A brief description to different multi-fractal behaviors of daily wind speed records over China

Scaling behaviors of the long daily wind speed records of four selected weather stations over China were analyzed by using Multi-Fractal Detrended Fluctuation Analysis (MF-DFA). The results indicated that all these four stations are characterized by long-range power-law correlations, but MF-DFA results showed non-universal multi-fractal behaviors over China. We fitted generalized Hurst exponent h(q) via a modified generalized binomial multiplicative cascade model, and different widths of the multi-fractal spectrum are estimated.     

Multifractal Detrended Cross-Correlation Analysis of sunspot numbers and river flow fluctuations

We use the Detrended Cross-Correlation Analysis (DCCA) to investigate the influence of sun activity represented by sunspot numbers on one of the climate indicators, specifically rivers, represented by river flow fluctuation for Daugava, Holston, Nolichucky and French Broad rivers. The Multifractal Detrended Cross-Correlation Analysis (MF-DXA) shows that there exist some crossovers in the cross-correlation fluctuation function versus time scale of the river flow and sunspot series. One of these crossovers corresponds to the well-known cycle of solar activity demonstrating a universal property of the mentioned rivers. The scaling exponent given by DCCA for original series at intermediate time scale, , is λ=1.17±0.04 which is almost similar for all underlying rivers at 1σ confidence interval showing the second universal behavior of river runoffs. To remove the sinusoidal trends embedded in data sets, we apply the Singular Value Decomposition (SVD) method. Our results show that there exists a long-range cross-correlation between the sunspot numbers and the underlying streamflow records. The magnitude of the scaling exponent and the corresponding cross-correlation exponent are λ∈(0.76,0.85) and γ_×∈(0.30,0.48), respectively. Different values for scaling and cross-correlation exponents may be related to local and external factors such as topography, drainage network morphology, human activity and so on. Multifractal cross-correlation analysis demonstrates that all underlying fluctuations have almost weak multifractal nature which is also a universal property for data series. In addition the empirical relation between scaling exponent derived by DCCA and Detrended Fluctuation Analysis (DFA), is confirmed.     

Dynamics of bid–ask spread return and volatility of the Chinese stock market

The bid–ask spread is taken as an important measure of the financial market liquidity. In this article, we study the dynamics of the spread return and the spread volatility of four liquid stocks in the Chinese stock market, including the memory effect and the multifractal nature. By investigating the autocorrelation function and the Detrended Fluctuation Analysis (DFA), we find that the spread return is the lack of long-range memory, while the spread volatility is long-range time correlated. Besides, the spread volatilities of different stocks present long-range cross-correlations. Moreover, by applying the Multifractal Detrended Fluctuation Analysis (MF-DFA), the spread return is observed to possess a strong multifractality, which is similar to the dynamics of a variety of financial quantities. Different from the spread return, the spread volatility exhibits a weak multifractal nature.     

Multifractal analysis of streamflow records of the East River basin (Pearl River), China

Scaling behaviors of the long daily streamflow series of four hydrological stations (Longchuan (1952-2002), Heyuan (1951-2002), Lingxia (1953-2002) and Boluo (1953-2002)) in the mainstream East River, one of the tributaries of the Pearl River (Zhujiang River) basin, were analyzed using multifractal detrended fluctuation analysis (MF-DFA). The research results indicated that streamflow series of the East River basin are characterized by anti-persistence. MF-DFA technique showed similar scaling properties in the streamflow series of the East River basin on shorter time scales, indicating universal scaling properties over the East River basin. Different intercept values of the fitted lines of log-log curve of F_q(s) versus s implied hydrological regulation of water reservoirs. Based on the numerical results, we suggested that regulation activities by water reservoirs could not impact the scaling properties of the streamflow series. The regulation activities by water reservoir only influenced the fluctuation magnitude. Therefore, we concluded that the streamflow variations were mainly the results of climate changes, and precipitation variations in particular. Strong dependence of generalized Hurst exponent h(q) on q demonstrated multifractal behavior of streamflow series of the East River basin, showing ‘universal’ multifractal behavior of river runoffs. The results of this study may provide valuable information for prediction and assessment of water resources under impacts of climatic changes and human activities in the East River basin.     

Chaotic SVD method for minimizing the effect of exponential trends in detrended fluctuation analysis

The Detrended Fluctuation Analysis (DFA) and its extensions (MF-DFA) have been used extensively to determine possible long-range correlations in self-affine signals. However, recent studies have reported the susceptibility of DFA to trends which give rise to spurious crossovers and prevent reliable estimation of the scaling exponents. In this study, a smoothing algorithm based on the Chaotic Singular-Value Decomposition (CSVD) is proposed to minimize the effect of exponential trends and distortion in the log-log plots obtained by DFA techniques. The effectiveness of the technique is demonstrated on monofractal and multifractal data corrupted with exponential trends.     

