticeable curvature exists. Figure 2 shows

p

y = -3,5491x + 10,182R2 = 0,9935

y = -1,0416x + 4,25R2 = 0,9968

0

1

2

3

4

5

6

7

8

9

10

0 0,5 1 1,5 2 2,5 3log rank

actual c=0.35, xo=3, x95=80

ultimate c=0.21, xo=0,02, x95=4

Fig. 4. US GOM OCS oil reserves: \stretched exponential".

the radio intensity of the galaxies raised to the powerc = 0:11 as a function of the (decimal) logarithm of therank n and the light intensity of the galaxies raised to thepower c = 0:04 as a function of the (decimal) logarithm ofthe rank n. The curves are convincingly linear over morethan ve decades in n.

The best t to the data for radiosources gives x0 4 108 for the radio emissions. From expression (8),hxi 9!x0 1:63:2102. Expression (9) gives x95% 9 104. The best t to the data gives x0 2 1034

for the light emissions. From expression (8), hxi 25!x0 1:6 1025 x0 3 109. Expression (9) gives


y = -3,7665x + 15,764R2 = 0,9981

6

7

8

9

10

11

12

13

14

15

16

0 0,5 1 1,5 2 2,5log rank

agglomerations 1996 c=0.165,xo=20, x95=15 500

Fig. 5. US urban agglomerations > 100 000 inhabitans:\stretched exponential".

x95% 2 1022. In these cases where the exponent c is

very small, the reference scale x0 is very small comparedto a typical scale obtained from hxi or x95%. The largeuncertainties are due to the smallness of the exponent cwhose inverse is thus poorly constrained.

Theories of these radio and light emissions are poorlyconstrained but, interpreted within the multiplicative cas-cade model [22], this indicates of the order of 10-20 (resp.25-50) levels in the cascade for radio (resp. light) emis-sions. The factor two in the range of this estimation ofthe number of levels stems from the choice of an exponen-tial versus a Gaussian for the distribution of the variablesthat enter in the multiplicative process (see [22] for moredetails).

2.2 US GOM OCS oil reserve sizes

The determination of the distribution of oil reserves inthe world is of great signicance for the assessment of thesustainability of energy consumption by mankind (and es-pecially by developed countries) in the future. The extrap-olation of the statistical data may bring useful inferenceon the rate of new discoveries that can be expected inthe future. It is thus particularly important to correctlycharacterize the distribution. The data are usually con-dential and politically sensitive in OPEC countries wherequotas depend of reserves. Most of reserve databases areunreliable. We have chosen the public data of the US OCS(outer continental shelf) published in open le by MMS(Mineral Management Service of the US Department ofInterior). We have found that it is important to select thedata within one unique natural domain. For oil reserves,the domain is the Petroleum System dened by its source-rock, i.e the genetic origin when most of previous studieswere carried out on tectonic classication.

In reference [21], it was noticed that the log-log rank-ordering plot exhibits a sizable downward curvature, indi-cating a signicant deviation from a power law distribu-tion.

Figure 3 shows the rank-ordering plot of oil eld sizes(in million barrels = Mb) in a log-log display by decades. Itis obvious that the largest elds were found rst and thatthe leftmost part of the curved plot for the smallest ranksdid not change during the last twenty years. This part ofthe curve can be easily extrapolated with a parabola whichaim to represent the ultimate distribution of oil reservesin the ground encompassing all smaller oil elds, many ofthose that have probably not yet been discovered.

Figure 4 represents the same data raised to the powerc = 0:35 as a function of the (decimal) logarithm of therank n. An excellent t to a straight line is found over al-most three decades in ranks which provides a characteris-tic size x0 = 31 using equation (5) and hxi x95% 80using expressions (8, 9).

