# On estimating the exponent of power-law frequency distributions.

**ABSTRACT** Power-law frequency distributions characterize a wide array of natural phenomena. In ecology, biology, and many physical and social sciences, the exponents of these power laws are estimated to draw inference about the processes underlying the phenomenon, to test theoretical models, and to scale up from local observations to global patterns. Therefore, it is essential that these exponents be estimated accurately. Unfortunately, the binning-based methods traditionally used in ecology and other disciplines perform quite poorly. Here we discuss more sophisticated methods for fitting these exponents based on cumulative distribution functions and maximum likelihood estimation. We illustrate their superior performance at estimating known exponents and provide details on how and when ecologists should use them. Our results confirm that maximum likelihood estimation outperforms other methods in both accuracy and precision. Because of the use of biased statistical methods for estimating the exponent, the conclusions of several recently published papers should be revisited.

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**ABSTRACT:**Scaling is central to ecology and Earth system sciences. However, the importance of scale (i.e., resolution and extent) for understanding carbon dynamics across scales is poorly understood and quantified. We simulated carbon dynamics under a wide range of combinations of resolution (nine spatial resolutions of 250 m, 500 m, 1 km, 2 km, 5 km, 10 km, 20 km, 50 km, and 100 km) and extent (57 geospatial extents ranging from 108 to 1,247,034 km(2) ) in the southeastern United States to explore the existence of scale dependence of the simulated regional carbon balance. Results clearly show the existence of a critical threshold resolution for estimating carbon sequestration within a given extent and an error limit. Further, an invariant power law scaling relationship was found between the critical resolution and the spatial extent as the critical resolution is proportional to A(n) (n is a constant, and A is the extent). Scale criticality and the power law relationship might be driven by the power law probability distributions of land surface and ecological quantities including disturbances at landscape to regional scales. The current overwhelming practices without considering scale criticality might have largely contributed to difficulties in balancing carbon budgets at regional and global scales. This article is protected by copyright. All rights reserved.Global Change Biology 05/2014; 20(7):2240-2251. · 8.22 Impact Factor - SourceAvailable from: Mirian Jiménez
##### Article: Scaling properties of tidal networks

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**ABSTRACT:**A new methodology is developed to extract tidal network from hydrodynamic conditions, and use data derived from numerical modelling or field observations to test the hypothesis that tidal networks are characterized by scale-invariant properties. Different tidal network configurations have been obtained from long-term numerical simulations in an idealized basin. These simulations show the influence of hydrodynamic conditions (tidal range, TR) and sediment (grain size sediment, D50) on the final configuration of the network. One of the signatures of scale- invariant behaviour is related to the presence of a power law relationship in the probability distribution of geometrical characteristics. For each model configuration and field site the probability distribution of drainage area and the drainage volume has been calculated, and in both cases tidal networks show scale invariant characteristics. After assessing the sensivity of the results, an energy expenditure analysis shows that tidal basins evolve towards a state with less morphodynamic activity, with a lower energy expenditure compare with the initial state.Water Resources Research 05/2014; 50(6):4585-4602. · 3.15 Impact Factor - SourceAvailable from: Oliver Lawrence PhillipsJames C. Stegen, Nathan G. Swenson, Brian J. Enquist, Ethan P. White, Oliver L. Phillips, Peter M. Jørgensen, Michael D. Weiser, Abel Monteagudo Mendoza, Percy Núñez Vargas[Show abstract] [Hide abstract]

