Sample and population exponents of generalized Taylor's law.
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 = aM(b). 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 b(jk) for the scaling of the kth vs. the jth cumulants. The sample exponent b(jk) depends predictably on the number of samples and for finite samples we obtain b(jk) ≃ 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.
Laboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland; Department of Aquatic Ecology, Eawag: Swiss Federal Institute of Aquatic Science and Technology, CH-8600 Dübendorf, Switzerland; andrea.rinaldo@epfl.ch andrea.giometto@epfl.ch marco.formentin@ruhr-uni-bochum.de.
Andrea
A
Giometto
Dipartimento di Fisica ed Astronomia, Università di Padova, I-35131 Padova, Italy; Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, CZ-18208 Prague, Czech Republic; andrea.rinaldo@epfl.ch andrea.giometto@epfl.ch marco.formentin@ruhr-uni-bochum.de.
Marco
M
Formentin
Laboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland; Dipartimento di Ingegneria Civile, Edile ed Ambientale, Università di Padova, I-35131 Padova, Italy; and andrea.rinaldo@epfl.ch andrea.giometto@epfl.ch marco.formentin@ruhr-uni-bochum.de.
Andrea
A
Rinaldo
Laboratory of Populations, The Rockefeller University and Columbia University, New York, NY 10065-6399.
Joel E
JE
Cohen
Dipartimento di Fisica ed Astronomia, Università di Padova, I-35131 Padova, Italy;
Amos
A
Maritan
D036262:Empirical Research; D008962:Models, Theoretical; D011157:Population Dynamics
Markovian environment; environmental stochasticity; fluctuation scaling; multiplicative growth; power law