Permutation Entropy (PE) has been shown to be a useful tool for time series analysis due to its low computational cost and noise robustness. This has drawn for its successful application in many fields. Some of these include damage detection, disease forecasting, and financial volatility analysis. However, to successfully use PE, an accurate selection of two parameters is needed: the permutation dimension $n$ and embedding delay $\tau$. These parameters are often suggested by experts based on a heuristic or by a trial and error approach. unfortunately, both of these methods can be time-consuming and lead to inaccurate results. To help combat this issue, in this paper we investigate multiple schemes for automatically selecting these parameters with only the corresponding time series as the input. Specifically, we develop a frequency-domain approach based on the least median of squares and the Fourier spectrum, as well as extend two existing methods: Permutation Auto-Mutual Information (PAMI) and Multi-scale Permutation Entropy (MPE) for determining $\tau$. We then compare our methods as well as current methods in the literature for obtaining both $\tau$ and $n$ against expert-suggested values in published works. We show that the success of any method in automatically generating the correct PE parameters depends on the category of the studied system. Specifically, for the delay parameter $\tau$, we show that our frequency approach provides accurate suggestions for periodic systems, nonlinear difference equations, and ECG/EEG data, while the mutual information function computed using adaptive partitions provides the most accurate results for chaotic differential equations. For the permutation dimension $n$, both False Nearest Neighbors and MPE provide accurate values for $n$ for most of the systems with $n = 5$ being suitable in most cases.

中文翻译：

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关于置换熵的自动参数选择

由于其低计算成本和噪声鲁棒性，置换熵 (PE) 已被证明是用于时间序列分析的有用工具。这吸引了它在许多领域的成功应用。其中一些包括损害检测、疾病预测和金融波动分析。但是，要成功使用 PE，需要准确选择两个参数：排列维度 $n$ 和嵌入延迟 $\tau$。这些参数通常由专家基于启发式或试错法建议。不幸的是，这两种方法都可能非常耗时并导致结果不准确。为了帮助解决这个问题，在本文中，我们研究了多种方案，用于仅以相应的时间序列作为输入自动选择这些参数。具体来说，我们开发了一种基于最小平方中值和傅立叶频谱的频域方法，并扩展了两种现有方法：用于确定 $\tau$ 的置换自动互信息 (PAMI) 和多尺度置换熵 (MPE) . 然后，我们将我们的方法以及文献中当前获取 $\tau$ 和 $n$ 的方法与已发表作品中专家建议的值进行比较。我们表明，任何自动生成正确 PE 参数的方法的成功都取决于所研究系统的类别。具体来说，对于延迟参数 $\tau$，我们表明我们的频率方法为周期系统、非线性差分方程和 ECG/EEG 数据提供了准确的建议，而使用自适应分区计算的互信息函数为混沌微分方程提供了最准确的结果。对于置换维度 $n$，False Nearest Neighbors 和 MPE 都为大多数系统提供了 $n$ 的准确值，其中 $n = 5$ 适用于大多数情况。