基于复杂度的置换熵：从确定性时间序列到白噪声
Complexity-based permutation entropies: from deterministic time series to white noise
摘要：
这是一个文件的交叉时间序列分析和复杂性理论，提出了新的成果，排列复杂性一般和排列熵特别。在这种情况下，排列复杂性是指通过顺序模式（排列），熵测度，缺失顺序模式的衰减率等来表征时间序列。自\textquotedblleft序数\textquotedblright \方法学诞生以来，它对任何类型的标量时间序列和实值过程的实际应用已被证明是简单而有用的。然而，理论方面仍然局限于无噪声的确定性序列和动力系统，主要障碍是当数据中存在随机性（也以观测噪声的形式）时，可见排列随长度的超指数增长。为了克服这个困难，我们采取了一种新的方法，通过复杂性类，这是精确定义的增长的可见排列长度，无论数据的确定性或噪声的性质。我们考虑三个主要类别：指数，子阶乘和阶乘。下一步是将Z-熵的概念适用于这些类中的每一个，我们称之为置换熵，因为它与指数类上的传统置换熵一致。Z-熵是一个群熵族，每个群熵在给定的复杂性类上都是广泛的。其结果是一个统一的方法，确定性和随机过程的顺序分析，从动力系统白噪声，新的概念和工具。数值模拟表明，置换熵区分时间序列的所有复杂性类。
英文摘要：
This is a paper in the intersection of time series analysis and complexity theory that presents new results on permutation complexity in general and permutation entropy in particular. In this context, permutation complexity refers to the characterization of time series by means of ordinal patterns (permutations), entropic measures, decay rates of missing ordinal patterns, and more. Since the inception of this \textquotedblleft ordinal\textquotedblright\ methodology, its practical application to any type of scalar time series and real-valued processes have proven to be simple and useful. However, the theoretical aspects have remained limited to noiseless deterministic series and dynamical systems, the main obstacle being the super-exponential growth of visible permutations with length when randomness (also in form of observational noise) is present in the data. To overcome this difficulty, we take a new approach through complexity classes, which are precisely defined by the growth of visible permutations with length, regardless of the deterministic or noisy nature of the data. We consider three major classes: exponential, sub-factorial and factorial. The next step is to adapt the concept of Z-entropy to each of those classes, which we call permutation entropy because it coincides with the conventional permutation entropy on the exponential class. Z-entropies are a family of group entropies, each of them extensive on a given complexity class. The result is a unified approach to the ordinal analysis of deterministic and random processes, from dynamical systems to white noise, with new concepts and tools. Numerical simulations show that permutation entropy discriminates time series from all complexity classes.
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