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 Jun;83(6 Pt 2):066115. doi: 10.1103/PhysRevE.83.066115. Epub 2011 Jun 22.
A power function relationship observed between the variance and the mean of many types of biological and physical systems has generated much debate as to its origins. This Taylor's law (or fluctuation scaling) has been recently hypothesized to result from the second law of thermodynamics and the behavior of the density of states. This hypothesis is predicated on physical quantities like free energy and an external field; the correspondence of these quantities with biological systems, though, remains unproven. Questions can be posed as to the applicability of this hypothesis to the diversity of observed phenomena as well as the range of spatial and temporal scales observed with Taylor's law. We note that the cumulant generating functions derived from this thermodynamic model correspond to those derived over a quarter century earlier for a class of probabilistic models known as the Tweedie exponential dispersion models. These latter models are characterized by variance-to-mean power functions; their phenomenological basis rests with a central-limit-theorem-like property that causes many statistical systems to converge mathematically toward a Tweedie form. We review evaluations of the Tweedie Poisson-gamma model for Taylor's law and provide three further cases to test: the clustering of single nucleotide polymorphisms (SNPs) within the horse chromosome 1, the clustering of genes within human chromosome 8, and the Mertens function. This latter case is a number theoretic function for which a thermodynamic model cannot explain Taylor's law, but where Tweedie convergence remains applicable. The Tweedie models are applicable to diverse biological, physical, and mathematical phenomena that express power variance functions over a wide range of measurement scales; they provide a probabilistic description for Taylor's law that allows mechanistic insight into complex systems without the assumption of a thermodynamic mechanism.
在许多生物和物理系统中，方差与均值之间存在幂函数关系，这一现象的起源引发了诸多争论。最近有人提出，这种泰勒定律（或波动标度）源于热力学第二定律和态密度的行为。这一假设基于自由能和外场等物理量；然而，这些量与生物系统的对应关系尚未得到证实。对于这一假设在各种观测现象中的适用性，以及用泰勒定律观测到的时空尺度范围，都可能存在疑问。我们注意到，从这个热力学模型导出的累积量生成函数，与二十多年前为一类称为Tweedie指数分散模型的概率模型所导出的函数相对应。后一类模型的特征是方差与均值的幂函数关系；它们的现象学基础在于一种类似于中心极限定理的性质，这种性质使得许多统计系统在数学上趋向于Tweedie形式。我们回顾了针对泰勒定律的Tweedie泊松 - 伽马模型的评估，并提供了另外三个案例进行检验：马1号染色体上单核苷酸多态性（SNP）的聚类、人类8号染色体上基因的聚类以及默滕斯函数。后一种情况是一个数论函数，对于它，热力学模型无法解释泰勒定律，但Tweedie收敛仍然适用。Tweedie模型适用于在广泛测量尺度上表现出幂方差函数的各种生物、物理和数学现象；它们为泰勒定律提供了一种概率描述，使得在不假设热力学机制的情况下，能够对复杂系统进行机理洞察。