**ABSTRACT:**Aim An understanding of the relationship between forest biomass and climate is needed to predict the impacts of climate change on carbon stores. Biomass patterns have been characterized at geographically or climatically restricted scales, making it unclear if biomass is limited by climate in any general way at continental to global scales. Using a dataset spanning multiple climatic regions we evaluate the generality of published biomass–climate correlations. We also combine metabolic theory and hydraulic limits to plant growth to first derive and then test predictions for how forest biomass should vary with maximum individual tree biomass and the ecosystem water deficit.Location Temperate forests and dry, moist and wet tropical forests across North, Central and South America.Methods A forest biomass model was derived from allometric functions and power‐law size distributions. Biomass and climate were correlated using extensive forest plot (276 0.1‐ha plots), wood density and climate datasets. Climate variables included mean annual temperature, annual precipitation, their ratio, precipitation of the driest quarter, potential and actual evapotranspiration, and the ecosystem water deficit. The water deficit uniquely summarizes water balance by integrating water inputs from precipitation with water losses due to solar energy.Results Climate generally explained little variation in forest biomass, and mixed support was found for published biomass–climate relationships. Our theory indicated that maximum individual biomass governs forest biomass and is constrained by water deficit. Indeed, forest biomass was tightly coupled to maximum individual biomass and the upper bound of maximum individual biomass declined steeply with water deficit. Water deficit similarly constrained the upper bound of forest biomass, with most forests below the constraint.Main conclusions The results suggest that: (1) biomass–climate models developed at restricted geographic/climatic scales may not hold at broader scales; (2) maximum individual biomass is strongly related to forest biomass, suggesting that process‐based models should focus on maximum individual biomass; (3) the ecosystem water deficit constrains biomass, but realized biomass often falls below the constraint; such that (4) biomass is not strongly limited by climate in most forests so that forest biomass may not predictably respond to changes in mean climate.Global Ecology and Biogeography 01/2011; 20(5). · 7.22 Impact Factor

Page 1

Ecology, 89(4), 2008, pp. 905–912

? 2008 by the Ecological Society of America

ON ESTIMATING THE EXPONENT OF POWER-LAW

FREQUENCY DISTRIBUTIONS

ETHAN P. WHITE,1,2,4BRIAN J. ENQUIST,2AND JESSICA L. GREEN3

1Department of Biology and the Ecology Center, Utah State University, Logan, Utah 84322 USA

2Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, Arizona 85721 USA

3Center for Ecology and Evolutionary Biology, University of Oregon, Eugene, Oregon 97403 USA

Abstract.

phenomena. In ecology, biology, and many physical and social sciences, the exponents of

these power laws are estimated to draw inference about the processes underlying the

phenomenon, to test theoretical models, and to scale up from local observations to global

patterns. Therefore, it is essential that these exponents be estimated accurately. Unfortunately,

the binning-based methods traditionally used in ecology and other disciplines perform quite

poorly. Here we discuss more sophisticated methods for fitting these exponents based on

cumulative distribution functions and maximum likelihood estimation. We illustrate their

superior performance at estimating known exponents and provide details on how and when

ecologists should use them. Our results confirm that maximum likelihood estimation

outperforms other methods in both accuracy and precision. Because of the use of biased

statistical methods for estimating the exponent, the conclusions of several recently published

papers should be revisited.

Power-law frequency distributions characterize a wide array of natural

Key words:binning; distribution; exponent; maximum likelihood estimation; power laws.

INTRODUCTION

Power laws have a long history in ecology and other

disciplines (Bak 1996, Brown et al. 2002, Newman 2005).

Power-law relationships appear in a wide variety of

physical, social, and biological systems and are often cited

as evidence for fundamental processes that underlie the

dynamics structuring these systems (Bak 1996, Brown et

al. 2002, Newman 2005). There are two major classes of

power laws commonly reported in the ecological litera-

ture. The first are bivariate relationships between two

variables. Examples of this type of relationship include

the species–area relationship and body-size allometries.

Standard approaches to analyzing this type of data are

generally reasonable and discussions of statistical issues

related to this kind of data are presented elsewhere (e.g.,

Warton et al. 2006). The second type of power law, and

the focus of this paper, is the frequency distribution,

where the frequency of some event (e.g., the number of

individuals) is related to the size, or magnitude, of that

event (e.g., the size of the individual).

Frequency distributions of a wide variety of ecological

phenomena tend to be, at least approximately, power-

law distributed. These phenomena include distributions

of species body sizes (Morse et al. 1985), individual body

sizes (Enquist and Niklas 2001), colony sizes (Jovani and

Tella 2007), abundance among species (Pueyo 2006),

trends in abundance of species through time (Keitt and

Stanley 1998), step lengths in animal search patterns

(i.e., Levy flights; Reynolds et al. 2007), fire magnitude

(Turcotte et al. 2002), island size (White and Brown

2005), lake size (Wetzel 1991), flood magnitude (Mala-

mud and Turcotte 2006), landslide magnitude (Guzzetti

et al. 2002), vegetation patch size (Kefi et al. 2007), and

fluctuations in metabolic rate (Labra et al. 2007).

