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

  • Cohen J
  • Xu M
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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): Approxi- mately, variance = a (mean) b , a > 0. Multiple mechanisms have been proposed to explain and interpret TL. Here, we show analyt- ically 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 uncer- tainty. 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 re- gression. 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 in- tervention of any biological or behavioral mechanisms. This find- ing 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.

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Authors

  • Joel E. Cohen

  • Meng Xu

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