Upper-truncated power laws in natural systems
Upper-truncated power laws in natural systems
复制标题
DOI：
10.1007/pl00001202
复制
发表时间：
2001-04-01
影响因子：
2
通讯作者：
Tebbens, SF
中科院分区：
文献类型：
--
作者：
Burroughs, SM;Tebbens, SF
当数据的累积数量-大小分布遵循幂定律时，数据集通常被认为是分形的，因为幂定律和分形图都是尺度不变的。随着测量对象尺寸的增加，许多自然现象的数据集的累积数量-尺寸分布呈现出从幂定律的“下降”。我们证明，当累积数据集在大对象大小下被截断时，这种下降是预期的。我们提供了一个广义方程，这里称为通用拟合函数(GFF)，它描述了基于幂定律的上截断累积数-尺寸分布。将GFF拟合到累积量-大小分布，得到了基本幂定律的系数和指数，以及表征上截断的参数。上截断的可能原因包括数据采样限制(空间或时间)和控制对象大小的物理变化。我们使用GFF方法分析了其他方法研究过的四个自然系统：澳大利亚首都地区的森林火区；Vernejul煤田的断层偏移量；Frio Strand Plain勘探区块的碳氢化合物体积；以及金星上的断层长度。我们证明，传统的将幂定律直接拟合到累积数-大小分布的方法对幂定律的指数估计太负，并高估了数据集的分维。我们考虑的四个系统用GFF方法进行了很好的拟合，表明它们具有上截断幂定律的性质。
When a cumulative number-size distribution of data follows a power law, the data set is often considered fractal since both power laws and fractals are scale invariant. Cumulative number-size distributions for data sets of many natural phenomena exhibit a "fall-off" from a power law as the measured object size increases. We demonstrate that this fall-off is expected when a cumulative data set is truncated at large object size. We provide a generalized equation, herein called the General Fitting Function (GFF), that describes an upper-truncated cumulative number-size distribution based on a power law. Fitting the GFF to a cumulative number-size distribution yields the coefficient and exponent of the underlying power law and a parameter that characterizes the upper truncation. Possible causes of upper truncation include data sampling limitations (spatial or temporal) and changes in the physics controlling the object sizes. We use the GFF method to analyze four natural systems that have been studied by other approaches: forest fire area in the Australian Capital Territory; fault offsets in the Vernejoul coal field; hydrocarbon volumes in the Frio Strand Plain exploration play; and fault lengths on Venus. We demonstrate that a traditional approach of fitting a power law directly to the cumulative number-size distribution estimates too negative an exponent for the power law and overestimates the fractal dimension of the data set. The four systems we consider are well fit by the GFF method, suggesting they have properties characterized by upper-truncated power laws.