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.
翻译:这是时间序列分析和复杂理论交汇过程中的一纸, 它在一般和具体变异复杂度方面呈现出新的结果。 在这种背景下, 变异复杂性是指时间序列的定性, 包括星系模式( 变异)、 温度测量、 缺失的星系模式的衰减率等等。 自从开始采用此\ textcolpoleft ordinal\ textcolentblight\ 方法以来, 它的实际应用到任何种类的变异性时间序列和真正估价的过程都证明是简单和有用的。 但是, 理论方面仍然局限于无噪音的变异性确定性序列和动态系统, 主要障碍是当数据中出现随机性( 也表现为观察性噪音) 时, 时序的变异性变化率和变异性( 变) 。 为了克服这一困难, 我们从一个新的变异性系统到一个变异性序列。 我们从每个变异性序列到下一个步骤, 由我们每个变动的变异性序列到每个变异性序列。