Sample and population exponents of generalized Taylor's law

TL;DR

This paper explains why empirical Taylor's law often shows an exponent around 2 by deriving a generalized law using large deviations theory, revealing the dependence of sample exponents on observation count.

Contribution

It introduces a generalized Taylor's law linking sample and population exponents and explains the observed empirical consistency of b≈2 through theoretical derivation.

Findings

Sample exponents depend predictably on the number of observations.

Empirical data supports the theoretical predictions.

Finite observations lead to sample exponents around 2 regardless of population exponents.

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 $b\simeq2$) independently of population exponents. Empirical analyses on two datasets support our theoretical results.