Sample and population exponents of generalized Taylor's law

Abstract

Taylor's law (TL) states that the variance $V$ of a non-negative random variable is a power function of its mean $M$, i.e. $V=a{M}^b$. The ubiquitous empirical verification of TL, typically displaying sample exponents $b\simeq 2$, suggests a context-independent mechanism. However, theoretical studies of population dynamics predict a broad range of values of $b$. Here, we explain this apparent contradiction by using large deviations theory to derive a generalized TL in terms of sample and populations exponents $b_{jk}$ for the scaling of the $k$-th vs the $j$-th cumulant (conventional TL is recovered for $b=b_{12}$), with the sample exponent found to depend predictably on the number of observed samples. Thus, for finite numbers of observations one observes sample exponents $b_{jk}\simeq k/j$ (thus $b\simeq 2$) independently of population exponents. Empirical analyses on two datasets support our theoretical results.

41 pages, 10 figures, 6 tables

References in corpus (2)

• The large deviation approach to statistical mechanics
• Fluctuation scaling in complex systems: Taylor's law and beyond

Cited by in corpus (14)

• Colloquium: Criticality and dynamical scaling in living systems
• Covariations in ecological scaling laws fostered by community dynamics
• Taylor's law in innovation processes
• Zipf's and Taylor's Laws
• Spatiotemporal fluctuation scaling law and metapopulation modeling of the novel coronavirus (COVID-19) and SARS outbreaks
• Joint assessment of density correlations and fluctuations for analysing spatial tree patterns
• Spatial patterns emerging from a stochastic process near criticality
• Statistical properties of fluctuations of time series representing the appearance of words in nationwide blog data and their applications: An example of observations and the modelling of fluctuation scalings of nonstationary time series
• Taylor's power law: before and after 50 years of scientific scrutiny
• Spatial and Temporal Taylor's Law in 1-Dim Chaotic Maps
• Taylor's Law of temporal fluctuation scaling in stock illiquidity
• Generalized Lotka-Volterra model with sparse interactions: non-Gaussian effects and topological multiple-equilibria phase
• How do simple evolutionary strategies and investment optimizations affect ecological patterns? The case of generalized Taylor's Law
• Waveform Proportionality and Taylor's Law Induced by Synchronization of Periodic and Chaotic Oscillators