One of the most widely verified empirical regularities of ecology is Taylor’s power law of fluctuation scaling, or simply Taylor’s law (TL). TL says that the logarithm of the variances of a set of random variables or a set of random samples is (exactly or approximately) a linear function of logarithm of the means of the corresponding random variables or random samples: logvariance = log a + b log mean, a > 0. Ecologists have argued about the interpretation of the intercept log a and slope b of TL and about what the values of these parameters reveal about the underlying probability distributions of the random samples. We show here that the form and the values of the parameters of TL and of any other variance function (relationship of variance to mean in a set of samples or a family of random variables) say nothing whatsoever about the underlying probability distributions of the random samples (or random variables) other than that they have finite mean and variance. Specifically, given any real-valued random variable with a finite mean and a finite variance, and given any variance function (e.g., TL with specified intercept log a and slope b), we construct a family of random variables with probability distributions of the same shape as the probability distribution of the given random variable (i.e., that are the same up to location and scale, or in the same “location-scale family”) and that obeys the given variance function exactly (e.g., TL exactly with the given intercept log a and slope b). Every variance function can be produced by the location-scale family of any random variable with finite positive mean and finite positive variance. We illustrate some consequences of these findings by examples (e.g., for presence-absence sampling in agricultural pest control).
Taylor’s power law Variance function Presence-absence Exponential dispersion model Tweedie distribution Power law Bartlett’s variance function Negative binomial distribution
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I thank Yannis Michalakis for a provocative question, Robin A. J. Taylor for identifying the source of an equation in his book, Alan Hastings and anonymous reviewers for helpful suggestions, and Roseanne Benjamin for assistance.