Tweedie Distributions - The Double Power Law

The Double Power Law

The eponym Taylor's power law has been applied to a wide range of data that manifests a variance-to-mean power function. However, subtle mathematical differences exist between the transformational properties of some of these data. A double power law, which includes Taylor’s original law, has been proposed to describe these differences. For a population count drawn from an area of size t with mean abundance per unit area µ, and where

,

we have for the double power law:

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The constant d∈ has been identified as a fractal exponent. The factor aµ p represents the original Taylor’s law, a function of the mean abundance per unit area; the last term t2-d describes how the power law scales with the enumerative bin size. This last term implies a statistically self-similar scaling of the spatial distribution of items of interest as the size of the bin changes. It is this second portion of the double power law that underlies the variance-to-mean power law reported in systems like regional blood flow heterogeneity, the genomic distribution of SNPs and genes, and number theoretic examples.