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In this paper, we introduce the composite multiscale cross-sample entropy (CMCSE) which may induce undefined entropies and then further propose the refined composite multiscale cross-sample entropy (RCMCSE) which modifies CMCSE. First, we apply multiscale cross-sample entropy (MCSE), CMCSE and RCMCSE methods to three types of artificial time series in order to test the validity and accuracy of these methods. Results show that RCMCSE reduces not only standard deviation, but also the probability of inducing undefined entropy effectively, which can provide better robustness and more accurate entropies. Then, these three methods are employed to investigate financial time series including US and Chinese stock indices. For the study between stock indices in the same region, some conclusions which are consistent with previous study are drawn by the RCMCSE results. Meanwhile, it can be found that undefined entropies are induced and the numbers of inducing undefined entropy by three methods for investigation between three US stock indices and two Chinese mainland stock indices are given. Compared with MCSE and CMCSE, RCMCSE method is capable of reducing the number of undefined entropy and providing more accurate entropies. Moreover, the differences on inducing undefined entropy between results for US stock indices & two Chinese mainland stock indices and results for US stock indices & HSI demonstrate a much closer relation between US stock markets and HSI than between US stock markets and two Chinese mainland stock markets. Hence, it can be concluded that RCMCSE is more applicable for the study between US and Chinese stock markets. [ABSTRACT FROM AUTHOR]

In this paper, we introduce weighted permutation entropy (WPE) and three different symbolic approaches to investigate the complexities of stock time series containing amplitude-coded information and explore the influence of using different symbolic approaches on obtained WPE results. We employ WPE based on symbolic approaches to the US and Chinese stock markets and make a comparison between the results of US and Chinese stock markets. Three symbolic approaches are able to help the complexity containing in the stock time series by WPE method drop whatever the embedding dimension is. The similarity between these stock markets can be detected by the WPE based on Binary Δ -coding-method, while the difference between them can be revealed by the WPE based on σ -method, Max–min-method. The combinations of the symbolic approaches: σ -method and Max–min-method, and WPE method are capable of reflecting the multiscale structure of complexity by different time delay and analyze the differences between complexities of stock time series in more detail and more accurately. Furthermore, the correlations between stock markets in the same region and the similarities hidden in the S&P500 and DJI, ShangZheng and ShenCheng are uncovered by the comparison of the WPE based on Binary Δ -coding-method of six stock markets. [ABSTRACT FROM AUTHOR]