Sample and population exponents of generalized Taylor’s law

Significance Taylor’s law (TL) has been verified very widely in the natural sciences, information technology, and finance. The widespread observation of TL suggests that a context-independent mechanism may be at work and stimulated the search for processes affecting the scaling of population fluctuations with population abundance. We show that limited sampling may explain why TL is often observed to have exponent b=2. Abrupt transitions in the TL exponent associated with smooth changes in the environment were recently discovered theoretically and comparable real-world transitions could harm fish populations, forests, and public health. Our study shows that limited sampling hinders the anticipation of such transitions and provides estimates for the number of samples required to reveal early warning signals of abrupt biotic change. Taylor’s law (TL) states that the variance V of a nonnegative random variable is a power function of its mean M; i.e., V=aMb. TL has been verified extensively in ecology, where it applies to population abundance, physics, and other natural sciences. Its ubiquitous empirical verification suggests a context-independent mechanism. Sample exponents b measured empirically via the scaling of sample mean and variance typically cluster around the value b=2. Some theoretical models of population growth, however, predict a broad range of values for the population exponent b pertaining to the mean and variance of population density, depending on details of the growth process. Is the widely reported sample exponent b≃2 the result of ecological processes or could it be a statistical artifact? Here, we apply large deviations theory and finite-sample arguments to show exactly that in a broad class of growth models the sample exponent is b≃2 regardless of the underlying population exponent. We derive a generalized TL in terms of sample and population exponents bjk for the scaling of the kth vs. the jth cumulants. The sample exponent bjk depends predictably on the number of samples and for finite samples we obtain bjk≃k/j asymptotically in time, a prediction that we verify in two empirical examples. Thus, the sample exponent b≃2 may indeed be a statistical artifact and not dependent on population dynamics under conditions that we specify exactly. Given the broad class of models investigated, our results apply to many fields where TL is used although inadequately understood.