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Complexity-entropy curves(CEC) is a useful tool to detect the structure of time series. It’s widely applied in many research areas since it can distinguish the chaotic system and the stochastic process well. However, the original permutation complexity-entropy curves(PCEC) based on permutation entropy(PE) has a defect for it can not take means and amplitudes of time series into consideration, which may lead to some errors when distinguishing the systems. In this paper, we propose dispersion complexity-entropy curves(DCEC) to overcome the defects of PCEC. In addition, we expand the curves from two-dimension to threedimension. We first compare DCEC with PCEC by simulated data. The result shows that DCEC are not only more distinguishable but also more robust with the change of parameters when detecting periodic series. Then, we apply our method to real-world data to illustrate its practicability. We propose a creative feature extraction method based on DCEC and combine it with MSVM to diagnose the different types of bearing fault, which obtains perfect results for the accuracy achieves 100%. We also apply DCEC to stock indices in different countries and different periods to analyze the complexity degree of financial markets. The results successfully detect American financial crisis in 2008 and the rapid development of the economy in China during 2014-2018. These demonstrate that DCEC can serve as an effective method to analyze complex systems.

In this paper, we propose a hybrid method called sliding-window amplitude-based dispersion entropy, which combines dispersion entropy with sliding-window amplitude, to characterize nonlinear time series. This hybrid method not only inherits the fast calculation speed and the ability to characterize nonlinear time series of dispersion entropy, but also has higher noise resistance than dispersion entropy. We firstly utilize three artificial data (logistic map, Hénon map, ARFIMA model) to qualify the effectiveness of the proposed method, results show that our method can correctly characterize the nonlinear time series, and has stronger robustness to noise. Next, the method is applied to analyze stock market system, the data of stock market are composed of six main indices from different countries, the result shows that the proposed method can easily distinguish the emerging markets and developed markets, and can reveal some features under the financial time series.

The ordinal pattern is an essential tool to extract the information in time series. However, little attention has been paid to the transition probability matrix of ordinal patterns, which affects the accuracy and comprehensiveness of the extracted information. In this paper, we propose transition permutation entropy (TPE) and transition dissimilarity measure (TDM) through the transition matrix. TPE can evaluate the complexity of systems. TDM measures the dissimilarity between systems through the dynamic transition and the probability distribution of ordinal patterns. The proposed methods are comprehensively evaluated by simulation experiments and vehicle dynamic response data. The results show that both TPE and TDM can distinguish complex systems and locate the rail corrugation. The combination of TDM, multidimensional scaling, and neural networks can be used for fault detection and is better than other distance calculation methods.