Multifractal Detrended Fluctuation Analysis of Interevent Time Series in a Modified OFC Model

We use multifractal detrended fluctuation analysis(MF-DFA) method to investigate the multifractal behavior of the interevent time series in a modified Olami-Feder-Christensen(OFC) earthquake model on assortative scale-free networks.We determine generalized Hurst exponent and singularity spectrum and find that these fluctuations have multifractal nature.Comparing the MF-DFA results for the original interevent time series with those for shuffled and surrogate series,we conclude that the origin of multifractality is due to both the broadness of probability density function and long-range correlation.     

On spurious and corrupted multifractality: The effects of additive noise, short-term memory and periodic trends

We study the performance of multifractal detrended fluctuation analysis (MF-DFA) applied to long-term correlated and multifractal data records in the presence of additive white noise, short-term memory and periodicities. Such additions and disturbances that can be typically found in the observational records of various complex systems ranging from climate dynamics to physiology, network traffic, and finance. In monofractal records, we find that (i) additive white noise hardly results in spurious multifractality, but causes underestimated generalized Hurst exponents h(q) for all q values; (ii) short-range correlations lead to pronounced crossovers in the generalized fluctuation functions F_q(s) at positions that decrease with increasing moment q, thus causing significantly overestimated h(q) for small q and spurious multifractality; (iii) periodicities like seasonal trends (with standard deviations comparable with the one of the studied process) result in spurious “reversed” multifractality where h(q) increases with increasing q (except for very short time windows). We also show that in multifractal cascades moderate additions of noise, short-range memory, or periodic trends cause flawed results for h(q) with q<2, while h(q) with q>2 remains nearly unchanged.     

Are developed and emerging agricultural futures markets multifractal? A comparative perspective

Although there are many reports on the empirical evidence of the existence of multifractality in various financial or commodity markets in current literature, few can be found to compare the multifractal properties of emerging and developed economies, especially for agricultural futures markets in those countries (regions). We therefore chose China as the representative of the transition and emerging economies, and USA as the representative of developed ones. We attempt to find the answers to the following questions: (1) Are all those different markets multifractal? (2) What are the dynamical causes for multifractality in those markets (if any)? (3) Are the multifractality strengths in those markets of the transition and emerging economies weaker (or stronger) than those of the developed ones? To answer these questions, Multifractal Detrended Fluctuation Analysis (MF-DFA) are applied to study some of the representative agricultural futures markets in China and USA, namely, wheat, soy meal, soybean and corn. Our results suggest that all the markets of China and USA exhibit multifractal properties except US soybean market, which is much closer to mono-fractal comparing with China’s soybean market. To investigate the sources of multifractality, shuffling and phase randomization procedures are applied to destroy the temporal correlations and non-Gaussian distributions respectively. We found that multifractality can be mainly attributed to the non-Gaussian probability distribution and secondarily to the nonlinear temporal correlation mechanism for all the markets, except US soybean and soy meal, which derives from some other unknown factors. Furthermore, the average of τ(q) are applied to obtain the multifractal spectra of the two markets as a whole. The results show that the width of the multifractal spectrum of US agricultural futures markets is significantly narrower than that of China’s. Based on our findings, we proposed a hypothesis that the strength of multifractality in developed economies may be weaker than that in emerging and transition ones.     

Long range correlation and possible electron conduction through DNA sequences

Long range correlation analysis and charge conductivity investigation are applied to sequences in 16 chromosomes in the Saccharomyces cerevisiae genome. DNA sequence data are analyzed via Hurst’s analysis and Detrended Fluctuation Analysis (DFA) analysis. Super diffusive nature of mapping sequences are evident with the measured Hurst exponent H to be around the value of 0.60 for all sequences in the 16 chromosomes. The DFA result is consistent with the result from the Hurst analysis. Tight binding models are applied for the investigation of charge conduction through DNA sequences. The overall averaged transmission coefficients, 〈T_N〉_av, calculated from sixteen chromosomes are shown to be significantly different from values calculated from random as well as periodic sequences. Sequences from the S. cerevisiae genome promise better charge conduction ability than random sequences. Finally, delocalized electronic wave function patterns are also shown through calculations using the tight binging model. Slightly delocalized electronic wavefunctions are seen on sequences in sixteen chromosomes, as compared with those obtained from random sequences on the same eigenenergies.     

