Applied Mathematics
Vol.5 No.13(2014), Article ID:47930,7 pages
DOI:10.4236/am.2014.513198
Heavy-Tailed Distributions Generated by Randomly Sampled Gaussian, Exponential and Power-Law Functions
Frederic von Wegner
Medical Biophysics Group, Institute of Physiology and Pathophysiology, University of Heidelberg, Heidelberg, Germany
Email: fwegner@physiologie.uni-heidelberg.de
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 11 April 2014; revised 21 May 2014; accepted 2 June 2014
ABSTRACT
A simple stochastic mechanism that produces exact and approximate power-law distributions
is presented. The model considers radially symmetric Gaussian, exponential and power-law
functions in n = 1, 2, 3 dimensions. Randomly sampling these functions with a radially
uniform sampling scheme produces heavy-tailed distributions. For two-dimensional
Gaussians and one-dimensional exponential functions, exact power-laws with exponent
−1 are obtained. In other cases, densities with an approximate power-law behaviour
close to the origin arise. These densities are analyzed using Padé approximants
in order to show the approximate power-law behaviour. If the sampled function itself
follows a power-law with exponent −α, random sampling leads to densities that
also follow an exact power-law, with exponent. The presented mechanism shows
that power-laws can arise in generic situations different from previously considered
specialized systems such as multi-particle systems close to phase transitions, dynamical
systems at bifurcation points or systems displaying self-organized criticality.
Thus, the presented mechanism may serve as an alternative hypothesis in system identification
problems.
Keywords:Heavy-Tailed Distributions, Random Sampling, Gaussian, Exponential, Power-Law
1. Introduction
Across scientific disciplines, heavy-tailed and in particular, power-law distributed quantities have received special attention due to their association with phenomena such as phase transitions, self-organized criticality and fractal patterns in space and time [1] -[5] . Power-laws are often contrasted with exponential and Gaussian distributions that typically occur in spatiotemporal correlation functions and as distributions of characteristic quantities in standard equilibrium kinetics [1] [6] . However, there exists no unique mechanism for the generation of power-law behaviour [2] [6] -[9] . Therefore, in the context of system identification, the occurrence of a powerlaw cannot be used to infer the mechanisms governing the generating process. We here present a simple mechanism producing exact and approximate power-law distributions. In the presented model, Gaussian, exponential and power-law functions in one, two and three dimensions are uniformly random-sampled. The resulting amplitude distributions of the random samples show exact and approximate power-law functional forms. Exact power-law distributions with exponent −1 are obtained for one-dimensional exponential and two-dimensional Gaussian distributions. Generalized power-laws with arbitrary scaling exponents are obtained from randomly sampled power-law functions. The presented mechanism can easily be imagined to occur in diverse experimental settings where a sensor at a fixed location samples a signal, of Gaussian shape for instance, which occurs at a random distance of the sensor site. Given this generic mechanism for the generation of power-law distributions, our model may serve as an alternative mechanism to be accounted for whenever a power-law distribution is found in an experimental setting.
2. Background
Let
be a random variable over
, where
represents the Borel sets over
and let
be the probability density of
. By conservation
of probability, for any monotonous, differentiable transformation
, the density
is obtained from the random variable transformation theorem
[10] :
(1.1)
where
is the continuous derivative of the inverse of
. In the following,
will be one of the functions (“signal shapes”)
to be randomly sampled, i.e. a Gaussian, an exponential or a power-law function
in one, two or three dimensions. In higher dimensions, these functions are assumed
to follow the given law in any direction, i.e. to have radial symmetry. In the context
of this article,
represents the radial variable, commonly denoted
as
, in polar or spherical coordinates.
We choose the following representations, valid in any dimension.
Gaussian:
(1.2)
Exponential:
(1.3)
Power-law:
(1.4)
For the shape parameters it is assumed that.
Assuming a radially uniform sampling on, we obtain the following expressions for
in
dimensions:
(1.5)
Padé approximants of the transformed densities
were calculated with the CAS maxima (http://maxima.sourceforge.net/).
3. Randomly Sampled Gaussian, Exponential and Power-Law Functions
3.1. Gaussians
We assume radially symmetric Gaussian functions in one, two and three dimensions.
