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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：

'''Tweedie分布'''是'''Exponential Dispersion Models 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、'''Poisson distribution 泊松分布'''和'''Gamma Distribution 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>，

<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。

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'''Fluctuation Scaling 涨落标度'''，在生态学文献中称为'''Taylor's Law 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>

Random sequences, governed by the Tweedie distributions and evaluated by the method of expanding bins exhibit a biconditional relationship between the variance to mean power law and power law autocorrelations.  The Wiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest 1/f noise.

Random sequences, governed by the Tweedie distributions and evaluated by the method of expanding bins exhibit a biconditional relationship between the variance to mean power law and power law autocorrelations.  The Wiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest 1/f noise.

随机序列，由 Tweedie 分布控制，并用扩箱法进行评价，表现出方差到均方幂律和幂律自相关的双重关系。维纳-钦钦定理进一步暗示，对于任何在这些条件下表现出方差到平均幂律的序列，也会表现出1/f 噪声。

随机序列由Tweedie分布控制，并通过展开箱的方法进行评估，在方差均值幂律和幂律自相关之间表现出双条件关系。维纳-钦钦定理进一步表明，在这些条件下，对于任何具有方差到均值幂律的序列，也会出现1/f噪声

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" />

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" />

The Tweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and 1/f noise. It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a variance function that comes within the domain of attraction of a Tweedie model.  Almost all distribution functions with finite 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 behavior, and  the Tweedie distributions become foci of convergence for a wide range of data types.

The Tweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and 1/f noise. It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a variance function that comes within the domain of attraction of a Tweedie model.  Almost all distribution functions with finite 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 behavior, and  the Tweedie distributions become foci of convergence for a wide range of data types.

Tweedie 收敛定理为涨落标度和1/f 噪声的广泛表现提供了一个假设性的解释。本质上，它要求任何渐近显示出均值幂律方差的指数离散模型都必须表示出一个方差函数，这个方差函数来自于 Tweedie 模型的吸引域。几乎所有具有有限累积量母函数的分布函数都可以称为指数离散模型，大多数指数离散模型都具有这种形式的方差函数。因此，许多概率分布具有表示这种渐近性的方差函数，而 Tweedie 分布成为各种数据类型收敛的焦点。

Tweedie收敛定理为波动缩放和1/f噪声的广泛表现提供了一个假设的解释。它要求，本质上，任何指数色散模型，渐近地显示一个方差到均值幂律，将需要表达一个方差函数，在特威迪模型的吸引范围内。几乎所有具有有限累积母函数的分布函数都符合指数离散模型，而大多数指数离散模型都表现出这种形式的方差函数。因此，许多概率分布都有表达这种渐近行为的方差函数，而Tweedie分布成为了广泛数据类型收敛的焦点。

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" />

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" />

Much as the central limit theorem requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express 1/f noise and fluctuation scaling.

Much as the central limit theorem requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express 1/f noise and fluctuation scaling.

正如中心极限定理需要某些类型的随机变量作为聚焦点来收敛正态分布和表示白噪声一样，Tweedie 收敛定理需要某些非高斯随机变量来表示1/f 噪声和波动尺度。

正如中心极限定理要求某些类型的随机变量作为收敛高斯分布和表达白噪声的焦点一样，Tweedie收敛定理要求某些非高斯随机变量来表达1/f噪声和波动缩放。

===Cosmology 宇宙学===<!-- This section is linked from [[Cosmic inflation]] -->

===Cosmology 宇宙学===<!-- This section is linked from [[Cosmic inflation]] -->