Weighted multifractal cross-correlation analysis based on Shannon entropy

New research for detecting complex associations between variables with randomness

Many correlation analysis methods can capture a wide range of functional types of variables. However, the influence of uncertainty and distribution status in data is not considered, which leads to the neglect of the regularity information between variables, so that the correlation of variables that contain functional relationship but subject to specific distributions cannot be well identified. Therefore, a novel correlation analysis framework for detecting associations between variables with randomness (RVCR-CA) is proposed. The new method calculates the normalized RMSE to evaluate the degree of functional relationship between variables, calculates entropy difference to measure the degree of uncertainty in variables and constructs the copula function to evaluate the degree of dependence on random variables with distributions. Then, the weighted sum method is performed to the above three indicators to obtain the final correlation coefficient R. In the study, which considers the degree of functional relationship between variables, the uncertainty in variables and the degree of dependence on the variables containing distributions, cannot only measure the correlation of functional relationship variables with specific distributions, but also can better evaluate the correlation of variables without clear functional relationships. In experiments on the data with functional relationship between variables that contain specific distributions, UCI data and synthetic data, the results show that the proposed method has more comprehensive evaluation ability and better evaluation effect than the traditional method of correlation analysis.

Investigation on Vibration Signal Characteristics in a Centrifugal Pump Using EMD-LS-MFDFA

Vibration signals from centrifugal pumps are nonlinear, non-smooth, and possess implied trend terms, which makes it difficult for traditional signal processing methods to accurately extract their fault characteristics and details. With a view to rectifying this, we introduced empirical mode decomposition (EMD) to extract the trend term signals. These were then refit using the least squares (LS) method. The result (EMD-LS) was then combined with multi-fractal theory to form a new signal identification method (EMD-LS-MFDFA), whose accuracy was verified with a binomial multi-fractal sequence (BMS). Then, based on the centrifugal pump test platform, the vibration signals of shell failures under different degrees of cavitation and separate states of loosened foot bolts were collected. The signals’ multi-fractal spectra parameters were analyzed using the EMD-LS-MFDFA method, from which five spectral parameters (Δα, Δf, α0, αmax, and αmin) were extracted for comparison and analysis. The results showed EMD-LS-MFDFA’s performance was closer to the BMS theoretical value than that of MFDFA, displayed high accuracy, and was fully capable of revealing the multiple fractal characteristics of the centrifugal pump fault vibration signal. Additionally, the mean values of the five types of multi-fractal spectral characteristic parameters it extracted were much greater than the normal state values. This indicates that the parameters could effectively distinguish the normal state and fault state of the centrifugal pump. Moreover, α0 and αmax had a smaller mean square than Δα, Δf and αmin, and their stability was higher. Thus, compared to the feature parameters extracted by MFDFA, our method could better realize the separation between the normal state, cavitation (whether slight, moderate, or severe), and when the anchor bolt was loose. This can be used to characterize centrifugal pump failure, quantify and characterize a pump’s different working states, and provide a meaningful reference for the diagnosis and study of pump faults.

Multifractal analysis of financial markets: a review

Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

Multiscale fractional order generalized information of financial time series based on similarity distribution entropy

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.

Analysis of Shannon-Fisher information plane in time series based on information entropy

In this paper, we propose a Shannon-Fisher information plane based on the information entropy to analyze financial stock markets. In order to evaluate the effectiveness of this method, we apply this method to two types of artificial time series: Autoregressive Fractionally Integrated Moving Average models and Chebyshev map model. The results show that with the embedding dimension m and the number of possible states of the system M increasing, the normalized Shannon entropy increases, and the Fisher information measure (FIM) decreases. When the parameter M is not so big, the embedding dimension m plays a leading role in determining the FIM. In addition, compared with the classical Shannon-Fisher information through permutation entropy, we conclude that the proposed approach can give us more accurate information on the classification of financial stock markets.