Efficient and robust synchronization estimation, namely refined cross-sample entropy (RCSE) measure, is presented in this study to analyze complex time series. Unlike the original cross-sample entropy (CSE) that relies on a fixed tolerance r, the proposed RCSE is based on a concept called the cumulative histogram method (CHM) to gain a range of entropy values with different r selections in a certain range. Moreover, the dissimilarity measure in RCSE is redefined, rather than the distance function used in the CSE. Trials are conducted over both simulated and real-world data for providing a comparative study. The original CSE is introduced as a comparison to testify that the proposed RCSE is capable of drawing more specific relationships from time series, and it is also a superior method to describe the synchronization between them. The results show that the new method is capable of distinguishing different kinds of time series and possesses robustness in the noise test. It is suggest that the proposed RCSE may potentially become a new reliable method for synchronization estimation of complex time series.

为了分析复杂的时间序列，本研究提出了一种有效且鲁棒的同步估计，即改进的交叉样本熵（RCSE）度量。与原始的交叉样本熵（CSE）依赖于固定的容差r不同，建议的RCSE基于称为累积直方图方法（CHM）的概念来获得具有不同r的一系列熵值在一定范围内选择。此外，重新定义了RCSE中的相异性度量，而不是CSE中使用的距离函数。对模拟和真实数据进行了试验，以提供比较研究。引入原始CSE作为比较，以证明所建议的RCSE能够从时间序列中得出更具体的关系，它也是描述它们之间的同步的一种较好方法。结果表明，该方法能够区分不同时间序列，在噪声测试中具有鲁棒性。建议所提出的RCSE可能成为复杂时间序列同步估计的一种新的可靠方法。