Frequency distributions are usually displayed as simple

histograms of the quantity of interest. If a distribution is

well-characterized by a power law then the frequency of

an event (e.g., the number of individuals with mass

between 10 and 20 g), f, is related to the size of that

event, x, by a function of the following form:

fðxÞ ¼ cxk

ð1Þ

where c and k are constants, and k is called the exponent

and is typically negative (i.e., k , 0). Because f(x) is a

probability density function (PDF) the value of c is a

simple function of k and the minimum and maximum

values of x (Table 1). The specific form of the PDF

depends on whether the data are continuous or discrete,

on the presence of minimum and maximum values, and

on whether k is ,?1 or .?1. The different forms are

often given distinct names for clarity (see Table 1).

There is substantial interest in using the parameters of

these power-law distributions to make inferences about

the processes underlying the distributions, to test

mechanistic models, and to estimate and predict patterns

and processes operating beyond the scope of the

observed data. For example, power-law species abun-

dance distributions with k ’ ?1 are considered to

represent evidence for the primary role of stochastic

Manuscript received 6 August 2007; revised 9 November

2007; accepted 20 November 2007. Corresponding Editor:

A. M. de Roos.

4E-mail: epwhite@biology.usu.edu

905

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birth–death processes, combined with species input, in

community assembly (Pueyo 2006, Zillio and Condit

2007); quantitative models of tree size distributions

make specific predictions (e.g., k ¼ ?2; Enquist and

Niklas 2001) that can be used to test these models

(Coomes et al. 2003, Muller-Landau et al. 2006); and

power-law frequency distributions of individual size

have been used to scale up from individual observations

to estimate ecosystem level processes (Enquist et al.

2003, Kerkhoff and Enquist 2006).

One concern when interpreting the exponents of these

distributions is that there are a wide variety of different

approaches currently being used to estimate the

exponents (Sims et al. 2007, White et al. 2007). These

include techniques based on: (1) binning (e.g., Enquist

and Niklas 2001, Meehan 2006, Kefi et al. 2007); (2) the

cumulative distribution function (e.g., Rinaldo et al.

2002); and (3) maximum likelihood estimation (e.g.,

Muller-Landau et al. 2006, Edwards et al. 2007, Zillio

and Condit 2007). There has been little discussion in the

ecological literature of how the choice of methodology

influences the parameter estimates, and methods other

than binning are rarely used. If different methods

produce different results this could have consequences

for the conclusions drawn about the ecology of the

system (Edwards et al. 2007, Sims et al. 2007).

Here, we: (1) describe the different approaches used to

quantify the exponents of power-law frequency distri-

butions; (2) show that some of these approaches give

biased estimates; (3) illustrate the superior performance

of some approaches using Monte Carlo methods; (4)

make recommendations for best estimating parameters

of power-law distributed data; and (5) show that some

of the conclusions of recent studies are effected by the

use of biased statistical techniques.

METHODS FOR ESTIMATING THE EXPONENT

Linear binning

Perhaps the most intuitive way to quantify an

empirical frequency distribution is to bin the observed

data using bins of constant linear width. This generates

the familiar histogram. Specifically, linear binning

entails choosing a bin i of constant width (w ¼ xiþ1?

xi), counting the number of observations in each bin

(i.e., with values of x between xiand xiþw), and plotting

this count against the value of x at the center of the bin

(xi/2þxiþ1/2). If the counts are divided by the sum of all

the counts, this plot is an estimate of the probability

density function, f(x). The traditional approach to

estimating the power-law exponent is to fit a linear

regression to log-transformed values of f(x) and x, with

the slope of the line giving an estimate of the exponent,

k. Bins with zero observations are excluded, because

log(0) is undefined, and sometimes bins with low counts

are also excluded (e.g., Enquist and Niklas 2001). While

in practice the choice of bin width is normally arbitrary,

this choice represents a trade-off between the number of

bins analyzed (i.e., the resolution of the frequency

distribution) and the accuracy with which each value of

f(x) is estimated (fewer observations per bin provide a

poorer density estimate; Pickering et al. 1995).

TABLE 1.

and parameter values over which it applies, its probability density function (PDF; or probability mass function) f(x), its

cumulative distribution function F(x), and the maximum likelihood estimate (MLE) for k based on the PDF.

Descriptions of different power-law frequency distributions, including the name of the distribution, the range of data

Distribution?

f(x)F(x)MLE for k

1) Pareto

Range a ? x , ‘

Parameters k , ?1, a . 0

2) Truncated Pareto?

