Multivariate weighted multiscale permutation entropy for complex time series

In this paper, we propose multivariate multiscale permutation entropy (MMPE) and multivariate weighted multiscale permutation entropy (MWMPE) to explore the complexity of the multivariate time series over multiple different time scales. First, we apply these methods to the simulated trivariate time series which is compose of white noise and 1/f noise to test the validity of multivariate methods. The standard deviations of weighted methods are bigger because of containing more amplitude information, while the standard deviations of multivariate method are smaller than the method for single channel. Hence, it can be found that MWMPE shows a better distinguish capacity, while it is able to measure the complexity of the multichannel data accurately and reflect more information about the multivariate time series as well as holds a better robustness. Then MMPE and MWMPE methods are employed to the financial time series: closing prices and trade volume, from different area. It can be verified that the methods for multichannel data analyze the properties of multivariate time series comprehensively. The entropy values taking the weight into account for both the multichannel and single channel amplify the local fluctuation and reflect more amplitude information. The MWMPE maintain the fluctuation characteristic of SCWMPE of both price and volume. The MWMPE results of these stock markets can be divided into three groups: (1) S&P500, FTSE, and HSI, (2) KOSPI, and (3) ShangZheng. The weighted contingency also shows the difference of inhomogenity of the distributions of ordinal patterns between these groups. Thus, MWMPE method is capable of differentiating these stock markets, detecting their multiscale structure and reflects more information containing in the financial time series.

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