Interpreted within the cascade multiplicative model,the value of the exponent c 1=3 corresponds to about3 to 6 generation levels in the generation of a typical oileld. On the same display, we have plotted the ultimatecurve of Figure 3 and obtain a linear t for c = 0:21.The ultimate curve is obtained by tting the parabolicfractal law to the rst ranks in Figure 3. In this way, weget a plausible asymptotic corresponding to the reliableknowledge of all oil reservoirs in the earth. In this casewhere the largest sizes are well known and the small oneshidden, it is easier to extrapolate the plot in a log-logdisplay than in a log-power. The stretched exponentialmay thus give us a clue as to the formation of the oilelds but the parabolic fractal is probably better suitedfor the extrapolation towards small oil elds. In particular,the existence of a maximum size that characterizes theparabolic fractal distribution is well-adapted to accountfor this real data set.

2.3 World, US and French urban agglomeration sizedistributions and UN population per country

Urban agglomerations provide an example of self-organization that has recently been studied from thepoint of view of statistical physics of complex growth pat-terns [38]. A model has been proposed [39] in terms ofstrongly correlated percolation in a gradient that accountsreasonably well for the morphology of large cities and theirgrowth as a function of time. The distribution of areas oc-cupied by secondary towns around a major metropole hasbeen found approximately to be a power law [39]. Here,we analyze the distribution of urban agglomeration sizes.The term agglomeration refers to the natural geographiclimits (dened by the continuity of the buildings (no gaplarger than 200 m) as opposed to the articial adminis-trative boundaries that give spurious results. We take thepopulation as the proxy for the agglomeration size.

Figure 5 shows the rank-ordering plot of agglomerationsizes (larger than 100 000 inhabitants) in the US (UnitedNation demographic yearbook 1989) raised to the powerc = 0:165 as a function of the (decimal) logarithm of therank n. An excellent t to a straight line is found overmore than two decades in ranks which provides a refer-ence scale x0 = 20 2 using equation (5) and the typical


0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7

log rank

shifted linear: S1*(n+A)^-a:A=5,5, a=1,27PF a=0,435, b=0,193

stret exp a=8.7, b=15.8,c=0,16, xo=20, x95=15 500data

cumulative Million inhabitantsaggl >100 000 1<


y = -1,4339x + 9,397R2 = 0,9972

3

4

5

6

7

8

9

10

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5log rank

agglomerations c=0,13,xo=0,03, x95=123

Fig. 8. Worlds urban agglomeration: \stretched exponential".

the typical agglomeration size hxi 2:1 104 x0 = 600and x95% 120, using expressions (8, 9). The fact thathxi is much larger than x0 reflects the fat tail nature ofthe distribution with the small exponent c.

Figure 9 shows the rank-ordering plot of country pop-ulation sizes in the world reported by the United Nations(Urban agglomeration 1996). Each country population hasbeen raised to the power c = 0:42. A good t to a straightline is found over more than two decades in ranks. Thet shown in the gure provides a reference scale x0 = 7million using equation (10) and the typical country sizehxi 19:5 million and x95% 91 million, using expres-sions (8, 9). Notice the existence of two outliers or \kings",China and India.

2.4 Daily Forex US-Mark and US-Franc price variations

Stock market prices fluctuate under the action of manyfactors and the precise characterization of the distributionof price variations has important applications for optionpricing, portfolio optimization and trading. In addition,from a theoretical point of view, it constraints the modelsof the stock market. Historically, the central limit theoremled to the rst paradigm in terms of Gaussian pdfs thatwas rst put in doubt by Mandelbrot [41] when he pro-posed to use Levy distributions, that are characterised bya fat tail decaying as a power law with index between0 and 2. Recently, physicists have characterised more pre-cisely the distribution of market price variations [42,43]and found that a power law truncated by an exponen-tial provides a reasonable t at short time scales (muchless than one day), while at larger time scales the dis-tributions cross over progressively to the Gaussian distri-bution which becomes approximately correct for monthlyand larger scale price variations. Alternative representa-tions exist in terms of a superposition of Gaussian pdfscorresponding to cascade models inspired from an anal-ogy with turbulence [44]. These two classes of descriptionscan only be distinguished using higher order statistics thatseem to favor the cascade description [45].

