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→Scale invariant Tweedie distributions 标度不变的Tweedie分布
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>
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>
Tweedie distributions are a special case of exponential dispersion models, a class of statistical models used to describe error distributions for the generalized linear model and characterized by closure under additive and reproductive convolution as well as under scale transformation. 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 distributions, and extreme stable distributions.
Tweedie distributions are a special case of exponential dispersion models, a class of statistical models used to describe error distributions for the generalized linear model and characterized by closure under additive and reproductive convolution as well as under scale transformation. 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 distributions, and extreme stable distributions. Consequent to their inherent scale invariance Tweedie random variables Y demonstrate a variance var(Y) to mean E(Y) power law:
Consequent to their inherent scale invariance Tweedie random variables Y demonstrate a variance var(Y) to mean E(Y) power law:
: \text{var}\,(Y) = a[\text{E}\,(Y)]^p,
: \text{var}\,(Y) = a[\text{E}\,(Y)]^p,
where a and p are positive constants. This variance to mean power law is known in the physics literature as fluctuation scaling, and in the ecology literature as Taylor's law.
where a and p are positive constants. This variance to mean power law is known in the physics literature as fluctuation scaling, and in the ecology literature as Taylor's law.
Tweedie分布是指数弥散模型的一种特殊情况，是一类用于描述广义线性模型误差分布的统计模型，在可加卷积和再生卷积以及尺度变换下具有闭包性。这包括一些常见的分布：正态分布、泊松分布和伽玛分布，以及其他一些非同寻常的分布，如复合泊松-伽玛分布、正稳定分布和极端稳定分布。由于它们固有的标度不变性，Tweedie随机变量 y 显示方差var(''Y'')与均值E(''Y'')之间服从幂律关系：
<math>\text{var}\,(Y) = a[\text{E}\,(Y)]^p</math>，其中 a 和 p 是正常数。这种均值幂律的变化在物理学文献中称为涨落标度，在生态学文献中称为泰勒定律。
<math>\text{var}\,(Y) = a[\text{E}\,(Y)]^p</math>，
其中a和p是正常数。这种方差-均值的幂律关系在物理学文献中称为涨落标度，在生态学文献中称为泰勒定律。
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>
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>