Multifractal properties of the Indian financial market

We investigate the multifractal properties of the logarithmic returns of the Indian financial indices (BSE & NSE) by applying the multifractal detrended fluctuation analysis. The results are compared with that of the US S&P 500 index. Numerically we find that qth-order generalized Hurst exponents h(q) and τ(q) change with the moments q. The nonlinear dependence of these scaling exponents and the singularity spectrum f(α) show that the returns possess multifractality. By comparing the MF-DFA results of the original series to those for the shuffled series, we find that the multifractality is due to the contributions of long-range correlations as well as the broad probability density function. The financial markets studied here are compared with the Binomial Multifractal Model (BMFM) and have a smaller multifractal strength than the BMFM.     

Multifractal analysis of polyalanines time series

Multifractal properties of the energy time series of short α-helix structures, specifically from a polyalanine family, are investigated through the MF-DFA technique (multifractal detrended fluctuation analysis). Estimates for the generalized Hurst exponent h(q) and its associated multifractal exponents τ(q) are obtained for several series generated by numerical simulations of molecular dynamics in different systems from distinct initial conformations. All simulations were performed using the GROMOS force field, implemented in the program THOR. The main results have shown that all series exhibit multifractal behavior depending on the number of residues and temperature. Moreover, the multifractal spectra reveal important aspects of the time evolution of the system and suggest that the nucleation process of the secondary structures during the visits on the energy hyper-surface is an essential feature of the folding process.     

Multifractal behavior of wild-land and forest fire time series in Brazil

We examine statistical properties of a daily hot pixel time series recorded in Brazil during the period 1998–2006, using Multifractal Detrended Fluctuation Analysis (MF-DFA). We find that generalized scaling exponent h(q)$h\left(q\right)$ is a decreasing function of q$q$, indicating multifractal behavior of hot pixel dynamics. We also calculate multifractal spectra f(α)$f\left(\alpha \right)$ and use fourth-degree polynomial regression to estimate complexity parameters that describe the degree of multifractality of the underlying process. After July 2002, when a significant increase of the number of hot pixel observations is recorded, the complexity of the series is reduced (manifested by the reduction of width of the f(α)$f\left(\alpha \right)$ spectrum), while small fluctuations increase their dominance over large scale fluctuations (manifested by the increase of skew parameter r$r$). These results should be taken into account when devising ecological and climatic models for Brazil, that contemplate the phenomena of wild-land and forest fires.     

Wavelet Leaders: A new method to estimate the multifractal singularity spectra

Wavelet Leaders is a novel alternative based on wavelet analysis for estimating the Multifractal Spectrum. It was proposed by Jaffard and co-workers improving the usual wavelet methods. In this work, we analyze and compare it with the well known Multifractal Detrended Fluctuation Analysis. The latter is a comprehensible and well adapted method for natural and weakly stationary signals. Alternatively, Wavelet Leaders exploits the wavelet self-similarity structures combined with the Multiresolution Analysis scheme. We give a brief introduction on the multifractal formalism and the particular implementation of the above methods and we compare their effectiveness. We expose several cases: Cantor measures, Binomial Multiplicative Cascades and also natural series from a tonic-clonic epileptic seizure. We analyze the results and extract the conclusions.     

Multifractal Behaviors of Stock Indices and Their Ability to Improve Forecasting in a Volatility Clustering Period

The financial market is a complex system, which has become more complicated due to the sudden impact of the COVID-19 pandemic in 2020. As a result there may be much higher degree of uncertainty and volatility clustering in stock markets. How does this “black swan” event affect the fractal behaviors of the stock market? How to improve the forecasting accuracy after that? Here we study the multifractal behaviors of 5-min time series of CSI300 and S&P500, which represents the two stock markets of China and United States. Using the Overlapped Sliding Window-based Multifractal Detrended Fluctuation Analysis (OSW-MF-DFA) method, we found that the two markets always have multifractal characteristics, and the degree of fractal intensified during the first panic period of pandemic. Based on the long and short-term memory which are described by fractal test results, we use the Gated Recurrent Unit (GRU) neural network model to forecast these indices. We found that during the large volatility clustering period, the prediction accuracy of the time series can be significantly improved by adding the time-varying Hurst index to the GRU neural network.     

Multifractal estimates of monofractality in RR-heart series in power spectrum ranges

Two popular estimators of multifractal properties: the Wavelet Transform Modulus Maxima method and Multifractal Detrended Fluctuation Analysis are applied to investigate signals consisting of normal RR-series in 39 healthy subjects and 90 patients suffering from systolic dysfunction of the left ventricle. However, differently from standards for multifractal analysis the scaling is performed separately in intervals corresponding to standard power spectral bands: low (LF), very low (VLF) and ultra low frequencies (ULF). Tests on fractional Brownian motions (fBm) are done to quantify properties of the estimators as detectors of monofractality in LF, VLF and ULF bands. Arguments are given that multifractal analysis of RR-series performed in these bands has a physiological meaning. The increased activation of the sympathetic nervous system caused by heart disease is detected evidently only by analysis in LF. The transition in multifractal characteristics between diurnal and nocturnal activity takes place when the analysis moves from LF and VLF to ULF. Only in ULF, the diurnal heart rate variability can be approximated by fBm with a self-similarity index of H=0.20.     