The daily time scale is the most used for practicalapplications but is unfortunately fully in the cross-overregime between the truncated Levy law at the shortesttime scales and the asymptotic Gaussian behavior at thelargest time scales. It has thus been poorly constrained.Here, we show that a stretched exponential pdf provides aparsimonious and accurate t to the full range of currencyprice variations at this daily intermediate time scale. Wepresent two dierent data dealing with foreign exchangerates. The foreign exchange market is slightly dierentfrom the other nancial markets such as the stock marketfor instance since one does not exchange a valuable againstmoney, but a currency for another currency. The foreignexchange is the most active market in the world with adaily turnover of more than a trillion US dollars, with alarge part of the trades being done for hedging and specu-lative purposes. These transactions concern only some ma-jor currencies with the US dollar implied in 80% of them.The deutschmark (DEM) is the second major currencywith 20% of the transactions are exchanges between USdollars and DEM. Contrary to the other nancial markets,the foreign exchange market is a 24 hours global market.See [46] for more informations.

The rst data is represented in Figure 10 which showsthe positive and negative variations of the US dollar ex-pressed in German marks during the period from 1989 tothe end of 1994. We use again the rank-ordering represen-tation and plot in Figure 10a the nth price variation (posi-tive with square symbols and negative with diamond sym-bols) in log-log coordinates. Figure 10b plots the nth pricevariation (positive with square symbols and negative withdiamond symbols) taken to the power c = 0:87 and 0.90respectively as a function of the decimal logarithm of therank. We observe an excellent description by the almostsame straight line over the full range of quotation varia-tions for both the positive and negative variations. Thisshows that the pdf is approximately symmetric: there isessentially the same probabibility for an appreciation or adepreciation of the US dollar with respect to the Germanmark. Notice that the apparent slight deviations abovethe straight line for the largest variations are completelywithin the expected error bars. The best t to both pos-itive and negative variations with equation (4) gives thesame consistent values a = 0:008 and b = 0:06 0:005.From this, we obtain x0 = 0:5%, and from expressions(8, 9), we get hxi 1:09x0 1% and x95% 1:7%.

Figure 11a plots the nth price variation for the FrenchFranc expressed in German marks (in the period from1989 to the end of 1994) with the positive variationsrepresented with square symbols and the negative varia-tions represented with diamond symbols) in log-log coor-dinates. Figure 11b plots the nth price variation (positivewith square symbols and negative with diamond symbols)taken to the power c = 0:72 and 0.64 respectively as afunction of the decimal logarithm of the rank. We observean excellent description with straight lines over the fullrange of quotation variations. Two important dierenceswith the $/Mark case are noteworthy. First, the expo-nent c is smaller, which corresponds to a \fatter" tail, i.e.


y = -5,0993x + 12,121R2 = 0,9946

0

2

4

6

8

10

12

14

16

18

20

0 0,5 1 1,5 2 2,5log rank

country c=0.42, xo=7, x95=91

Kings

China

India

USA

Fig. 9. UN 1996 population by country: stretched exponential.

0,0001

0,001

0,01

0,11 10 100 1000 10000

rank

positivenegative

+ 174 zero values

(a)

y = -0,0177x + 0,0596R2 = 0,9967

y = -0,0198x + 0,0666R2 = 0,9948

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0 0,5 1 1,5 2 2,5 3 3,5log rank

positive c=0.87, xo=0.004, x95=0.015

negative c=0.9, xo=0.004, x95=0.015

(b)

Fig. 10. (a) Variation of DM/$: parabolic fractal. (b) Varia-tions of DM/$: stretched exponential.

the existence of larger variations. The dynamics ofFranc/Mark exchange rate is thus wilder than that of thetwo stronger currencies $/Mark. Secondly, the coecientsa and b of the positive and negative exchange rate varia-tions are dierent, characterizing a clear asymmetry withlarger negative variations of the Franc expressed in Marks.This asymmetry corresponds to a progressive depreciationof the Franc with respect to the Mark. One could haveimagined that such a depreciation would correspond to asteady drift on which are superimposed symmetric varia-tions. We nd something else: the depreciation is puttingits imprints at all scales of price variations and is simplyquantied, not by a drift, but by a dierent reference scalex0.

