Weighted multiscale permutation entropy of financial time series

Abstract

In this paper, we propose the weighted multiscale permutation entropy (WMPE) as a novel measure to quantify the complexity of nonlinear time series containing amplitude-coded information. WMPE is different from multiscale permutation entropy (MPE) in the sense that it suits better signals having considerable amplitude information and also succeed in accounting for the multiple time scales inherent in the financial systems. We first perform the MPE and WMPE methods on synthetic data showing the power of WMPE. Then, we apply the MPE and WMPE methods to the US and Chinese stock markets and make a comparison to discuss their differences and similarities between these different markets. The WMPE of each US and Chinese stock market points out the necessity of applying permutation entropy on multiple scales and reveals amplitude-coded information contained in the signals and immunity to degradation by noise when $$m = 5-7$$ . Some new conclusions are gotten by the new characteristics we detected in the comparison. WMPE method can distinguish ShangZheng and ShenCheng from these markets and imply that HSI is more similar to the US markets. WMPE moves the fluctuation range of entropy values of different market down differently reflecting more accurate complexity of different stock markets containing amplitude information. Compared WMPE of ShangZheng and ShenCheng with the WMPE of US markets and HSI, US stock market and HSI may have more amplitude information carried by the signals of stock market. Furthermore, compared with MPE, WMPE can reduce the standard deviation which ensures the results more robust. The conclusion that $$m = 7$$ is the best embedding dimension to investigate the WMPE results can be drawn because the WMPE tends to be stable in a certain range and reflects the necessity of investigation on multiscale and advantages of adding different weight, and it can distinguish these markets while reducing the standard deviations of all the markets.