Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra, E-08193 Bellaterra, Catalonia, Spain.
Barcelona Graduate School of Mathematics, Edifici C, Campus Bellaterra, E-08193 Barcelona, Spain.
Phys Rev E. 2019 Dec;100(6-1):062106. doi: 10.1103/PhysRevE.100.062106.
Power-law-type distributions are extensively found when studying the behavior of many complex systems. However, due to limitations in data acquisition, empirical datasets often only cover a narrow range of observation, making it difficult to establish power-law behavior unambiguously. In this work we present a statistical procedure to merge different datasets, with two different aims. First, we obtain a broader fitting range for the statistics of different experiments or observations of the same system. Second, we establish whether two or more different systems may belong to the same universality class. By means of maximum likelihood estimation, this methodology provides rigorous statistical information to discern whether power-law exponents characterizing different datasets can be considered equal among them or not. This procedure is applied to the Gutenberg-Richter law for earthquakes and for synthetic earthquakes (acoustic emission events) generated in the laboratory: labquakes. Different earthquake catalogs have been merged finding a Gutenberg-Richter law holding for more than eight orders of magnitude in seismic moment. The value of the exponent of the energy distribution of labquakes depends on the material used in the compression experiments. By means of the procedure proposed in this manuscript, we find that the Gutenberg-Richter law for earthquakes and charcoal labquakes can be characterized by the same power-law exponent, whereas Vycor labquakes exhibit a significantly different exponent.
当研究许多复杂系统的行为时，会广泛发现幂律型分布。然而，由于数据采集的限制，经验数据集通常只覆盖观察的一个狭窄范围，使得明确建立幂律行为变得困难。在这项工作中，我们提出了一种统计程序来合并不同的数据集，具有两个不同的目的。首先，我们获得了更广泛的统计拟合范围，用于不同实验或同一系统的不同观察。其次，我们确定两个或更多不同的系统是否可能属于同一普遍性类别。通过最大似然估计，这种方法为辨别不同数据集的特征幂律指数是否可以认为彼此相等提供了严格的统计信息。该程序应用于地震的古登堡-里希特定律和实验室产生的合成地震（声发射事件）：实验室地震。通过合并不同的地震目录，我们发现地震矩的古登堡-里希特定律在超过八个数量级的范围内成立。实验室地震能量分布的指数值取决于压缩实验中使用的材料。通过本文提出的程序，我们发现地震和木炭实验室地震的古登堡-里希特定律可以用相同的幂律指数来描述，而 Vycor 实验室地震则表现出明显不同的指数。