The best t to the positive variations of theFranc/Mark exchange rate with equation (4) gives a =0:014 and b = 0:10. From this, we obtain x0 = 0:12%,and from expressions (8, 9), we get hxi 1:4x0 0:17%and x95% 5:6x0 0:7%. The dierence between hxi andx95% illustrates clearly the wilder character of the fat tailof the Frank-Mark exchange rate variations compared tothe $/Mark.

The best t to the negative variations of theFranc/Mark exchange rate with equation (4) givesa = 0:0095 and b = 0:07. From this, we obtain x0 = 0:16%,and from expressions (8, 9), we get hxi 0:2% andx95% 4:6x0 0:7%. The dierence between hxi andx95% illustrates clearly the wilder character of the fat tailof the Frank-Mark exchange rate variations compared tothe $/Mark and the fact that the depreciation of the Franccan occur by large and sudden drops rather than accordingto a steady drift.

To sum up this subsection, we have found that thestretched exponential quanties in a remarkably simpleand illuminating way the dierence between the exchange


0,0001

0,001

0,01

0,11 10 100 1000 10000

rank

positive variationnegative variation

+ 1882 zero values

(a)

y = -0,0313x + 0,1004R2 = 0,9719

y = -0,0219x + 0,0687R2 = 0,9941

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0 0,5 1 1,5 2 2,5 3 3,5log rank

positive c=0.64, xo=0.001, x95=0.007

negative c=0.72, xo=0.002, x95=0.007

(b)

Fig. 11. (a) Variations of M/F: Parabolic fractal. (b) Varia-tions of Mark/Franc: stretched exponential.

rate between two strong currencies and between a strongand a weaker currency. This quantication uses only thetwo adjustable parameters c and x0 that represent thefatness of the tail of the stretched exponential pdf and itsreference scale.

2.5 Raup-Sepkoskis kill curve

It has been argued that the histogram of biological extinc-tion events over the last 600 million years obtained fromthe fossil record \can be reasonably well tted to a powerlaw with exponent between 1 and 3" (see [16] p. 165).Figure 12a reproduces the data from Sepkoskis compila-tion [47] in the log-log plot of the cumulative distributionwith inverted axis (a rank-ordering plot). Notice that therank does not start at n = 1 but at the rank of the orderof 60 because there are about 60 Genera that have a lifespan larger than 150 millions years and the data are notprecise enough to distinguish between these 60 Genera.The plot does not show any straight line whatsoever but

y = -0,222x2 + 0,8708x + 1,3462R2 = 0,9978

1

1,25

1,5

1,75

2

2,25

2 2,5 3 3,5 4 4,5log cumulative number of genera (rank)

(a)

y = -32,222x + 142,05R2 = 0,9764

0

10

20

30

40

50

60

70

80

90

1,5 2 2,5 3 3,5 4 4,5log rank

life span c=0,85, xo=22, x95=81

(b)

Fig. 12. (a) Raups kill curve: Bak \How nature works?" Fig-ure 39 p. 165. (b) Raups kill curve: \stretched exponential".

rather a systematic downward curvature (very close to aparabola) implying a much \thinner" tail than predictedby a power law. The t is carried out using the parabolicfractal given by equation (10). The t is very good andsuggest that there might be a maximum lifespan of about350 millions years for species as predicted from equation(12). Figure 12b shows the alternative rank-ordering plotof the lifespans measured in million years raised to thepower c = 0:85 as a function of the (decimal) logarithmof the rank n.

The curve is less convincing than for the previous ex-amples as it exhibits a sigmoidal shape, but the data aremuch more dicult to obtain and probably much less re-liable. However, a straight line provides a reasonable tover about two decades in ranks. This provides a refer-ence scale x0 = 22 million years using equation (5) and


the typical lifespans hxi 1:09x0 25 million years andx95% 3:6x0 82 million years, using expressions (8)and (9). These numbers seem very reasonable when look-ing directly at gure 39 of reference [16] p. 165.

