Kendal Wayne S, Jørgensen Bent
Division of Radiation Oncology, University of Ottawa, 501 Smyth Road, Ottawa, Ontario, Canada K1H 8L6.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Dec;84(6 Pt 2):066120. doi: 10.1103/PhysRevE.84.066120. Epub 2011 Dec 27.
Plants and animals of a given species tend to cluster within their habitats in accordance with a power function between their mean density and the variance. This relationship, Taylor's power law, has been variously explained by ecologists in terms of animal behavior, interspecies interactions, demographic effects, etc., all without consensus. Taylor's law also manifests within a wide range of other biological and physical processes, sometimes being referred to as fluctuation scaling and attributed to effects of the second law of thermodynamics. 1/f noise refers to power spectra that have an approximately inverse dependence on frequency. Like Taylor's law these spectra manifest from a wide range of biological and physical processes, without general agreement as to cause. One contemporary paradigm for 1/f noise has been based on the physics of self-organized criticality. We show here that Taylor's law (when derived from sequential data using the method of expanding bins) implies 1/f noise, and that both phenomena can be explained by a central limit-like effect that establishes the class of Tweedie exponential dispersion models as foci for this convergence. These Tweedie models are probabilistic models characterized by closure under additive and reproductive convolution as well as under scale transformation, and consequently manifest a variance to mean power function. We provide examples of Taylor's law, 1/f noise, and multifractality within the eigenvalue deviations of the Gaussian unitary and orthogonal ensembles, and show that these deviations conform to the Tweedie compound Poisson distribution. The Tweedie convergence theorem provides a unified mathematical explanation for the origin of Taylor's law and 1/f noise applicable to a wide range of biological, physical, and mathematical processes, as well as to multifractality.
特定物种的植物和动物往往会根据其平均密度与方差之间的幂函数，在其栖息地内聚集。这种关系，即泰勒幂定律，生态学家们从动物行为、种间相互作用、人口统计学效应等方面进行了各种解释，但都没有达成共识。泰勒定律也体现在广泛的其他生物和物理过程中，有时被称为波动标度，并归因于热力学第二定律的影响。1/f噪声是指功率谱对频率具有近似反比依赖性的情况。与泰勒定律一样，这些谱出现在广泛的生物和物理过程中，但其成因尚无普遍共识。一种关于1/f噪声的当代范式基于自组织临界性的物理学。我们在此表明，泰勒定律（当使用扩展箱方法从序列数据推导时）意味着1/f噪声，并且这两种现象都可以通过一种类似中心极限的效应来解释，该效应将Tweedie指数分散模型类确立为这种收敛的焦点。这些Tweedie模型是概率模型，其特征在于在加法和繁殖卷积以及尺度变换下封闭，因此表现出方差与均值的幂函数关系。我们在高斯酉系综和正交系综的特征值偏差中提供了泰勒定律、1/f噪声和多重分形的示例，并表明这些偏差符合Tweedie复合泊松分布。Tweedie收敛定理为泰勒定律和1/f噪声的起源提供了一个统一的数学解释，适用于广泛的生物、物理和数学过程以及多重分形。
