Inferring the parameters of Taylor’s power law of fluctuation scaling

Taylor’s power law of fluctuation scaling has been well approximated empirically in many fields, including physics, meteorology, computer science, finance and ecology. In ecology, it describes well the abundances of many species, humans and non-humans. Taylor’s law asserts that, in a collection of probability distributions, the variance of each distribution is directly proportional to a power of the mean of each distribution, exactly for population moments and, whether or not population moments exist, approximately for sample moments. Linear regression of log variance as a function of log mean is widely used to estimate the parameters of Taylor’s power law. We provide large-sample asymptotics for this kind of inference under general conditions, and we derive confidence intervals for the parameters. In many ecological applications, the means and variances are estimated over time or across space from arrays of abundance data collected at different locations and time points. Our results depend on the asymptotic relation between the time-series length and the number of spatial points. When their ratio converges to a constant as both become large, the usual normalized statistics are asymptotically biased. We provide a bias correction to get correct confidence intervals. Taylor’s law, widely studied in multiple sciences, is a source of challenging new statistical problems in a non-stationary spatio-temporal framework. We illustrate our results with both simulated and real data.