2.6 Earthquake size and fault displacementdistributions

The well-known Gutenberg-Richter law gives the numberof earthquakes in a given region (possibly the world) withmagnitude larger than a given value. Translated into seis-mic moment (roughly proportional to energy released),the Gutenberg-Richter law corresponds to a power lawdistribution (1) with an exponent close to 2/3. beingsmaller than 1, the average energy released by earthquakesis mathematically innite, or in other words it is controlledby the largest events. There must thus be a cross overto another regime falling faster and several models havepreviously been discussed, in terms of another power lawfor the largest earthquakes with exponent larger than 1[31,48] or in terms of a Gamma distribution correspond-ing approximately to an exponential tail [49]. It is thussometimes found that the Gutenberg-Richter law is toomuch linear [50]! Here, we reexamine the worldwide Har-vard catalog (see [31] for a description and an analysisin terms of rank-ordering with power law distributions)containing the 1300 largest earthquakes in the world from1977 to 1992. Figure 13a shows the rank-ordering plot ofthe nth seismic moment as a function of the rank n in theusual log-log representation. A clear bending is observedthat can be well-tted by the parabolic fractal distribu-tion (formula (10)). Figure 13b shows the rank-orderingplot of the nth seismic moment raised to the power c = 0:1as a function of the logarithm of the rank n. A t by astretched exponential distribution is also very good. Bothmodels t very well and are similar for the extrapolationtowards the smallest events. The choice of one model ishowever of signicant consequence for the prediction of thesize of the next largest earthquake. The parabolic fractalis in the present case ill-suited as it predicts a maximumsize close to the largest observed event. This is probablyunderestimated and a stretched exponential extrapolationis probably to be prefered.

The distribution of fault displacements has been stud-ied quantitatively to go beyond the usual geometrical de-scription and quantify the relative activity of faults withincomplex fault networks. Fault displacements provide along-term measurement of seismicity and are thus impor-tant for seismic hazard assessment and long-term predic-tion of earthquakes [51]. In contrast to a widespread be-lief, the distribution is not a power law as seen in gure14a which represents the rank-ordering plot of the nthlargest displacement of seismic faults [52] as a function ofthe nth rank in log-log coordinates. The curvature is verystrong and is tted to the parabolic fractal. Figure 14bqualies essentially an exponential distribution since thenth largest displacement raised to a power close to 1 islinear in the logarithm (decimal) of the nth rank. The

y = -0,1606x2 - 0,5312x + 2,5829R2 = 0,999

-1

-0,5

0

0,5

1

1,5

2

2,5

3

0 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 3,25

log rank

parabolic

(a)

y = -0,3219x + 1,8657R2 = 0,998

0,8

0,9

1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

0 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 3,25log rank

seismic moments c=0.1, xo=3E-9, x95=2E-4

(b)

Fig. 13. (a) Seismic moments. (b) Seismic moments: stretchedexponential.

characteristic displacement is essentially given by the co-ecient a of the t equal to about 400= ln(10) 180 m.

2.7 Temperature data over 400 000 years from Vostoknear the south pole

Isotope concentrations in ice cores measured at Vostoknear the south pole provide proxies for the earth temper-ature time series over the last 400 000 years, with morethan 2600 data points [53]. A large research eort is fo-cused at improving the reliability of this proxy and ana-lyzing it to detect trends and oscillatory components thatmight be useful for climate modelling and for the assess-ment of present temperature warming trends [54]. To pre-pare the following plots, we normalize the temperaturevariations by the corresponding time interval. Figure 15arepresents the log-log rank-ordering plot of the nth largest


10

100

1000

1 10 100 1000rank

parabola S1=750, a=0,01,b=0,2fault displacement

(a)

y = -405,62x + 987,8R2 = 0,991

0

100

200

300

400

500

600

700

800

900

1000

0 0,5 1 1,5 2 2,5log rank

displacement c=1,035, xo=148, x95=428

(b)

Fig. 14. (a) Seismic fault displacement: parabolic fractal. (b)Seismic fault displacement: \stretched exponential".

