The coupling analysis between stock market indices based on permutation measures

• Research on coupling relationship between financial time series based on cross-permutation entropy, which is a simple, stable and efficient method.
• Proposed two improvements of this permutation idea for analyzing spiky signals as well as short length time series.
• Statistical distribution plot, hypothesis testing and standard deviation theories are applied to verify the effectiveness of our methods.
• Test our methods on synthetic data and model system first, and then applied it to financial dataset.

Many information-theoretic methods have been proposed for analyzing the coupling dependence between time series. And it is significant to quantify the correlation relationship between financial sequences since the financial market is a complex evolved dynamic system. Recently, we developed a new permutation-based entropy, called cross-permutation entropy (CPE), to detect the coupling structures between two synchronous time series. In this paper, we extend the CPE method to weighted cross-permutation entropy (WCPE), to address some of CPE’s limitations, mainly its inability to differentiate between distinct patterns of a certain motif and the sensitivity of patterns close to the noise floor. It shows more stable and reliable results than CPE does when applied it to spiky data and AR(1) processes. Besides, we adapt the CPE method to infer the complexity of short-length time series by freely changing the time delay, and test it with Gaussian random series and random walks. The modified method shows the advantages in reducing deviations of entropy estimation compared with the conventional one. Finally, the weighted cross-permutation entropy of eight important stock indices from the world financial markets is investigated, and some useful and interesting empirical results are obtained.