Range a ? x ? b

Parameters k 6¼ ?1, a ? 0, b ? 0

?(k þ 1)a?(kþ1)xk

1 ? a?(kþ1)xkþ1

^k ¼ ?1 ?

1

n

X

n

i¼1

logxi

a

? ?

"#?1

(k þ 1)(bkþ1? akþ1)?1xk

xkþ1? akþ1

bkþ1? akþ1

X

fð?k;aÞ

ln x ¼

?1

ð^k þ 1Þþb^kþ1ln b ? a^kþ1ln a

b^kþ1? a^kþ1

3) Discrete Pareto?,§

Range x ¼ a, a þ 1, a þ 2, ... ‘

Parameters k , ?1, a ? 1

4) Power Functionjj

Range 0 ? x ? b

Parameters k . ?1, b . 0

xk

fð?k;aÞ

x

j¼a

jk

ln x ¼?f0ð?^k;aÞ

fð?^k;aÞ

X

(k þ 1)b?(kþ1)xk

(x/b)kþ1

^k ¼ logðbÞ ?1

n

n

i¼1

logðxiÞ

"#?1

? 1

Notes: The minimum value of x for which a distribution is valid is given by a, which is defined to be greater than zero. The

maximum value of x for which a distribution is valid is given by b, which is defined to be less than infinity.

? Sources are as follows: Pareto (Johnson et al. 1994); Truncated Pareto (Page 1968); Discrete Pareto (Clauset et al. 2007); Power

Function (Evans et al. 2000). There is an error in the MLE solution given by Evans et al. (2000) that has been corrected. Note that

MLEs are only guaranteed to be minimum variance unbiased estimators in the limit of large n. If n is small, corrections to the MLE

are available (Johnson et al. 1994, Clark et al. 1999, Clauset et al. 2007). All solutions assume that a and b are known.

? The MLE equations for these distributions cannot be solved analytically forˆk, so they must be solved using numerical methods.

§ f(?k, a) ¼ R‘

jj The Power Function distribution is often ignored in discussions of power-law distributions because it rarely occurs in natural

systems (Newman 2005). We include it here for completeness and because it has been suggested that in some groups individual size

distributions based on mass may be approximately power-law distributed with k . ?1 (e.g., Enquist and Niklas 2001).

k¼0(k þ a)kis the generalized zeta function and f0(?k, a) is its derivative with respect to ?k.

ETHAN P. WHITE ET AL.906Ecology, Vol. 89, No. 4

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Logarithmic binning

Simple logarithmic binning.—This approach is similar

to linear binning, except that instead of the bins having

constant linear width, they have constant logarithmic

width, b ¼ log(xiþ1) ? log(xi). The estimate of k is

obtained by log-transforming the values of x and

following the procedure described in the previous

section. Since the x data are transformed to begin with,

it is not necessary to transform the bin centers again

prior to fitting the regression. For power-law-like

distributions, an advantage of logarithmic binning is

the reduction of the number of zero and low-count bins

at larger values of x because the linear width of a bin

increases linearly with x; i.e., wi¼ xi(eb? 1). However,

this means that the number of observations within each

bin is determined not only by x, but also by the linear

width of the bin. Therefore, the slope of the regression

will give an estimate of k þ 1, not k (Appendix A; Han

and Straskraba 1998, Bonnet et al. 2001, Sims et al.

2007).

Normalized logarithmic binning.—The problem of

increasing linear width of logarithmic bins can be dealt

with by normalizing the number of observations in each

bin by the linear width of the bin, w. This converts the

counts into densities (number of observations per unit of

x; Bonnet et al. 2001, Christensen and Moloney 2005).

The linear width of a logarithmic bin can be calculated

as xi(eb? 1) (Appendix A). This normalization approach

is typically used in the characterization of aquatic size-

spectra and power-law distributions in physics (Kerr

and Dickie 2001, Christensen and Moloney 2005). It

removes the artifact from traditional logarithmic bin-

ning while maintaining the advantage of using larger

bins where there are fewer values of x. An alternative

approach is to use simple logarithmic binning and

subtract one from the estimated exponent (Han and

Straskraba 1998, Bonnet et al. 2001).

