Amigó José M, Dale Roberto, Tempesta Piergiulio
Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain.
Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, 28040 Madrid, Spain and Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, No. 13-15, 28049 Madrid, Spain.
Chaos. 2021 Jan;31(1):013115. doi: 10.1063/5.0023419.
Permutation entropy measures the complexity of a deterministic time series via a data symbolic quantization consisting of rank vectors called ordinal patterns or simply permutations. Reasons for the increasing popularity of this entropy in time series analysis include that (i) it converges to the Kolmogorov-Sinai entropy of the underlying dynamics in the limit of ever longer permutations and (ii) its computation dispenses with generating and ad hoc partitions. However, permutation entropy diverges when the number of allowed permutations grows super-exponentially with their length, as happens when time series are output by dynamical systems with observational or dynamical noise or purely random processes. In this paper, we propose a generalized permutation entropy, belonging to the class of group entropies, that is finite in that situation, which is actually the one found in practice. The theoretical results are illustrated numerically by random processes with short- and long-term dependencies, as well as by noisy deterministic signals.
排列熵通过由称为序数模式或简称为排列的秩向量组成的数据符号量化来度量确定性时间序列的复杂性。这种熵在时间序列分析中越来越受欢迎的原因包括：（i）在排列长度不断增加的极限情况下，它收敛于基础动力学的柯尔莫哥洛夫 - 西奈熵；（ii）其计算无需生成和特设划分。然而，当允许的排列数量随着其长度呈超指数增长时，排列熵会发散，在时间序列由具有观测或动态噪声的动力系统或纯随机过程输出时就会出现这种情况。在本文中，我们提出了一种广义排列熵，它属于群熵类别，在那种情况下是有限的，而这实际上就是在实践中发现的情况。理论结果通过具有短期和长期依赖性的随机过程以及有噪声的确定性信号进行了数值说明。