normalized temperature variations (positive and negativevariations are treated separately) as a function of the nthrank. The curvature is very strong and clearly excludes apower law distribution. Figure 15b shows that a stretchedexponential distribution can account reasonably well forboth the distribution of positive and negative variations.The dierence in the exponent c for the positive and nega-tive temperature variations is not statistically signicant:c 0:65. The same holds true for the characteristic valuex0 13. We obtain, from expressions (8, 9) hxi = 17 andx95% = 70. From this one-point statistical analysis, thereis not much dierence between the positive and negativetemperature variations, with however a perceptible ten-dancy for observing more often larger negative tempera-ture variations (the curve for the negative variations is sys-tematically above that for the negative variations). Overthis 400 000 year period and within this one-point statisti-cal analysis, the temperature trend, if any, is more towards

0,1

1

10

100

1000

1 10 100 1000 10000

rank

positivenegative

+ 321 zero values

(a)

y = -13,378x + 43,774R2 = 0,986

y = -11,521x + 38,054R2 = 0,98

0

5

10

15

20

2530

3540

4550

0 0,5 1 1,5 2 2,5 3 3,5log rank

positive c=0.64, xo=12, x95=69

negative c=0.67, xo=14, x95=71

(b)

Fig. 15. (a) Vostok: variations of temperatures versus time:parabolic fractal. (b) Vostok: variations temperatures versustime; stretched exponential.

cooling than warming. Higher-order statistics, taking intoaccount correlations between successive times such as bystudying the time series directly, are needed to ascertainany recent warming trend.

2.8 Citations of the 1120 most cited physicistsover the period 1981-June 1997

D.A. Pendlebury from the Institute for Scientic Informa-tion has recently ranked by total citations the 1120 mostcited physicists over the period 1981-June 1997. The datarepresent citations recorded over 1981-June 1997 to ISI-indexed physics papers 1981-June 1997, and do not repre-sent citations to books, to pre-1980 papers indexed by ISI,or to any papers not indexed by ISI during 1981-June 97.Figure 16a shows in log-log plot the dependence of Sn asa function of rank n, where Sn is the total number of cita-tions of the nth most cited physicist. One observes clearlya curvature both for the smallest and the largest ranks,excluding a power law distribution. If however one insists


1 0 0 0

1 04

1 1 0 1 0 0 1 0 0 0

cita

tion

num

ber

Rank

Y = M0*XM1

30235M0-0.36158M1

0.9834R

(a)

1 0

1 2

1 4

1 6

1 8

2 0

2 2

0 1 2 3 4 5 6 7

(citat

ion nu

mber)

^0.3

ln(Rank)

Y = M0 + M1*X19.695M0

-1.3485M10.97399R

(b)

Fig. 16. (a) Log-log plot of the dependence of the nth rankSn as a function of rank n, where Sn is the total number ofcitations of the nth most cited physicist; (b) The total numberSn of citations of the nth most cited physicist raised to thepower c = 0:3 as a function of the natural logarithm of therank.

in tting this curve with a power law, one gets an appar-ent average exponent around 1=0:36 3. Figure 16bshows Sn raised to the power c = 0:3 as a function ofthe natural logarithm of the rank. The whole range (overthree decades in rank) is well-described by the stretchedexponential. From the parameters of the t shown onthe gure, we obtain x0 = 2:7 using equation (5) andthe typical citation numbers hxi 25 and x95% 105.For the physicists (to whom the present authors belong)who do not appear in this aristocracy of the 1120 mostcited, it should come as a small satisfaction that it suf-ces to have more than about 105 citations to be amongthe 5% most cited physicists in the world! The 1120thrank has 2328 citations (while the rst rank has ten timesmore): inserting this number 2328 in equation (3) and us-ing the value x0 = 2:7, this shows that the 1120 mostcited physicists correspond to the fraction 5 104 of the

total physicist population, a number that appears quitereasonable.

