Tweedie convergence: A mathematical basis for Taylor's power law, $1/f$ noise, and multifractality

Tweedie convergence: A mathematical basis for Taylor's power law, $1 / f$ noise, and multifractality

Wayne S. Kendal and Bent Jørgensen

Phys. Rev. E 84, 066120 – Published 27 December 2011

Abstract

Plants and animals of a given species tend to cluster within their habitats in accordance with a power function between their mean density and the variance. This relationship, Taylor's power law, has been variously explained by ecologists in terms of animal behavior, interspecies interactions, demographic effects, etc., all without consensus. Taylor's law also manifests within a wide range of other biological and physical processes, sometimes being referred to as fluctuation scaling and attributed to effects of the second law of thermodynamics. 1/ $f$ noise refers to power spectra that have an approximately inverse dependence on frequency. Like Taylor's law these spectra manifest from a wide range of biological and physical processes, without general agreement as to cause. One contemporary paradigm for 1/ $f$ noise has been based on the physics of self-organized criticality. We show here that Taylor's law (when derived from sequential data using the method of expanding bins) implies 1/ $f$ noise, and that both phenomena can be explained by a central limit-like effect that establishes the class of Tweedie exponential dispersion models as foci for this convergence. These Tweedie models are probabilistic models characterized by closure under additive and reproductive convolution as well as under scale transformation, and consequently manifest a variance to mean power function. We provide examples of Taylor's law, 1/ $f$ noise, and multifractality within the eigenvalue deviations of the Gaussian unitary and orthogonal ensembles, and show that these deviations conform to the Tweedie compound Poisson distribution. The Tweedie convergence theorem provides a unified mathematical explanation for the origin of Taylor's law and 1/ $f$ noise applicable to a wide range of biological, physical, and mathematical processes, as well as to multifractality.

Received 15 July 2011

DOI:https://doi.org/10.1103/PhysRevE.84.066120

Authors & Affiliations

Wayne S. Kendal^1,* and Bent Jørgensen^2,†

¹Division of Radiation Oncology, University of Ottawa, 501 Smyth Road, Ottawa, Ontario, Canada K1H 8L6
²Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

^* wkendal@ottawahospital.on.ca
^† bentj@stat.sdu.dk

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Vol. 84, Iss. 6 — December 2011

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Figure 1
(a) GUE deviations. The deviations $|{\bar{D}}_{n}|$ between the observed and predicted number of eigenvalues below the size $E_{n}$ were estimated for a $10 000 \times 10 000$ matrix from the GUE, and are plotted here. (b) Variance function. The data sequence $|{\bar{D}}_{n}|$ was divided into a series of sequential, nonoverlapping and equal-sized counting bins; the values within each bin were summed and the mean and variance of these summed values were determined for the series of bins. The process was repeated for larger and larger bin sizes and the variances so obtained were plotted against their respective means on logarithmic axes. A power function relationship was obtained with $a = 0.483, p = 1.736, and r^{2} = 0.999 .$ (c) Power spectrum for a GUE ensemble. For each of ten repetitions the sequence of deviations was packed with zeros to yield $16 384$ points and the mean and trend were subtracted. A 49 point Hamming window was used to smooth these data and a fast Fourier transform was used to estimate the power spectrum by squaring the modulus of the transform. An ensemble average of the spectra was computed and then a log-log plot of these data revealed a power spectrum of the form $1 / f^{0.82}$ on linear regression. An initial flattening is evident over about the lowermost ten points of the spectrum, comprising about 0.1 $%$ of the data sequence. This flattening can be attributed to an artifact of the discrete Fourier transform caused by the low frequency component introduced by the rectangular data window.Reuse & Permissions
Figure 2
(a) GOE deviations. The deviations $| {\bar{D}}_{n} |$ between the observed and predicted number of eigenvalues below a specified size were estimated from a $10 000 \times 10 000$ matrix from the GUE, and are plotted here. (b) Variance function. A variance to mean power function was obtained with the values $a = 0.558, p = 1.856, and r^{2} = 1.000$ (c) Power spectrum for a GOE ensemble. Ten data sets were used to estimate an ensemble power spectrum, which on linear regression approximated the form $1 / f^{0.91}$ .Reuse & Permissions
Figure 3
Influence of matrix size on the power-law exponent. (a) GUE. Taylor's law exponent $p$ was estimated for the deviations $| {\bar{D}}_{n} |$ of various sized matrices from the GUE. For each matrix size the calculations were repeated ten times with independently derived data. The mean value of $p$ (solid dots) are provided along with their respective 95 $%$ CIs (error bars). For 30 × 30 matrices through to $10 000 \times 10 000$ matrices the mean values for $p$ ranged between 1.54 and 1.72. (b) GOE. Taylor's law exponent $p$ for the GOE deviations $| {\bar{D}}_{n} |$ were estimated for a range of matrix sizes and repeated ten times with independently derived data. The mean values of $p$ , and their corresponding 95 $%$ CIs, are provided graphically. These mean values ranged between 1.59 and 1.73 indicating that, as in the case of the GUE, the manifestations of Taylor's law were essentially independent of matrix size.Reuse & Permissions
Figure 4
Probability-probability plots. (a) GUE. An empirical CDF was derived from the data of Fig. 1(a), and a theoretical CDF, based on the Tweedie PG distribution [Eq. (25)], was fitted to the empirical CDF ( $θ = - 3.717, λ = 2.186, and α = - 0.333$ so that $p = 1.75$ ). The empirical CDF was plotted versus the theoretical CDF to yield a linear relationship, demonstrating consistency of the Tweedie model with the GUE data. (b) GOE. The empirical CDF derived from the data of Fig. 2(a) was plotted versus a theoretical PG CDF ( $θ = - 2.054, λ = 1.885, and α = - 0.358$ so that $p = 1.74$ ). The PG model was consistent with these data.Reuse & Permissions
Figure 5
Multifractal analysis. (a) Multifractal spectrum of the GUE deviations. The deviations $| {\bar{D}}_{n} |$ were estimated from a $10 000 \times 10 000$ matrix of the GUE and a multifractal analysis was conducted using analyzing wavelets based on the sixth derivative of the Gaussian function (software from http://www.physionet.org/physiotools/multifractal/). Here the multifractal scale exponent $τ (q)$ is plotted versus the dimensional index $q$ . (b) Singularity spectrum of the GUE deviations. The $D$ ( $h$ ) spectrum was obtained by Legendre transform of the numerical data in (a), and is plotted versus the Hölder exponent $h$ . (c) Multifractal spectrum of the GOE deviations. $| {\bar{D}}_{n} |$ were estimated from a $10 000 \times 10 000$ matrix of the GOE and analysis was conducted using analyzing wavelets based on the sixth derivative of the Gaussian function. (d) Singularity spectrum of the GOE deviations. This spectrum was obtained by Legendre transform of the numerical data in (c).Reuse & Permissions

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