Multifractal detrended fluctuation analysis of sheep livestock prices in origin

The multifractal detrended fluctuation analysis (MF-DFA) is used to verify whether or not the returns of time series of prices paid to farmers in original markets can be described by the multifractal approach. By way of example, 5 weekly time series of prices of different breeds, slaughter weight and market differentiation from 2000 to 2012 are analyzed. Results obtained from the multifractal parameters and multifractal spectra show that the price series of livestock products are of a multifractal nature. The Hurst exponent shows that these time series are stationary signals, some of which exhibit long memory (Merino milk-fed in Seville and Segureña paschal in Jaen), short memory (Merino paschal in Cordoba and Segureña milk-fed in Jaen) or even are close to an uncorrelated signals (Merino paschal in Seville). MF-DFA is able to discern the different underlying dynamics that play an important role in different types of sheep livestock markets, such as degree and source of multifractality. In addition, the main source of multifractality of these time series is due to the broadness of the probability function, instead of the long-range correlation properties between small and large fluctuations, which play a clearly secondary role.     

Degree distributions of the visibility graphs mapped from fractional Brownian motions and multifractal random walks

The dynamics of a complex system is usually recorded in the form of time series, which can be studied through its visibility graph from a complex network perspective. We investigate the visibility graphs extracted from fractional Brownian motions and multifractal random walks, and find that the degree distributions exhibit power-law behaviors, in which the power-law exponent α is a linear function of the Hurst index H of the time series. We also find that the degree distribution of the visibility graph is mainly determined by the temporal correlation of the original time series with minor influence from the possible multifractal nature. As an example, we study the visibility graphs constructed from three Chinese stock market indexes and unveil that the degree distributions have power-law tails, where the tail exponents of the visibility graphs and the Hurst indexes of the indexes are close to the α∼H linear relationship.     

A hybrid detrending method for fractional Gaussian noise

Determining trend and implementing detrending operations are important steps in data analysis. Yet there is neither precise definition of “trend” nor any logical algorithm for extracting it. In this paper, we propose a Hybrid Detrending Method (HDM) which is based on the Empirical Mode Decomposition (EMD) and the Detrended Fluctuation Analysis (DFA). Our method can remove the polynomial and sinusoidal trends from the fractional Gaussian noise (fGn) background. We illustrate the method by selected examples from artificial time series and climate data. In comparison with existing frequency domain based detrending methods, our method is a posteriori, the trend defined by our method is only derived from the data. Further, our method also preserves the scaling behavior of the original signals.     

Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis (MF-DFA) is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis (DFA), which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression τ(q)=qh(q)-1 stipulating the relationship between the multifractal exponent τ(q) and the generalized Hurst exponent h(q). As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as τ(q)=qh(q)-qH'-1, where H' is the nonconservation parameter in the universal multifractal formalism. The singular spectra, α and f(α), are also derived according to this new relationship.     

Generalized Structure Functions and Multifractal Detrended Fluctuation Analysis Applied to Vegetation Index Time Series: An Arid Rangeland Study

Estimates suggest that more than 70% of the world’s rangelands are degraded. The Normalized Difference Vegetation Index (NDVI) is commonly used by ecologists and agriculturalists to monitor vegetation and contribute to more sustainable rangeland management. This paper aims to explore the scaling character of NDVI and NDVI anomaly (NDVIa) time series by applying three fractal analyses: generalized structure function (GSF), multifractal detrended fluctuation analysis (MF-DFA), and Hurst index (HI). The study was conducted in four study areas in Southeastern Spain. Results suggest a multifractal character influenced by different land uses and spatial diversity. MF-DFA indicated an antipersistent character in study areas, while GSF and HI results indicated a persistent character. Different behaviors of generalized Hurst and scaling exponents were found between herbaceous and tree dominated areas. MF-DFA and surrogate and shuffle series allow us to study multifractal sources, reflecting the importance of long-range correlations in these areas. Two types of long-range correlation appear to be in place due to short-term memory reflecting seasonality and longer-term memory based on a time scale of a year or longer. The comparison of these series also provides us with a differentiating profile to distinguish among our four study areas that can improve land use and risk management in arid rangelands.