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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.

Cumulative residual entropy (CRE) is a measure of uncertainty and departs from other entropy in that it is established on cumulative residual distribution function instead of density function. In this paper, we prove some important properties of discrete CRE and propose fractional multiscale cumulative residual entropy (FMCRE) as a function of fractional order α, which combines CRE with fractional calculus, probability of permutation ordinal patterns, and multiscale to overcome the limitation of CRE. After adding amplitude information through weighted permutation ordinal patterns, we get fractional weighted multiscale cumulative residual entropy (FWMCRE). FMCRE and FWMCRE extend CRE into a continuous family and can be used in more situations with a suitable parameter. Moreover, they can capture long-range phenomena more clearly and have higher sensitivity to the signal evolution. Results from simulated data verify that FMCRE and FWMCRE can identify time series accurately and have immunity to noise. We confirm that the length of time series has little effect on the accuracy of distinguishing data, and even short series can get results exactly. Finally, we apply FMCRE and FWMCRE on stock data and confirm that they can be used as metrics to measure uncertainty of the system as well as distinguishing signals. FWMCRE can also track changes in stock markets and whether adding amplitude information must be decided by the characteristics of data.

This paper addresses a novel multiscale fractional order distribution entropy based on a similarity matrix (MFS-DistEn) approach to quantify the information of time series on multiple time scales. It improves the metric method of distance matrix in the original DistEn algorithm and further defines the similarity degree between each vector so that we could measure the probability density distribution more accurately. Besides, the multiscale distribution entropy based on similarity matrix combines the advantages of both the multiscale analysis and DistEn and is able to identify dynamical and scale-dependent information. Inspired by the properties of Fractional Calculus, we select the MFS-DistEn notation as the main indicator to present the relevant properties. The characteristics of the generalized MFS-DistEn are tested in both simulated nonlinear signals generated by the autoregressive fractionally integrated moving-average process, logistic map, and real world data series. The results demonstrate the superior performance of the new algorithm and reveal that tuning the fractional order allows a high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The improved similarity DistEn still has relatively lower sensitivity to the predetermined parameters and decreases with an increase of scale.

Permutation entropy (PE) is a novel measure to quantify the complexity of nonlinear time series. In this paper, we propose a generalized permutation entropy ( PEq,δ) based on the recently postulated entropic form, Sq,δ, which was proposed as an unification of the well-known Sq of nonextensive-statistical mechanics and Sδ, a possibly appropriate candidate for the black-hole entropy. We find that PEq,δ with appropriate parameters can amplify minor changes and trends of complexities in comparison to PE. Experiments with this generalized permutation entropy method are performed with both synthetic and stock data showing its power. Results show that PEq,δ is an exponential function of q and the power ( k(δ)) is a constant if δ is determined. Some discussions about k(δ) are provided. Besides, we also find some interesting results about power law.

The paper proposes the asymmetric multiscale cross-sample entropy (AMCSE) method and applies it to analyze the financial time series of US, Chinese, and European stock markets. The asynchronies of these time series in USA, China, and Europe all decrease (the correlations increase) with the increase in scale which declares that taking into account bigger time scale to study these financial time series is capable of revealing the intrinsic relations between these stock markets. Meanwhile, we find that there is a crossover between the upwards and the downwards in these AMCSE results, which indicates that when the scale reach a certain value, the asynchronies of the upwards and the downwards for these stock markets are equal and symmetric. But for the other scales, the asynchronies of the upwards and the downwards are different from each other indicating the necessity and importance of multiscale analysis for revealing the most comprehensive information of stock markets. The series with a positive trend have a higher decreasing pace on asynchrony than those with a negative trend, while the asynchrony between the series with a positive or negative trend is lower than that between the original series. Moreover, it is noticeable that there are some small abnormal rises at some abnormal scales. We find that the asynchronies are the highest at scales smaller than 2 when investigating the time series of stock markets with a negative trend. The existences of asymmetries declare the inaccuracy and weakness of multiscale cross-sample entropy, while by comparing the asymmetries of US, Chinese, and European markets, similar conclusions can be drawn and we acquire that the asymmetries of Chinese markets are the smallest and the asymmetries of European markets are the biggest. Thus, it is of great value and benefit to investigate the series with different trends using AMCSE method.

