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.

Multifractal analysis of financial markets

Investigating the influence of water on swelling deformation and mechanical behavior of mudstone considering water softening effect

We propose an approach for analyzing signals with long-range correlations by decomposing the signal increment series into magnitude and sign series and analyzing their scaling properties. We show that signals with identical long-range correlations can exhibit different time organization for the magnitude and sign. We find that the magnitude series relates to the nonlinear properties of the original time series, while the sign series relates to the linear properties. We apply our approach to the heartbeat interval series and find that the magnitude series is long-range correlated, while the sign series is anticorrelated and that both magnitude and sign series may have clinical applications.

Magnitude and Sign Correlations in Heartbeat Fluctuations

Hurst exponent dynamics of S&P 500 returns: Implications for market efficiency, long memory, multifractality and financial crises predictability by application of a nonlinear dynamics analysis framework

Optimum spacing and rock breaking efficiency of TBM double disc cutters penetrating in water-soaked mudstone with FDEM

Here we propose the new method DH-MMA, based on multiscale multifractal detrended fluctuation analysis(MMA), to investigate the scaling properties in stock markets. It is demonstrated that our approach can provide a more stable and faithful description of the scaling properties in comprehensive range rather than fixing the window length and slide length. It allows the assessment of more universal and subtle scaling characteristics. We illustrate DH-MMA by selecting power-law artificial data sets and six stock markets from US and China. The US stocks exhibit very strong multifractality for positive values of q , however, the Chinese stocks show stronger multifractality for negative q than positive q . In general, the US stock markets show similar behaviors, but Chinese stock markets display distinguishing characteristics.

The scaling properties of stock markets based on modified multiscale multifractal detrended fluctuation analysis

A Two-Dimensional Moisture Diffusion Continuous Model for Simulating Dry Shrinkage and Cracking of Soil

Multivariate multiscale sample entropy (MMSE) has recently been proposed to evaluate complexity of time series. However, the results of estimation of complexity by MMSE method may be inaccurate as the coarse-graining procedure reduces the length of the time series at a large scale. In addition, MMSE has some limitations, mainly its inability to detect abrupt changes in the signal and ignore the difference between distinct patterns. In order to overcome those above limitations of MMSE, this paper introduces the weighted multivariate composite multiscale sample entropy (WMCMSE) as a measure to characterize the complexity of nonlinear time series. And we illustrate the necessity of WMCMSE method by comparing WMCMSE results with multivariate multiscale sample entropy (MMSE), weighted multivariate multiscale sample entropy (WMMSE), multivariate composite multiscale sample entropy (MCMSE) on random series. Then, WMCMSE method is employed to study the complexity of traffic speed and volume time series of Beijing Ring 2, 3, 4 roads, which are from August 11th to October 20th, 2012. The results of WMCMSE show that the WMCMSE method can distinguish the behavior of the flow time series, which means that we can detect the congested traffic system. We found that road condition of ring 2, 3, 4 road is very different. Compared to the ring 4 road, the values of the WMCMSE of the ring 2 and 3 roads are higher, indicating that the traffic flow on Ring 2 and Ring 3 road are more complex compared to Ring 4 road. The higher the complexity, the more traffic jams. Government departments can judge the traffic congestion of that time period and that road according to the complexity differences between different time periods and different roads, and then take different measures.

Weighted multivariate composite multiscale sample entropy analysis for the complexity of nonlinear times series

The Detrended Fluctuation Analysis (DFA) and its extensions (MF-DFA) have been proposed as robust techniques to determine possible long-range correlations in self-affine signals. However, many studies have reported the susceptibility of DFA to trends which give rise to spurious crossovers and prevent reliable estimations of the scaling exponents. Lately, several modifications of the DFA method have been reported with many different techniques for eliminating the monotonous and periodic trends. In this study, a smoothing algorithm based on the Orthogonal V-system (OVS) is proposed to minimize the effect of power-law trends, periodic trends, assembled trends and piecewise function trends. The effectiveness of the new method is demonstrated on monofractal data and multifractal data corrupted with different trends.

The orthogonal v-system detrended fluctuation analysis

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The scaling properties of stock markets based on modified multiscale multifractal detrended fluctuation analysis

A Two-Dimensional Moisture Diffusion Continuous Model for Simulating Dry Shrinkage and Cracking of Soil

Weighted multivariate composite multiscale sample entropy analysis for the complexity of nonlinear times series

The orthogonal v-system detrended fluctuation analysis