The radial distribution in arbitrary dimensions is given by (1.2). Let us now assume
the Gaussian function is randomly sampled with the radially uniform sampling scheme
(1.5), where the sampling volume is given by. The function
is a continuous, bijective mapping with inverse
and derivative
In
dimensions, random variable transformation (1.1) yields the densities
:
We observe an exact power-law distribution
with power-law exponent
in
dimensions. Figure 1 shows the randomly sampled
densities
in
dimensions.
3.2. Exponentials
In this section, radially symmetric exponential shapes as given by (1.3) in
dimensions are analyzed. The exponential defines a continuous, bijective mapping
. The inverse is given by
and the derivative of the inverse by
In
dimensions, random variable transformation (1.1) yields the densities
:
Figure 1. Radially uniform random sampling of Gaussian functions in n = 1, 2, 3 dimensions yield exact and approximate power-law distributions (black curves). In the case n = 1, an exact power-law with exponent −1 is obtained. The blue curves are the Padé approximants to the exact distrubutions P(y). For visualization purposes, the blue curves are offset by a fixed amount.
In the exponential case, an exact power-law distribution
with exponent
is obtained in one dimension (in n = 1). In
dimensions, the distributions
approximately follow a power-law for
. This behaviour is visualized in Figure 2 and analyzed quantitatively using Padé approximants
further below. Figure 2 shows the randomly sampled
densities
in
dimensions.
3.3. Power-Laws
Finally, we ask which amplitude distribution
is obtained by randomly sampling functions that already follow a power-law. The
function
in arbitrary dimensions is given by (1.4). In order to obtain a continuous, bijective
mapping, domain and co-domain are set accordingly,
, where
. The inverse is given
by
and the derivative of the inverse by
Random variable transformation in
dimensions yields the three densities
Figure 2.
Random sampling of exponential functions in n = 1, 2, 3 dimensions yield an exact
power-law distribution with exponent −1 for n = 1. For n = 2, 3, an approximate
power-law behaviour is observed for. Blue curves are the Padé approximants
to the exact distrubutions P(y). For visualization purposes, the blue curves are
offset by a fixed amount.
In this case, exact power-laws with exponents
are obtained in any dimension
. Figure 3
shows the randomly sampled densities.
4. Padé Approximants
In Figure 1 and Figure
4, an approximate power-law behaviour of
is observed for small values of
. In order to quantify
this behaviour, we computed Padé approximants of order
of the densities
[11] . The approximants were calculated from the
Taylor expansions of
at the left border of the codomain of
. For first order approximations,
the densities
can be approximated by functions of the form
. As derived above, sampling a Gaussian in two
dimensions or an exponential in one dimension, exact power-laws with exponent
are obtained. In these cases, the Padé approximants yield the exact result. In the
other cases, the Padé approximants yield functions that follow the density
close to the origin. The Padé coefficients are given in
Table1
5. A Numerical Example
A small numerical example is presented to illustrate the connection between the
theoretically derived results and possible implications for experimental data. Consider
an experiment where Gaussian shaped signals occur at a random distance
to a fixed sensor. This situation is illustrated in the left panel of Figure 3, with the sensor
at the center. In analogy to the analytical derivations, all events are assumed
to occur within a two-dimensional disc of radius
. The amplitude y of the
event measured at site
depends on the random distance
between the sensor and the center of the event (dashed line). We simulated
events at random distances from S and recorded the amplitude y as measured at S.
The right panel of Figure 4 shows the resulting
empirical distribution (blue circles) in double logarithmic coordinates. The linear
shape suggests a power-law behaviour of the distribution. Estimating the exponent
yields
(fitted distribution as black solid line
Figure 3.
Random sampling of power-law functions in n dimensions produces exact power-law
distributions P(y) with exponent.
