Topological entropy and geometric entropy and their application to the horizontal visibility graph for financial time series

In this paper, we introduce topological entropy (TE) based on time series, which characterizes the total exponential complexity of a quantified system with a single number. Combined with multiscale theory, we propose geometric entropy (GE), aiming to examine the correlation among different time series. In order to detect the properties of TE and GE, we apply them to an original symbolic method utilized to measure time series irreversibility, namely horizontal visibility algorithm. On this basis, we propose a time series irreversibility measure, i.e., normalized index. Then, we employ TE and GE based on the horizontal visibility graph symbolic algorithm to simulated time series, which is generated by the logistic map with different parameters. Through the comparison of the results, we find out that different simulated data have the same variation tendency of TE, which means that TE is capable of reflecting the similarity among different time series. On the basic of these results, we further analyze the irreversibility of simulated data and also get some interesting findings. From the GE results comparison, we conclude that the GE method can distinguish different time series and expose their correlation efficiently. As a farther validation, we explore the effects of these methods on the analysis of different stock time series. Results show that they can reflect a large number of interrelationships, and successfully quantify the changes in the complexity of different stock market data.