Multifractal Phenomena Research Articles

The Multifractal Phenomenon of Stock Price Caused by “Tesla Rights Defense Event”

In this paper, we explore the periodicity of the multifractal phenomenon of stock price caused by public relations events using the multifractal detrended fluctuation analysis (MF-DFA) method. The Tesla stock closing price in the context of the Tesla rights defense event is used to obtain the generalized Hurst exponent and multifractal spectrum. We find that the stock price exhibits multifractal characteristics. Afterwards, we conduct a multifractal time-varying analysis using high-frequency data for 12 time periods before and after the Tesla rights defense event, which happened on 19 April 2021. We calculate the [Formula: see text] and [Formula: see text] before and after the rights defense event to measure the degree of multifractal of Tesla stock price. In addition, we also compute the difference between the two [Formula: see text] and two [Formula: see text], which are regarded as [Formula: see text] and [Formula: see text]. The results show that the Tesla stock market efficiency degree raised as the period time increases until the eighth period. And after the eighth period, both [Formula: see text] and [Formula: see text] tend to be stable, and the degree of market efficiency caused by the Tesla rights defense event remains almost unchanged. In addition, we use three broad market indices in the context of the Russia–Ukraine conflict to test the universality. The three broad market indices are Dow Jones Industrial Average (DJIA), Shanghai Stock Exchange Composite Index (SSEC) and Nikkei 225 ETF (N225). The results demonstrate that with the extending of time period, [Formula: see text] and [Formula: see text] gradually stabilize, indicating that public memorability will diminish and the market efficiency almost remains unchanged.

Speech and Music – Nonlinear Acoustical Decoding in Neurocognitive Scenario

Speech and music signals are multifractal phenomena. The time displacement profile of speech and music signal show strikingly different scaling behaviour. However, a full complexity analysis of their frequency and amplitude has not been made so far. We propose a novel complex network based approach (Visibility Graph) to study the scaling behaviour of frequency wise amplitude variation of speech and music signals over time and then extract their PSVG (Power of Scale freeness of Visibility Graph). From this analysis it emerges that the scaling behaviour of amplitude-profile of music varies a lot from frequency to frequency whereas it’s almost consistent for the speech signal. Our left auditory cortical areas are proposed to be neurocognitively specialised in speech perception and right ones in music. Hence we can conclude that human brain might have adapted to the distinctly different scaling behaviour of speech and music signals and developed different decoding mechanisms, as if following the so called Fractal Darwinism. Using this method, we can capture all non-stationary aspects of the acoustic properties of the source signal to the deepest level, which has huge neurocognitive significance. Further, we propose a novel non-invasive application to detect neurological illness (here autism spectrum disorder, ASD), using the quantitative parameters deduced from the variation of scaling behaviour for speech and music.

Topological Vortexes, Asymptotic Freedom, and Multifractals

In this paper, we study the Kelvinons, which are monopole ring solutions to the Euler equations, regularized as the Burgers vortex in the viscous core. There is finite anomalous dissipation in the inviscid limit. However, in the anomalous Hamiltonian, some terms are growing as logarithms of Reynolds number; these terms come from the core of the Burgers vortex. In our theory, the turbulent multifractal phenomenon is similar to asymptotic freedom in QCD, with these logarithmic terms summed up by an RG equation. The small effective coupling does not imply small velocity; on the contrary, velocity is large compared to its fluctuations, which opens the way for a quantitative theory. In the leading order in the perturbation theory in this effective coupling constant, we compute running multifractal dimensions for high moments of velocity circulation, which is in good agreement with the data for quantum Turbulence and available data for classical Turbulence. The logarithmic dependence of fractal dimensions on the loop size comes from the running coupling in anomalous dimensions. This slow logarithmic drift of fractal dimensions would be barely observable at Reynolds numbers achievable at modern DNS.

Explicit construction of joint multipoint statistics in complex systems

Complex systems often involve random fluctuations for which self-similar properties in space and time play an important role. Fractional Brownian motions, characterized by a single scaling exponent, the Hurst exponent H, provide a convenient tool to construct synthetic signals that capture the statistical properties of many processes in the physical sciences and beyond. However, in certain strongly interacting systems, e.g., turbulent flows, stock market indices, or cardiac interbeats, multiscale interactions lead to significant deviations from self-similarity and may therefore require a more elaborate description. In the context of turbulence, the Kolmogorov–Oboukhov model (K62) describes anomalous scaling, albeit explicit constructions of a turbulent signal by this model are not available yet. Here, we derive an explicit formula for the joint multipoint probability density function of a multifractal field. To this end, we consider a scale mixture of fractional Ornstein–Uhlenbeck processes and introduce a fluctuating length scale in the corresponding covariance function. In deriving the complete statistical properties of the field, we are able to systematically model synthetic multifractal phenomena. We conclude by giving a brief outlook on potential applications which range from specific tailoring or stochastic interpolation of wind fields to the modeling of financial data or non-Gaussian features in geophysical or geospatial settings.

