Editing Scale invariance (section)

===Scale-invariant Tweedie distributions===
'''[[Tweedie distributions]]''' are a special case of '''[[exponential dispersion model]]s''', a class of statistical models used to describe error distributions for the [[generalized linear model]] and characterized by [[Closure (mathematics)|closure]] under additive and reproductive convolution as well as under scale transformation.<ref name="Jørgensen1997">{{cite book |last=Jørgensen |first=B. |year=1997 |title=The Theory of Dispersion Models |publisher=Chapman & Hall |location=London |isbn=978-0412997112 }}</ref>  These include a number of common distributions: the [[normal distribution]], [[Poisson distribution]] and [[gamma distribution]], as well as more unusual distributions like the compound Poisson-gamma distribution, positive [[stable distribution]]s, and extreme stable distributions.
Consequent to their inherent scale invariance Tweedie [[random variable]]s ''Y'' demonstrate a [[variance]] var(''Y'') to [[mean]] E(''Y'')  power law:
: <math>\text{var}\,(Y) = a[\text{E}\,(Y)]^p</math>,
where ''a'' and ''p'' are positive constants. This variance to mean power law is known in the physics literature as '''fluctuation scaling''',<ref name="Eisler2008">{{cite journal |last1=Eisler |first1=Z. |last2=Bartos |first2=I. |last3=Kertész |first3=J. |year=2008 |title=Fluctuation scaling in complex systems: Taylor's law and beyond |journal=[[Advances in Physics|Adv Phys]] |volume=57 |issue=1 |pages=89–142 |doi=10.1080/00018730801893043 |arxiv = 0708.2053 |bibcode = 2008AdPhy..57...89E |s2cid=119608542 }}</ref> and in the ecology literature as [[Taylor's law]].<ref name="Kendal2011a">{{cite journal |last1=Kendal |first1=W. S. |last2=Jørgensen |first2=B. |year=2011 |title=Taylor's power law and fluctuation scaling explained by a central-limit-like convergence |journal=Phys. Rev. E |volume=83 |issue=6 |pages=066115 |doi=10.1103/PhysRevE.83.066115 |pmid=21797449 |bibcode = 2011PhRvE..83f6115K }}</ref>

Random sequences, governed by the Tweedie distributions and evaluated by the [[Tweedie distributions|method of expanding bins]] exhibit a [[Logical biconditional|biconditional]] relationship between the variance to mean power law and power law [[autocorrelation]]s.  The [[Wiener–Khinchin theorem]] further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest [[pink noise|''1/f'' noise]].<ref name="Kendal2011">{{cite journal |last1=Kendal |first1=W. S. |last2=Jørgensen |first2=B. |year=2011 |title=Tweedie convergence: A mathematical basis for Taylor's power law, 1/''f'' noise, and multifractality |journal=Phys. Rev. E |volume=84 |issue=6 |pages=066120 |doi=10.1103/PhysRevE.84.066120 |bibcode = 2011PhRvE..84f6120K |pmid=22304168|url=https://findresearcher.sdu.dk:8443/ws/files/55639035/e066120.pdf }}</ref>

The [[Tweedie distributions|'''Tweedie convergence theorem''']] provides a hypothetical explanation for the wide manifestation of fluctuation scaling and ''1/f'' noise.<ref name="Jørgensen1994">{{cite journal |last1=Jørgensen |first1=B. |last2=Martinez |first2=J. R. |last3=Tsao |first3=M. |year=1994 |title=Asymptotic behaviour of the variance function |journal=[[Scandinavian Journal of Statistics|Scand J Statist]] |volume=21 |issue=3 |pages=223–243 |jstor=4616314 }}</ref> It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a [[natural exponential family|variance function]] that comes within the [[Attractor|domain of attraction]] of a Tweedie model.  Almost all distribution functions with finite [[cumulant|cumulant generating functions]] qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form.  Hence many probability distributions have variance functions that express this [[Asymptotic expansion|asymptotic behavior]], and  the Tweedie distributions become foci of convergence for a wide range of data types.<ref name="Kendal2011" />

Much as the [[central limit theorem]] requires certain kinds of random variables to have as a focus of convergence the [[normal distribution|Gaussian distribution]] and express [[white noise]], the Tweedie convergence theorem requires certain non-Gaussian random variables to express ''1/f'' noise and fluctuation scaling.<ref name="Kendal2011" />