Fitting the cumulative distribution function

An alternative to binning methods is to work with the

cumulative distribution function (CDF). The CDF

describes the probability that a random variable, X,

drawn from f(x) is ?x. The CDF is straightforward to

construct for a set of observed data, and no binning is

required. To construct the CDF, first rank the n

observed values (xi) from smallest to largest (i ¼

1 ... n). The probability that an observation is less than

or equal to xi(the CDF) is then estimated as i/n (this is

the Kaplan-Meier estimate; Evans et al. 2000). Analyz-

ing the CDF avoids the subjective influence of the choice

of bin width and the problem of empty bins. Having

determined the CDF for a power-law distribution, the

exponent, k, of the probability density function (PDF)

can be estimated using regression. The traditional

approach is to transform the equation for the CDF

such that the slope of a linear equation is a function of k.

The linearized equation differs among distributions

(Appendix A). The slope of the regression will be equal

to k þ 1, making it necessary to subtract 1 to obtain k

(Bonnet et al. 2001, Rinaldo et al. 2002).

Maximum likelihood estimation

Maximum likelihood estimation (MLE) is one of the

preferred approaches for estimating frequency distribu-

tion parameters (e.g., Rice 1994). MLE determines the

parameter values that maximize the likelihood of the

model (in this case, a power law with an unknown

exponent) given the observed data. Specifically, MLE

finds the value of k that maximizes the product of the

probabilities of each observed value of x (i.e., the

product of f(x) evaluated at each data point; see Rice

[1994] for a good introduction to maximum likelihood

methods). The specific solution for the maximum

likelihood estimate of k and whether the solution is

closed form or requires numerical methods to solve

depends on the minimum and maximum values of x and

on the value of k (Table 1). Alternatively, the likelihood

can be maximized directly using numerical methods

(Clauset et al. 2007, Zillio and Condit 2007). While

MLE does not provide an opportunity for visual

inspection of the distribution to determine if the

assumption of the power-law functional form is

reasonable, the validity of this assumption can be

assessed using simple goodness-of-fit tests such as the

Chi-square on binned data (Clark et al. 1999, Clauset et

al. 2007, Edwards et al. 2007), or by visually assessing

the linearity of binned data, or the CDF (Benhamou

2007), under the appropriate transformation.

COMPARING THE METHODS

While uncorrected simple logarithmic binning clearly

provides incorrect estimates of k, the alternative

approaches discussed above all seem reasonable and

intuitive. However, the different approaches do not

perform equally well, and some produce biased esti-

mates of the exponent (e.g., Pickering et al. 1995, Clark

et al. 1999, Sims et al. 2007). We applied Monte Carlo

methods to illustrate the advantages and disadvantages

of the various approaches and to explore cases relevant

to ecology that have not been previously addressed.

Monte Carlo methods generate data that are, by

definition, power-law distributed with known exponents,

making it possible to compare the performance of the

different techniques in estimating the value of k.

We generated power-law distributed random numbers

using the inverse transformation method for the Pareto

distribution (Ross 2006), and using the rejection method

for the discrete Pareto distribution (Devroye 1986).

Each analysis consisted of the following: (1) generating

10000 Monte Carlo data sets for each point in the

analysis (e.g., for each sample size), (2) estimating the

exponent for each data set using the methods described

previously, and (3) comparing the performance of the

methods based on bias (i.e., accuracy) and on the

variance in the estimate (i.e., precision). We report on

simulated distributions generated using k¼?2 and a¼1.

April 2008907ON ESTIMATING POWER-LAW EXPONENTS

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The results for other combinations of parameters are

qualitatively similar. We also evaluated the influence of

sample size on the various estimation techniques, and

for binning-based approaches we evaluated the effect of

bin width on the analysis.

GENERAL RULES

Uncorrected simple logarithmic binning gives the wrong

exponent.—Non-normalized logarithmic binning does

not estimate k; it estimates k þ 1 (Han and Straskraba

1998, Bonnet et al. 2001, Sims et al. 2007). Therefore if

simple logarithmic binning is used, and an estimate of k

is the desired result, then it is necessary to subtract 1

from the slope of the logarithmically binned data. Not

doing so will give the wrong value for the exponent.

Binning-based approaches perform poorly.—Linear

binning performs poorly by practically any measure.

In most cases it produces biased estimates of the

exponent and its estimates are highly variable (Figs. 1

and 2). In addition, the estimated exponent is highly

dependent on the choice of bin width, and this

dependency varies as a function of sample size (Fig. 3).

