Inverse sample entropy analysis for stock markets

Abstract

Entropy has been an important tool for the complexity analysis of time series from various fields. Based on studying all the template mismatches, a modified sample entropy (SE) method, named as inverse sample entropy (ISE), for investigating the complexity of financial time series is proposed in this paper. Different from SE, ISE considers the far neighbors of templates; it also provides more comprehensive information combined with SE. Stock markets usually fluctuate with the economy policies; ISE allows us to detect the financial crisis by the change of complexity. By experiments on both simulated data and real-world stock data, ISE shows that the threshold $r$ is more flexible compared with that of SE, which allows ISE to be applied not only to limited type of data. Besides, it is more robust to high dimension $m$, so ISE can be extended to the application of high dimension analysis. For studying the impact of embedding dimension $m$ under multiple scales on both artificial and real-world data, we made a comparison on the use of SE and ISE. Both SE and ISE are able to distinguish time series with different features and characteristics. While SE is sensitive to high dimension analysis, ISE shows robustness.

Appendix

Logistic map, defined as $x_{i + 1} = \lambda x_{i} \left( {1 - x_{i} } \right)$, shows time series of various features with the parameter $\lambda$.

The values in the period-doubling cascade behavior are also called stable fixed points or attractors or solutions in statistical physics [36, 37]. For $\lambda < 3.4495$, the fixed points simply oscillate between the two points which are the intersections of a vertical line through the $\lambda$-value. Figure 

Fig. 14

The period-doubling cascade behavior for logistic map with $\lambda$ from 2 to 4

14 shows the attractors under different $\lambda$ values. At $\lambda = 3.4495$, the mechanism that produces the period-2 solution from the period-1 solution is repeated: each of the period-2 points is destabilized, producing two additional solutions. A period-4 solution therefore appears at the point $\lambda = 3.4495$. Thus if $\lambda_{4} = 3.4495 < \lambda < \lambda_{8} = 3.54409$, where $\lambda_{8}$ is the bifurcation value to a period-8 solution, $x_{i}$ exhibits a period-4 solution with the values given by the intersection of the curve of equilibrium states with the vertical line through the $\lambda$-value in Fig. 14.

With $\lambda$ increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly; the ratio between the lengths of two successive bifurcation intervals approaches the Feigenbaum constant $\delta \approx 4.66920$. This behavior is an example of a period-doubling cascade.

At $\lambda \approx 3.56995$ is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, we no longer see oscillations of finite period. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.