We propose the asymmetric multiscale detrended cross-correlation analysis (MS-ADCCA) method and apply MS-ADCCA method to explore the existence of asymmetric cross-correlation for daily price returns in US and Chinese stock markets and to assess the properties of these asymmetric cross-correlations. The results all show the existences of asymmetric cross-correlations, while small asymmetries at small scales and larger asymmetries at larger scales are also displayed. There is a strong similarity between S&P500 and DJI, and we reveal that the asymmetries depend more on the cross-correlations of S&P500 vs. DJI, S&P500 vs. NQCI, DJI vs. NQCI, and ShangZheng vs. ShenCheng when the market is falling than rising, respectively. By comparing the spectra of S&P500 vs. NQCI and DJI vs. NQCI with uptrends and downtrends, we detect some new characteristics which lead to some new conclusions. Likewise, some new conclusions also can be drawn by the new characteristics displayed through the comparison between the spectra of ShangZheng vs. HSI and ShenCheng vs. HSI. Obviously, we conclude that although the overall spectra are similar and one market has the same effect when it is rising and falling in the study of asymmetric cross-correlations between it and different markets, the cross-correlations and asymmetries on the trends of the different markets are all different. MS-ADCCA method can detect the differences on the asymmetric cross-correlations by different trends of markets. Moreover, the uniqueness of cross-correlation between NQCI and HSI can be detected in the study of the asymmetric cross-correlations, which confirms that HSI is unique in the Chinese stock markets and NQCI is unique in the US stock markets further.

Traffic systems, especially urban traffic systems, are regulated by different kinds of interacting mechanisms which operate across multiple spatial and temporal scales. Traditional approaches fail to account for the multiple time scales inherent in time series, such as empirical probability distribution function and detrended fluctuation analysis, which have lead to different results. The role of multiscale analytical method in traffic time series is a frontier area of investigation. In this paper, our main purpose is to introduce a new method-multiscale time irreversibility, which is helpful to extract information from traffic time series we studied. In addition, to analyse the complexity of traffic volume time series of Beijing Ring 2, 3, 4 roads between workdays and weekends, which are from August 18, 2012 to October 26, 2012, we also compare the results by this new method and multiscale entropy method we have known well. The results show that the higher asymmetry index we get, the higher traffic congestion level will be, and accord with those which are obtained by multiscale entropy.

In this paper, we apply the detrended fluctuation analysis (DFA), local scaling detrended fluctuation analysis (LSDFA), and detrended cross-correlation analysis (DCCA) to investigate correlations of several stock markets. DFA method is for the detection of long-range correlations used in time series. LSDFA method is to show more local properties by using local scale exponents. DCCA method is a developed method to quantify the cross-correlation of two non-stationary time series. We report the results of auto-correlation and cross-correlation behaviors in three western countries and three Chinese stock markets in periods 2004-2006 (before the global financial crisis), 2007-2009 (during the global financial crisis), and 2010-2012 (after the global financial crisis) by using DFA, LSDFA, and DCCA method. The findings are that correlations of stocks are influenced by the economic systems of different countries and the financial crisis. The results indicate that there are stronger auto-correlations in Chinese stocks than western stocks in any period and stronger auto-correlations after the global financial crisis for every stock except Shen Cheng; The LSDFA shows more comprehensive and detailed features than traditional DFA method and the integration of China and the world in economy after the global financial crisis; When it turns to cross-correlations, it shows different properties for six stock markets, while for three Chinese stocks, it reaches the weakest cross-correlations during the global financial crisis.