As a measure of complexity, information entropy is frequently used to categorize time series, such as machinery failure diagnostics, biological signal identification, etc., and is thought of as a characteristic of dynamic systems. Many entropies, however, are ineffective for multivariate scenarios due to correlations. In this paper, we propose a local structure entropy (LSE) based on the idea of a recurrence network. Given certain tolerance and scales, LSE values can distinguish multivariate chaotic sequences between stochastic signals. Three financial market indices are used to evaluate the proposed LSE. The results show that the $LS{E}^{FSTE100}$ and $LS{E}^{S\&P500}$ are higher than $LS{E}^{SZI}$, which indicates that the European and American stock markets are more sophisticated than the Chinese stock market. Additionally, using decision trees as the classifiers, LSE is employed to detect bearing faults. LSE performs higher on recognition accuracy when compared to permutation entropy.

As effective representations of complex systems, complex networks have attracted scholarly attention for their many practical applications. They also represent a new tool for time series analysis. In order to characterize the underlying dynamic features, the structure of transformed networks should be encoded with the systematic evolution information that always hides behind the time series data. Thus, the way of mapping segments of the time series into nodes of the network is particularly crucial, but it is liable to be unstable under noise and missing values. In this paper, we propose a coarse-graining based on statistics of segments (CBS) founded complex network method, which can make it immune to interference to a certain degree. The time series is divided into many segments by a slide window, of which the width is determined by the multi-scale entropy of the data. We use a multi-dimensional symbol to represent the motion state of every segment. Due to the utilization of the distribution information of the fragments’ numerical characteristics, the coarse-graining process is self-adaptive to some extent. The complex network is then established based on the adjacent relations of the symbolic sequence. With our method, the differences in the network measurements between the periodic and chaotic motion is easily observable. Furthermore, we investigated the robustness of CBS by adding noise and missing values. We found that CBS is still valid, even with strong noise and 15% missing values, and simulation shows that it is more robust than the VG and LS approaches. By mapping a time series into a complex network, we provide a new tool for understanding the dynamic evolution mechanism of a complex system. This method has been applied in various fields, such as physics, engineering, medicine and economics. However, the interference of noise may greatly affects the reliability of judgment, which is based on the structures of transformed networks. An insufficient robustness is mostly to blame for the transformation from a time series to a symbolic sequence. In this paper, we suggest a new approach to the coarse-graining process which is self-adaptive for threshold choosing. Simulations show that even with strong disturbances, our network structure is easily distinguishable under different dynamic mechanisms.

Distinguishing between chaotic and stochastic dynamics given an input series is a widely studied topic within the time series analysis due to high demand from the practitioners in various fields. Due to one of the fundamental properties of chaotic systems, namely, being sensitive to parameters and initial conditions, chaotic time series exhibit features also observed in randomly generated signals. In this paper, we introduce distance as a measure of similarity between segments based on the ordinal structure. Furthermore, we introduce a new fuzzy entropy, Fuzzy Permutation Entropy (FPE), which can be used to detect determinism in time series. FPE immunes from repeated equal values in signals to some extent, especially for chaotic series. With specific embedding dimensions, it can be employed to distinguish chaotic signals from noise. We show an example for white Gaussian noise, autoregressive moving-average, continuous or discrete chaotic time series, and test FPE’s performance with additive observational noise. We show an application of FPE on rolling bearings’ fault diagnosis.

A method based on coarse-graining to construct a directed weighted complex network which models the transformation of the trading data of an individual stock is introduced. The degree (strength) distribution of derived network follows a power-law. A moderated regression equation with interaction effects of average return and out-degree (in-degree) on out-strength (in-strength) is established. Moreover, we found that the differences of nodes affect the network’s structure and average return level impacts nodes’ eigenvector centrality and pagerank, significantly.

The polarized images of a synchrotron emitting ring are studied in the spacetime of a rotating black hole in the Scalar-Tensor-Vector-Gravity (STVG) theory. The black hole owns an additional dimensionless MOG parameter described its deviation from Kerr black hole. The effects of the MOG parameter on the observed polarization vector and Strokes $Q-U$ loops depend heavily on the spin parameter, the magnetic field configuration, the fluid velocity and the observation inclination angle. For the fixed MOG parameter, the changes of the polarization vector in the image plane are similar to those in the Kerr black hole case. The comparison of the polarization images between Kerr-MOG black hole and M87* implies that there remains some possibility for the STVG-MOG theory.