A modified multiscale sample entropy measure based on symbolic representation and similarity (MSEBSS) is proposed in this paper to research the complexity of stock markets. The modified algorithm reduces the probability of inducing undefined entropies and is confirmed to be robust to strong noise. Considering the validity and accuracy, MSEBSS is more reliable than Multiscale entropy (MSE) for time series mingled with much noise like financial time series. We apply MSEBSS to financial markets and results show American stock markets have the lowest complexity compared with European and Asian markets. There are exceptions to the regularity that stock markets show a decreasing complexity over the time scale, indicating a periodicity at certain scales. Based on MSEBSS, we introduce the modified multiscale cross-sample entropy measure based on symbolic representation and similarity (MCSEBSS) to consider the degree of the asynchrony between distinct time series. Stock markets from the same area have higher synchrony than those from different areas. And for stock markets having relative high synchrony, the entropy values will decrease with the increasing scale factor. While for stock markets having high asynchrony, the entropy values will not decrease with the increasing scale factor sometimes they tend to increase. So both MSEBSS and MCSEBSS are able to distinguish stock markets of different areas, and they are more helpful if used together for studying other features of financial time series.

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synchrony strong noise complexity undefined entropies features of financial time series symbolic representation and similarity msebss algorithm periodicity similarity mcsebss modified multiscale crosssample entropy measure asynchrony

Pengjian Shang,Yue Wu, Yilong Li,.Multiscale sample entropy and cross-sample entropy based on symbolic representation and similarity of stock markets. 56 (0),.

Pengjian Shang,Yue Wu, Yilong Li,"Multiscale sample entropy and cross-sample entropy based on symbolic representation and similarity of stock markets" 56

Pengjian Shang,Yue Wu, Yilong Li,"Multiscale sample entropy and cross-sample entropy based on symbolic representation and similarity of stock markets"