Random sampling of skewed distributions implies Taylor’s power law of fluctuation scaling

Abstract

Taylor's law (TL), a widely verified quantitative pattern in ecology and other sciences, describes the variance in a species' population density (or other nonnegative quantity) as a power-law function of the mean density (or other nonnegative quantity): Approximately, variance = a(mean) b , a > 0. Multiple mechanisms have been proposed to explain and interpret TL. Here, we show analytically that observations randomly sampled in blocks from any skewed frequency distribution with four finite moments give rise to TL. We do not claim this is the only way TL arises. We give approximate formulae for the TL parameters and their uncertainty. In computer simulations and an empirical example using basal area densities of red oak trees from Black Rock Forest, our formulae agree with the estimates obtained by least-squares regression. Our results show that the correlated sampling variation of the mean and variance of skewed distributions is statistically sufficient to explain TL under random sampling, without the intervention of any biological or behavioral mechanisms. This finding connects TL with the underlying distribution of population density (or other nonnegative quantity) and provides a baseline against which more complex mechanisms of TL can be compared. or equivalently as a linear function when mean and variance are logarithmically transformed:Eqs. 1 and 2 may be exact if the mean and variance are population moments calculated from certain parametric families of probability distributions (e.g., a = 1 and b = 1 for a Poisson distribution). Eqs. 1 and 2 may be approximate if the mean and variance are sample moments based on finite random samples of observations. Most empirical tests of TL have not specified the random error associated with Eqs. 1 or 2. TL has been verified for hundreds of biological species and nonbiological quantities in more than a thousand papers in ecology, epidemiology, biomedical sciences, and other fields (2-4). Recently, examples of TL were found in bacterial microcosms (5, 6), forest trees (7, 8), human populations (9), coral reef fish populations (10), and barnacles (11,12). TL has been used practically in the design of sampling plans for the control of insect pests of soybeans (13,14) and cotton (15).Scientific studies of TL largely focus on the power-law exponent b (or slope b in the linear form), which Taylor believed to contain information about how populations of a species aggregate in space (1). Empirically, b often lies between 1 and 2 (16). Ballantyne and Kerkhoff (17) suggested that individuals' reproductive correlation determines the size of b. Ballantyne (18) proposed that b = 2 is a consequence of deterministic population growth. Cohen (19) showed that b = 2 arose from exponentially growing, noninteracting clones. Kilpatrick and Ives (20) proposed that interspecific competition could reduce the value of b. Other models that implied TL were the exponential dispersion model (21-23), models of spatially distributed colonies (24, 25), a stochastic version of logistic population dyn...

Synchrony affects Taylor’s law in theory and data

“…The linearity hypothesis of TL was usually, but not always, an adequate approximation in that linearity and homoscedasticity could not be rejected statistically (SI Appendix, S6 has details on how this was tested). In agreement with our theorem and Cohen and Xu (15), when a shifted normal distribution (which has skewness 0) was used for Y i , b ≈ 0 for all values of Ω. For skewed distributions, the slope b was generally smaller for larger values of Ω, confirming the prediction that b depends on synchrony.…”

“…2 was computed analytically (i.e., with formulas) for gamma, exponential, χ 2 , normal, and log-normal examples, and the formulas were compared with numerical results. For some distributions and parameters, the approximation was very accurate, and it was always at least qualitatively accurate (in the sense that it showed similar declines of b with increasing synchrony), except for the lognormal distribution, for which it was very inaccurate for some parameters because of insufficient sampling as previously observed (15). As expected from the theorem, Eq.…”

“…This theorem extends a theorem by Cohen and Xu (15), which assumes that the Y i are independent and identically distributed (iid). In that case, the second factor on the right of Eq.…”

“…Independence of the Y i is not necessary here: the same formula holds if ρ ij = 0 for i ≠ j and ρ ijk = 0 whenever i, j, and k are not all equal. Cohen and Xu (15) concluded that, in the iid case, skewness of Y i is necessary and sufficient for TL to have slope b ≠ 0. Our theorem extends this result to the case of identically distributed Y i that may be nonindependent.…”

“…TL has been generalized (10) and applied or proposed for application to fisheries management (3,4), estimation of species persistence times (11), and agriculture (2,12,13). Potential mechanisms of TL have been explored extensively (9,14,15). Because of its ubiquity, it has been suggested that TL could be another "universal law," like the central limit theorem (16).…”