Modified generalized multiscale sample entropy and surrogate data analysis for financial time series

Abstract

Complexity in time series is an intriguing feature of living dynamical systems such as financial systems, with potential use for identification of system state. Multiscale sample entropy (MSE) is a popular method of assessing the complexity in various fields. Inspired by Tsallis generalized entropy, we rewrite MSE as the function of q parameter called generalized multiscale sample entropy and surrogate data analysis ( $$q\hbox {MSE}$$ ). qMSDiff curves are calculated with two parameters q and scale factor $$\tau $$ , which consist of differences between original and surrogate series $$q\hbox {MSE}$$ . However, the distance measure shows some limitation in detecting the complexity of stock markets. Further, we propose and discuss a modified method of generalized multiscale sample entropy and surrogate data analysis ( $$q\hbox {MSE}_\mathrm{SS}$$ ) to measure the complexity in financial time series. The new method based on similarity and symbolic representation ( $$q\hbox {MSDiff}_\mathrm{SS}$$ ) presents a different way of time series patterns match showing distinct behaviors of complexity and $$q\hbox {MSDiff}_\mathrm{SS}$$ curves are also presented in two ways since there are two influence factors. Simulations are conducted over both synthetic and real-world data for providing a comparative study. The evaluations show that the modified method not only reduces the probability of inducing undefined entropies, but also performs more sensitive to series with different features and is confirmed to be robust to strong noise. Besides, it has smaller discrete degree for independent noise samples, indicating that the estimation accuracy may be better than the original method. Considering the validity and accuracy, the modified method is more reliable than the original one for time series mingled with much noise like financial time series. Also, we evaluate $$q\hbox {MSDiff}_\mathrm{SS}$$ for different areas of financial markets. The curves versus q of Asia are greater than those of America and Europe. Moreover, American stock markets have the lowest $$q\hbox {MSDiff}_\mathrm{SS}$$ , indicating that they are of low complexity. While the curves versus $$\tau $$ help to research their complexity from a different aspect, the modified method makes us have access to analyzing complexity of financial time series and distinguishing them.