While normalized logarithmic binning performs better

than linear binning, its estimates are also dependent on

the choice of bin width and are more variable than

alternate approaches. Our results are based on recom-

mended practices in binning analyses (following Picker-

ing et al. 1995). Many alternative approaches to

constructing bins and performing regressions on binned

data are conceivable, and it is possible that some of these

may improve the performance of the estimates. Howev-

er, this highlights the fact that binning-based methods

are sensitive to a variety of decisions, and it appears that

no amount of tweaking will be able to produce a

consistent binning-based method for estimating the

exponent. In general, binning results in a loss of

information about the distributions of points within a

bin and is thus expected to perform poorly (Clauset et al.

2007, Edwards et al. 2007). Therefore, while binning is

useful for visualizing the frequency distribution, and

normalized logarithmic binning performs well at this

task, binning-based approaches should be avoided for

parameter estimation (Clauset et al. 2007).

Maximum likelihood estimation performs best.—While

fitting the cumulative distribution function (CDF)

generally produces good results, estimates of k using

the CDF approach are often biased at small sample sizes

and are consistently more variable than those using

maximum likelihood estimation (MLE; Fig. 2; Clark et

al. 1999, Newman 2005). This probably results because

the logarithmic transformation used in fitting the CDF

weights a small number of points more heavily, and

because the points in the CDF are not independent thus

violating regression assumptions (see Clauset et al.

[2007] for other issues with regression-based approach-

es). While alternative approaches to fitting the CDF

(e.g., nonlinear regression) could improve the perfor-

mance of this estimator, MLE has been shown

mathematically to be the single best approach for

estimating power-law exponents (i.e., it is the minimum

variance unbiased estimator; Johnson et al. 1994, Clark

et al. 1999, Newman 2005). In addition, MLE produces

valid confidence intervals for the estimated exponent

FIG. 1.

methods of fitting the power-law exponent. (a) A single Monte

Carlo sample from a Pareto distribution plotted as 1 minus the

cumulative distribution, F(x). Data are plotted as gray open

circles along with the fits to the data using the four different

methods: linear binning (red; linear), normalized logarithmic

binning (blue; Nlog), maximum likelihood estimation (green;

MLE), and cumulative distribution function fitting (black;

CDF). (b) Kernel density estimates of the distribution of

exponents from 10000 Monte Carlo runs. Line colors are the

same as for (a), and the value of k used to generate the data is

indicated by the dashed line. Parameter values: n¼500 random

points in each simulation; k¼?2; 1 ? x , ‘; linear bin width¼

3; logarithmic bin width ¼ 0.3. The binning analyses used the

minimum value of x and excluded the last bin and bins

containing ?1 individual. Exclusion of the last bin is not

necessary, but it improves the performance of binning-based

approaches and is thus conservative in the context of our

conclusions. The single sample for (a) was chosen to illustrate

the general results shown in (b). Binning methods generate

biased estimates of the exponent and result in more variable

estimates than approaches based on MLE and CDF.

Example of Monte Carlo results for the different

ETHAN P. WHITE ET AL. 908Ecology, Vol. 89, No. 4

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Page 5

(Appendix A), which the other methods do not (Clark et

al. 1999, Newman 2005, Clauset et al. 2007).

COMPLICATIONS

Minimum and maximum values.—Minimum and

maximum attainable values of ecological quantities can

result either from natural limits on the quantity being

measured (e.g., trees cannot grow above some maximum

size), or from methodological limits on the values that

can be observed (e.g., fires ,1 ha are not recorded). In

addition, the power-law form of the distribution may

not hold over the entire range of x, making it necessary

to select a restricted range of x on which to estimate the

exponent. While binning-based approaches do not

assume particular limits on x (but see Pickering et al.

1995), CDF and MLE approaches assume the minimum

and maximum attainable values of x given in Table 1. In

some cases these limits may be known, but if not, it may

be necessary to estimate them (e.g., Kijko 2004, Clauset

et al. 2007). Because maximum likelihood estimation for

the truncated Pareto requires numerical methods, it has

been suggested that in some cases with both a minimum

and maximum value that the error introduced by

assuming that there is no maximum is small enough

that it is reasonable to estimate the exponent using the

maximum likelihood estimate for the Pareto distribu-

tion. Clark et al. (1999) suggest this approximation in

cases where the maximum value is at least two orders of

FIG. 3.

(a) linear and (b) normalized logarithmic binning for three

different sample sizes: n ¼ 200 (solid black line), n ¼ 500 (solid

red line), and n ¼ 1000 (solid blue line); based on 1000 Monte

Carlo runs from the Pareto distribution per point. Parameter

values were k ¼?2 (dashed black line) and 1 ? x , ‘. Error

bars are 62 SE. Other binning methods are as in Fig. 1.