Figure 4. Numerical example. In the left panel, a generic
experimental setting is illustrated. A sensor (S) is placed at a fixed location
and Gaussian shaped events occur at random distances x from the sensor S, within
a disc shaped 2D region of radius R. The amplitude of the Gaussian y measured at
the sensor site decreases with increasing distance x. The right panel shows the
empirical distribution of event amplitudes P(y) (blue circles, n = 104
samples) in double logarithmic coordinate axes to emphasize the exact power-law
character of the empirical distribution. A power-law fit to the data (black solid
line) yields an exponent of, a close fit to the theoretically
derived exponent α = −1.
in the right panel of Figure 4), a result close
to the theoretically derived distribution
with exponent
.
6. Discussion
In the present work, a simple mechanism for the generation of power-law distributions is derived. The idea is
Table 1 . First-order Padé approximants of the densities
P(y) are given by. The table shows the coefficients for Gaussian
and exponential functions in n dimensions, denoted Gaussian-n and Exponential-n.
based upon a realistic scenario in experimental sciences. A signal of a given shape,
e.g. a Gaussian or an exponential, is measured by a sensor at a random distance
x to the signal maximum. Random sampling arises when the Gaussian or exponentially
shaped signal occurs randomly distributed across space (with density) and the sensor
resides at a fixed site. Our derivation shows that two-dimensional Gaussians and
one-dimensional exponentials lead to exact power-law densities with exponent
and that approximate power-law densities arise in other dimensions. Indeed, this
mechanism has been observed experimentally in dynamic fluorescence microscopy of
subcellular calcium currents [12] . The result
is of interest as it provides a simple and realistic mechanism that produces exact
power-laws. Power-laws are often associated with specialized mechanisms such as
phase transitions in complex systems, bifurcation points of dynamical systems or
systems displaying selforganized criticality and relatively few authors have considered
alternative mechanisms [6] . The mechnism presented
here is generic and may serve as an alternative hypothesis in cases where power-law
distributions are observed in experimental settings.
References
- Hohenberg, P.C. and Halperin, B.I. (1977) Theory of Dynamic Critical Phenomena. Reviews of Modern Physics, 49, 435-479. http://dx.doi.org/10.1103/RevModPhys.49.435
- Mitzenmacher, M. (2003) A Brief History of Generative Models for Power Law and Lognormal Distributions. Internet Mathematics, 1, 226-251. http://dx.doi.org/10.1080/15427951.2004.10129088
- Montroll, M. and Shlesinger, M.F. (1983) Maximum Entropy Formalism, Fractals, Scaling Phenomena, and 1/f Noise: A Tale of Tails. The Journal of Chemical Physics, 32, 209-230. http://dx.doi.org/10.1007/BF01012708
- Newman, M.E.J. (2005) Power Laws, Pareto Distributions and Zipfs Law, Contemporary Physics.http://dx.doi.org/10.1080/00107510500052444
- Stanley, H.E. (1999) Scaling, Universality, and Renormalization: Three Pillars of Modern Critical Phenomena. Reviews of Modern Physics, 71, S358-S366. http://dx.doi.org/10.1103/RevModPhys.71.S358
- Sornette, D. (2004) Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization, and Disorder: Concepts and Tools. Springer, New York.
- Malmgren, R.D., Stouffer, D.B., Motter, A.E. and Amaral, L.A.N. (2008) A Poissonian Explanation for Heavy Tails in E-Mail Communication. Proceedings of National Academy Science of USA, 105, 18153-18158.http://dx.doi.org/10.1073/pnas.0800332105
- Sornette, D. (1998) Multiplicative Processes and Power Laws. Physical Review E, 57, 4811-4813.http://dx.doi.org/10.1103/PhysRevE.57.4811
- Touboul, J. and Destexhe, A. (2010) Can Power-Law Scaling and Neuronal Avalanches Arise from Stochastic Dynamics? PLoS One, 5, e8982.
- Ramachandran, K.M. and Tsokos, C.P. (2009) Mathematical Statistics with Applications. Academic Press.
- Jr. Baker, G.A. and Graves-Morris, P. (1996) Padé Approximants. Cambridge University Press, New York.
- Ríos, E., Shirokova, N., Kirsch, W.G., Pizarro, G., Stern, M.D., Cheng, H. and González, A. (2001) A Preferred Amplitude of Calcium Sparks in Skeletal Muscle. Biophysical Journal, 80, 169-183. http://dx.doi.org/10.1016/S0006-3495(01)76005-5