Characterizing dynamics of time series via Hill-index complexity measure

The ordinal patterns of time series provide a variety of information about the dynamic characteristics of the underlying process. With the ordinal symbolic approach, recently, a permutation-based Hill's diversity index has been proposed as a useful tool for complex system analysis. However, this method just measures the complexity of the system from the perspective of uncertainty, while the underlying structural information of the system dynamics can hardly be captured. To overcome this deficiency, this paper introduces a Hill-index complexity measure (HICM). The numerical applications suggest that the proposed HICM has greater discriminating power than the conventional statistical complexity measure (SCM). With parameter r, even between chaotic systems with extremely similar dynamics, HICM can make a clear differentiation, which can barely be achieved by SCM. Besides, HICM with appropriate r is relatively more robust against noise than SCM. In the empirical application to financial markets, the approach adopted in this paper could differentiate the stage of stock market development. Combined with multidimensional scaling based on the dynamic time wrapping distance, the method could reveal the potential similarities among the stock market dynamics and give a refined classification of these world indices.