We propose to rationalize the stretched exponentialagain using the results of [22] for multiplicative processes.It appears reasonable to involve a multiplication of factorsto account for the impact of a scientist. An author has alarge citation impact if he/her has (1) the ability to selectan good problem for investigation, (2) the competence towork on it and carry out the work to completion, (3) theability to create or belong to a research group and make itwork eciently, (4) the ability of recognizing a surprisingand worthwhile result, (5) gifts for writing clear and livelypapers, (6) the expertise, salemanship and dedication toadvertise his/her results. Similar ideas were put forwardby Shochley [55] who analyzed in 1957 the scientic outputof 88 research sta members of the Brookhaven NationalLaboratory in the USA. He found a log-normal distribu-tion which is the center of the limit distribution for theproduct of a large number of random variables. The factthat the extreme tail of the distribution that we analyzehere is a stretched exponential is of no surprise within thismodel in view of the extreme deviation theorem [22].

3 Conclusion

Power laws are generally used to represent natural distri-butions, often claimed to be power laws which representlinear regressions in log-log plots. In reality however, theplots often display linearity over a limited range of scalesand/or exhibit noticeable curvature. In some cases as in oileld reserves and earthquake sizes, the small value of theexponent would imply a diverging average, a result ruledout by the nite size of the earth. Here, we have investi-gated the relevance for a set of ten dierent data sets ofthe family of stretched exponential distributions. We havealso compared the ts with a natural extension of the lin-ear ts in log-log plots using a quadratic correction, whichleads to the so-called parabolic fractal distribution. Thestretched exponential seems to provide a reasonable t toall the data sets and has the advantage of a sound theoreti-cal foundation. We have been however surprised to realizethat the stretched exponential pdfs, that are supposedtheoretically to apply better for the rarest events, seem toaccount remarkably well for the center of most of the ana-lyzed distributions. Stretched exponential have a tail thatis \fatter" than the exponential but much less so that apure power law distribution. They thus provide a kind ofcompromise between the two descriptions. Stretched ex-ponentials have also the advantage of being economicalin their number of ajustable parameters. The parabolicfractal is also a natural parametric representation, thatsometimes perform better for the natural (non-economic)data sets and exhibits robustness in its parameters aand b.

We thank D. Stauer for very useful comments and sugges-tions and D. A. Pendlebury from the Institute for ScienticInformation for the data on physicist citations.


Appendix

Using the parametrization (2)

P (x)dx = c(xc1=xc0) exp[(x=x0)c]dx; (A.1)

we have

hxi = x0(1=c) (1=c); (A.2)

and

hx2i = x20(2=c) (2=c); (A.3)

where (x) is the Gamma function.Let us now give the most probable determination of

the parameters x0 and c. Using the maximum likelihoodmethod, we nd

xc0 = (1=n)nXi=1

(Y ci Ycn ) (A.4)

where Y1 > Y2 > > Yi > > Yn are the n largestobserved values. This expression (A.4) provides the mostprobable value for x0 conditioned on the knowledge ofthe exponent c. The method of maximum likelihood alsoallows us to get an equation for the most probable valueof the exponent c:

1=c =

nXi=1

(Y ci lnYi Ycn lnYn)

=

nXi=1

(Y ci Ycn )

(1=n)nXi=1

lnYi

which is implicit as c appears on both sides of the equality.We now provide the distribution of extreme variations. Weask what is the probability P (xmax > x

) that the largestvalue xmax among N realizations be greater than x

:

P (xmax > x) = 1

1 exp[(x=x0)

c

N= 1 expfN exp[(x=x0)

c]g: (A.5)

Denote p the chosen level of probability of tolerance forthe largest value xmax, i.e. P (xmax > x

) = p. Inverting(A5) yields

x = x0fln[N= ln(1 p)]g1=c: (A.6)

Table 1 is calculated for c = 0:7, x0 = 0:027 and N =7500. The value x = 0:062 is the usual estimate of thetypical largest value, corresponding to a probability of37%.

Table 1.

p 1 1=e = 0:63 1/2 0.1 0.01 0.001

x=x0 22.9 24.2 31.8 41.5 52.2

x 0.062 0.065 0.086 0.112 0.141

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