Changing bin width changes the estimated exponent for all

sample sizes.

Effect of bin width on the estimated exponents for

FIG. 2.

exponent and (b) the variance of that exponent, for the four

estimation methods: linear binning (red), normalized logarith-

mic binning (blue), maximum likelihood estimation (green), and

cumulative distribution function fitting (black). Values for each

sample size were generated using 10000 Monte Carlo runs from

the Pareto distribution with parameter values of k¼?2 (dashed

line), and 1 ? x , ‘; linear bin width¼7.5, and logarithmic bin

width ¼ 0.75. Other binning methods are as in Fig. 1. Linear

binning fails to converge to the correct estimate. While the

other methods all appear to converge at large sample sizes,

maximum likelihood estimation always yields the lowest

variance in the estimated exponent.

(a) Effect of sample size on the mean estimated

April 2008 909ON ESTIMATING POWER-LAW EXPONENTS

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magnitude greater than the minimum; that is, max (x) .

100 3 min (x).

Deviations from the power law.—Empirical data are

rarely perfectly power-law distributed over the entire

range of x (Brown et al. 2002, Newman 2005). MLE and

CDF approaches respond to deviations differently

because the traditional MLE analysis implicitly weights

data on a linear scale while the traditional CDF

approach weights it on a logarithmic scale (McGill

2003). The CDF approach will therefore respond more

strongly to deviations from the power law at large values

of x (such as those observed in individual size

distributions; e.g., Coomes et al. 2003) than the MLE

approach, whereas MLE will respond more strongly to

deviations at small values of x (commonly observed in

many power-law distributions; e.g., Newman 2005). It is

common to truncate data in the tails that exhibit

deviations from the power law before fitting the

exponent (e.g., Newman 2005). However, these devia-

tions also should not be ignored, as they may help

identify important biological processes (e.g., Coomes et

al. 2003). In some cases deviations may suggest that the

power law is in fact not the appropriate model for the

data. This can be evaluated using goodness-of-fit tests

on binned data (Clark et al. 1999, Clauset et al. 2007,

Edwards et al. 2007) or by using model selection

techniques to compare the power-law to alternative

distributions (Muller-Landau et al. 2006, Clauset et al.

2007, Edwards et al. 2007).

Discrete data.—Most of the MLE and CDF methods

presented here assume that the data are continuously

distributed, as is often the case (e.g., body size).

However, some ecological patterns (e.g., species-abun-

dance distributions) are comprised of discrete observa-

tions (e.g., it is impossible to census 4.3 individuals). It is

therefore necessary to use analogous discrete distribu-

tions. In the case of the Pareto distribution a discrete

analog exists in the form of the aptly named discrete

Pareto distribution (Johnson et al. 2005, Newman 2005;

Table 1; also called the Zipf or Riemann-zeta distribu-

tion). In some cases continuous distributions can

reasonably approximate discrete data; but in the case

of the Pareto, using the continuous maximum likelihood

estimate instead of that derived from the discrete

distribution produces strongly biased results and should

be avoided (Appendix C; Clauset et al. 2007).

IMPLICATIONS FOR PUBLISHED RESULTS

One of the most important implications for published

results is that studies that have estimated exponents

using uncorrected simple logarithmic binning (e.g.,

Morse et al. 1988, Meehan 2006) have reported the

wrong exponent. This is particularly important in cases

where the exponent is used to test quantitative

predictions. For example, an analysis in Meehan

(2006) evaluates whether observed individual size

distribution exponents were consistent with those

predicted, using simple logarithmic binning. Meehan

concluded that the empirical data matched the predic-

tions (Fig. 4a). However, since the reported exponents

are equal to k þ 1, the analysis suggests that the size

distribution is substantially steeper than expected, thus

refuting, rather than supporting, the hypothesized

mechanism (Fig. 4a; Appendix B).

Analyses based on linearly binned data should also be

revisited due to the potential for biased estimates and

the strong influence of bin width on the estimated

exponent. In particular, studies that have used linear

binning to test the predictions of theoretical models or

FIG. 4.

