They have the property that for n independent random variables Yi ~ ED(μ,σ2/wi), with weighting factors wi and
Since L. R. Taylor described this law in 1961 there have been many different explanations offered to explain it, ranging from animal behavior,[9] a random walk model,[10] a stochastic birth, death, immigration and emigration model,[11] to a consequence of equilibrium and non-equilibrium statistical mechanics.
[13][14] The majority of the observed values for the power-law exponent p have fallen in the interval (1,2) and so the Tweedie compound Poisson–gamma distribution would seem applicable.
as k→∞ and where L(k) is a slowly varying function at large values of k, this sequence is called a self-similar process.
Provided the autocorrelation function exhibits the same behavior, the additive sequences will obey the relationship
[8] The Tweedie convergence theorem thus provides an alternative explanation for the origin of 1/f noise, based its central limit-like effect.
A one-dimensional data sequence of self-similar data may demonstrate a variance-to-mean power law with local variations in the value of p and hence in the value of D. When fractal structures manifest local variations in fractal dimension, they are said to be multifractals.
Examples of data sequences that exhibit local variations in p like this include the eigenvalue deviations of the Gaussian Orthogonal and Unitary Ensembles.
The amount of radioactivity within each cube is taken to reflect the blood flow through that sample at the time of injection.
It is possible to evaluate adjacent cubes from an organ in order to additively determine the blood flow through larger regions.
Regional organ blood flow can thus be modelled by the Tweedie compound Poisson–gamma distribution.,[23] In this model tissue sample could be considered to contain a random (Poisson) distributed number of entrapment sites, each with gamma distributed blood flow.
Blood flow at this microcirculatory level has been observed to obey a gamma distribution,[24] thus providing support for this hypothesis.
The "experimental cancer metastasis assay"[25] has some resemblance to the above method to measure regional blood flow.
Groups of syngeneic and age matched mice are given intravenous injections of equal-sized aliquots of suspensions of cloned cancer cells and then after a set period of time their lungs are removed and the number of cancer metastases enumerated within each pair of lungs.
It has been long recognized that there can be considerable intraclonal variation in the numbers of metastases per mouse despite the best attempts to keep the experimental conditions within each clonal group uniform.
[28] Since hematogenous metastasis occurs in direct relationship to regional blood flow[29] and videomicroscopic studies indicate that the passage and entrapment of cancer cells within the circulation appears analogous to the microsphere experiments[30] it seemed plausible to propose that the variation in numbers of hematogenous metastases could reflect heterogeneity in regional organ blood flow.
For sparse data, however, this discrete variance-to-mean relationship would behave more like that of a Poisson distribution where the variance equaled the mean.
The local density of Single Nucleotide Polymorphisms (SNPs) within the human genome, as well as that of genes, appears to cluster in accord with the variance-to-mean power law and the Tweedie compound Poisson–gamma distribution.
[32][33] In the case of SNPs their observed density reflects the assessment techniques, the availability of genomic sequences for analysis, and the nucleotide heterozygosity.
R R Hudson has proposed a model where recombination could cause variation in the time to most common recent ancestor for different genomic segments.
[35] A high recombination rate could cause a chromosome to contain a large number of small segments with less correlated genealogies.
Assuming a constant background rate of mutation the number of SNPs per genomic segment would accumulate proportionately to the time to the most recent common ancestor.
The distribution of genes within the human genome also demonstrated a variance-to-mean power law, when the method of expanding bins was used to determine the corresponding variances and means.
[33] Similarly the number of genes per enumerative bin was found to obey a Tweedie compound Poisson–gamma distribution.
This probability distribution was deemed compatible with two different biological models: the microarrangement model where the number of genes per unit genomic length was determined by the sum of a random number of smaller genomic segments derived by random breakage and reconstruction of protochormosomes.
Over large evolutionary timescales there would occur tandem duplication, mutations, insertions, deletions and rearrangements that could affect the genes through a stochastic birth, death and immigration process to yield the Tweedie compound Poisson–gamma distribution.
Both these mechanisms would implicate neutral evolutionary processes that would result in regional clustering of genes.
The Gaussian unitary ensemble (GUE) consists of complex Hermitian matrices that are invariant under unitary transformations whereas the Gaussian orthogonal ensemble (GOE) consists of real symmetric matrices invariant under orthogonal transformations.
we obtain a sequence of eigenvalue fluctuations which, using the method of expanding bins, reveals a variance-to-mean power law.
[citation needed] Moreover, these deviations correspond to the Tweedie compound Poisson-gamma distribution and they exhibit 1/f noise.