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More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. More context is needed here. Probability and entropy are definitely related to choosing a particular configuration, but it is not obvious how scale invariance is connected to this. This likelihood is given by the [[probability distribution]].

More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. More context is needed here. Probability and entropy are definitely related to choosing a particular configuration, but it is not obvious how scale invariance is connected to this. This likelihood is given by the [[probability distribution]].

更准确地说，随机系统中的标度变化涉及从所有可能的随机排列中选择一个特定排列的可能性。这一可能性可由概率分布给出。此外还需要更多的背景内容。概率和熵必然与一个特定排列的选定有关，但标度不变性与它们之间的相互联系还不明显。

更准确地说，随机系统中的标度变化涉及从所有可能的随机排列中选择一个特定排列的可能性。这一可能性可由概率分布给出。此外还需要更多的背景内容。概率和熵必然与一个特定排列的选定有关，但标度不变性与它们之间的相互联系还不明显。

Examples of scale-invariant distributions are the [[Pareto distribution]] and the [[Zipfian distribution]].

Examples of scale-invariant distributions are the [[Pareto distribution]] and the [[Zipfian distribution]].

标度不变分布的例子还有'''Pareto distribution 帕累托分布'''和'''Zipfian distribution 齐夫分布'''。

标度不变分布的例子还有'''Pareto distribution 帕累托分布'''和'''Zipfian distribution 齐夫分布'''。

Consequent to their inherent scale invariance Tweedie [[random variable]]s ''Y'' demonstrate a [[variance]] var(''Y'') to [[mean]] E(''Y'')  power law:

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>,

: <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>

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">Eisler, Z.; Bartos, I.; Kertész, J. (2008). "Fluctuation scaling in complex systems: Taylor's law and beyond". ''Adv Phys''. '''57''' (1): 89–142. arXiv:0708.2053. Bibcode:2008AdPhy..57...89E. doi:10.1080/00018730801893043. 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：

'''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标度变换下具有闭包性<ref name="Jørgensen1997" />。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'''Fluctuation Scaling 涨落标度'''，在生态学文献中称为'''Taylor's Law 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'''<ref name="Eisler2008" />，在生态学文献中称为'''Taylor's Law 泰勒定律<ref name="Kendal2011a" />'''。

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>

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。'''Wiener–Khinchin Theorem 维纳-辛钦定理'''进一步表明，在这些条件下，对于任何具有方差-均值幂律的序列，也会出现1/f噪声<ref name="Kendal2011" />。

 

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。'''Wiener–Khinchin Theorem 维纳-辛钦定理'''进一步表明，在这些条件下，对于任何具有方差-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" />

'''Tweedie Convergence Theorem Tweedie 收敛定理'''为涨落标度和1/f噪声的广泛出现提供了一个假设性解释。本质上，它要求任何一个可以渐近地显示方差-均值幂律的指数弥散模型，需要在Tweedie模型的吸引域内表达一个方差函数。几乎所有具有有限累积母函数的分布函数都符合指数弥散模型，而大多数指数弥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分布成为了不同数据类型收敛的焦点。

'''Tweedie Convergence Theorem Tweedie 收敛定理'''为涨落标度和1/f噪声的广泛出现提供了一个假设性解释。本质上，它要求任何一个可以渐近地显示方差-均值幂律的指数弥散模型，需要在Tweedie模型的吸引域内表达一个方差函数。几乎所有具有有限累积母函数的分布函数都符合指数弥散模型，而大多数指数弥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分布成为了不同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" />

正如中心极限定理要求某些类型的随机变量以高斯分布为收敛焦点并表示白噪声一样，Tweedie收敛定理要求某些非高斯随机变量来表达1/f噪声和涨落标度。

正如中心极限定理要求某些类型的随机变量以高斯分布为收敛焦点并表示白噪声一样，Tweedie收敛定理要求某些非高斯随机变量来表达1/f噪声和涨落标度。