(a) Meehan (2006) and (b) Enquist and Niklas (2001), using less

biased methods. Plots are probability densities of the estimated

exponents using the original methodology from these studies

(red line; simple logarithmic binning in Meehan, linear binning

in Enquist and Niklas), and using less biased methods (blue

line; normalized logarithmic binning for Meehan, maximum

likelihood estimation for Enquist and Niklas). Both studies

purported to support a theoretically derived exponent (dotted

line). However, when the exponents are estimated using less

biased methods it becomes clear that the observed data deviate

significantly from the theoretical prediction.

Reanalysis of individual size distribution data from

ETHAN P. WHITE ET AL. 910 Ecology, Vol. 89, No. 4

Reports

Page 7

compare exponents from different data sets (e.g.,

Enquist and Niklas 2001, Coomes et al. 2003, Niklas

et al. 2003, Kefi et al. 2007) may have reached incorrect

conclusions. We reanalyzed the original data from

Enquist and Niklas (2001) and found that while the

original linear binning analyses suggested that observed

diameter distribution exponents were near the theoret-

ical prediction of ?2, MLE suggests that the observed

exponents are actually closer, on average, to ?2.5 (Fig.

4b; Appendix B). Our reanalysis indicates that the size–

frequency distributions in Gentry’s plots are not, in

general, adequately represented by a power law with an

exponent of ?2, as originally claimed by Enquist and

Niklas (2001; see Appendix B for an important caveat).

While normalized logarithmic binning performs better

than linear binning, it can still introduce biases of ;10%

depending on the bin width. While many analyses based

on normalized logarithmic binning are probably reason-

able, the recent suggestion that normalized logarithmic

binning is the best approach for fitting exponents (Sims

et al. 2007) is unwarranted, and MLE should be used

whenever possible (Clark et al. 1999, Clauset et al. 2007).

Compared to binning-based approaches, results from

fitting the CDF are probably reasonable. In cases with

low sample sizes, where small errors in the estimated

exponent could influence the conclusions of the study, or

where minimum or maximum attainable values of x

have been ignored (see Pickering et al. 1995), it may be

worth checking the results using MLE. Regardless,

MLE is the single best method for estimating exponents

and should be used in future studies.

CONCLUSIONS

The vast majority of ecological studies that estimate

exponents for power-law-like distributions use ap-

proaches based on binning the empirical data (e.g.,

Morse et al. 1988, Enquist and Niklas 2001, Coomes et

al. 2003, Niklas et al. 2003, Meehan 2006, Jovani and

Tella 2007, Kefi et al. 2007, Reynolds et al. 2007, Sims et

al. 2007). These binning-based methods tend to produce

results that are biased, have high variance, and are

contingent on the choice of bin width. Instead of

binning, maximum likelihood estimation should be used

when fitting power-law exponents to empirical data

(Clark et al. 1999, Newman 2005, Edwards et al. 2007).

We have focused on power laws because they, at least

approximately, characterize a number of distributions of

interest to ecologists. The issues raised here, and the

conclusions discussed, should apply broadly to frequen-

cy distributions in general, and in particular to other

distributions with heavy tails. Paying careful attention

to fitting methodologies and consultation of statistical

references (e.g., Johnson et al. 1994) should help

improve the estimation of distributional parameters.

ACKNOWLEDGMENTS

We particularly thank Tim Meehan for generously sharing

his data with us and for comments on the manuscript. For the

Gentry data we thank Alwyn Gentry, the Missouri Botanical

Garden, and collectors who assisted Gentry or contributed data

for specific sites. We also thank David Coomes, Susan Durham,

Jim Haefner, Brian McGill, and Tommaso Zillio for comments

on the manuscript. This work was funded by the National

Science Foundation through a Postdoctoral Fellowship in

Biological Informatics to E. P. White (DBI-0532847). We used

the first–last author emphasis approach (sensu Tscharntke et al.

2007) for the sequence of authors.

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APPENDIX A

Details of logarithmic binning, cumulative distribution function fitting, and maximum likelihood estimation (Ecological Archives

E089-052-A1).

APPENDIX B

Details of reanalysis of results published in Meehan (2006) and Enquist and Niklas (2001) (Ecological Archives E089-052-A2).

APPENDIX C

Example of bias resulting from fitting discrete data with continuous maximum likelihood solutions (Ecological Archives E089-

052-A3).

SUPPLEMENT

Matlab files for fitting power-law exponents using different methods (Ecological Archives E089-052-S1).

ETHAN P. WHITE ET AL.912Ecology, Vol. 89, No. 4

Reports

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