Multifractal Analysis of Pore Structure and Evaluation of Deep-Buried Cambrian Dolomite Reservoir with Image Processing: A Case from Tarim Basin, NW China

Reservoir pore space assessment is of great significance for petroleum exploration and production. However, it is difficult to describe the pore characteristics of deep-buried dolomite reservoirs with the traditional linear method because these rocks have undergone strong modification by tectonic activity and diagenesis and show significant pore space heterogeneity. In this study, 38 dolostone samples from 4 Cambrian formations of Tarim Basin in NW China were collected and 135 thin section images were analyzed. Multifractal theory was used for evaluation of pore space heterogeneity in deep-buried dolostone based on thin section image analysis. The physical parameters, pore structure parameters, and multifractal characteristic parameters were obtained from the digital images. Then, the relationships between lithology and these parameters were discussed. In addition, the pore structure was classified into four categories using K-means clustering analysis based on multifractal parameters. The results show that the multifractal phenomenon generally exists in the pore space of deep-buried dolomite and that multifractal analysis can be used to characterize the heterogeneity of pore space in deep-buried dolomite. For these samples, multifractal parameters, such as αmin, αmax, ΔαL, ΔαR, Δf, and AI, correlate strongly with porosity but only slightly with permeability. However, the parameter Δα, which is usually used to reveal heterogeneity, does not show an obvious link with petrophysical properties. Of dolomites with different fabrics, fine crystalline dolomite and medium crystalline dolomite show the best petrophysical properties and show significant differences in multifractal parameters compared to other dolomites. More accurate porosity estimations were obtained with the multifractal generalized fractal dimension, which provides a new method for porosity prediction. The various categories derived from the K-means clustering analysis of multifractal parameters show distinct differences in petrophysical properties. This proves that reservoir evaluation and pore structure classification can be accurately performed with the K-means clustering analysis method based on multifractal parameters of pore space in deep-buried dolomite reservoirs.

Multifractal phenomena and packing dimension

We undertake a general study of multifractal phenomena for functions or measures. We show that the existence of several kinds of multifractal functions or measures can be easily deduced from an abstract statement, leading to new results. This general approach does not work for Fourier or Dirichlet series. Using careful constructions, we extend our results to these cases.

Quantitative approach to multifractality induced by correlations and broad distribution of data

We analyze quantitatively the effect of spurious multifractality induced by the presence of fat-tailed symmetric and asymmetric probability distributions of fluctuations in time series. In the presented approach different kinds of symmetric and asymmetric broad probability distributions of synthetic data are examined starting from Levy regime up to those with finite variance. We use nonextensive Tsallis statistics to construct all considered data in order to have good analytical description of frequencies of fluctuations in the whole range of their magnitude and simultaneously the full control over exponent of power-law decay for tails of probability distribution. The semi-analytical compact formulas are then provided to express the level of spurious multifractality generated by the presence of fat tails in terms of Tsallis parameter q̃ and the scaling exponent β of the asymptotic decay of cumulated probability density function (CDF). The results are presented in Hurst and Hölder languages — more often used in study of multifractal phenomena. According to the provided semi-analytical relations, it is argued how one can make a clear quantitative distinction for any real data between true multifractality caused by the presence of nonlinear correlations, spurious multifractality generated by fat-tailed shape of distributions — eventually with their asymmetry, and the correction due to linear autocorrelations in analyzed time series of finite length. In particular, the spurious multifractal effect of fat tails is found basic for proper quantitative estimation of all spurious multifractal effects. Examples from stock market data are presented to support these findings.

COMPLEXITY AND MULTIFRACTAL OF VOLATILITY DURATION FOR AGENT-BASED FINANCIAL DYNAMICS AND REAL MARKETS

A random agent-based financial model is developed and investigated by the finite-range multitype contact dynamic system, in an attempt to reproduce and study the dynamics of financial markets. And an analysis method of detecting duration and intensity relationship in volatility series is introduced, called the volatility duration analysis. Then the auto-correlation analysis suggests that there exists evident volatility clustering feature in absolute volatility durations for the simulation data and the real data. Besides, the Lempel–Ziv complexity analysis is applied to study the complexity of the returns, the corresponding absolute returns and the volatility duration returns, which can reflect the fluctuation behaviors, the volatility behaviors and the volatility duration behaviors. At last, the multifractal phenomena of volatility durations of returns are comparatively studied for Shanghai Composite Index and the proposed model by multifractal detrended fluctuation analysis.

Multi-fractal scaling comparison of the Air Temperature and the Surface Temperature over China

Abstract The spatial and temporal multi-scaling behaviors between the daily Air Temperature (AT) and the Surface Temperature (ST) over China are compared in about 60-yr observations by Multi-fractal Detrended Fluctuation Analysis (MF-DFA) method. The different fractal phenomena and diversity features in the geographic distribution are found for the AT and ST series using MF-DFA. There are more multi-fractal features for the AT records but less for ST. The respective geographic sites show important scaling differences when compared to the multi-fractal signatures of AT with ST. An interval threshold for 95% confidence level is obtained by shuffling the AT records and the ST records. For the AT records, 93% of all observed stations shows the strong multi-fractal behaviors. In addition, the multi-fractal characteristics decrease with increasing latitude in South China and are obviously strong along the coast. The multi-fractal behaviors of the AT records between the Yangtze River and Yellow River basin and in most regions of Northwest China seem to be weak and not significant, even single mono-fractal features. However, for the ST records, the geographical distributions of multi-fractal phenomenon seem to be in disorder which account for 81% of the stations. The weak multi-fractal behaviors of the ST records are concentrated in North China, most regions of Northeast China.

Nonlinear multi-analysis of agent-based financial market dynamics by epidemic system.

Based on the epidemic dynamical system, we construct a new agent-based financial time series model. In order to check and testify its rationality, we compare the statistical properties of the time series model with the real stock market indices, Shanghai Stock Exchange Composite Index and Shenzhen Stock Exchange Component Index. For analyzing the statistical properties, we combine the multi-parameter analysis with the tail distribution analysis, the modified rescaled range analysis, and the multifractal detrended fluctuation analysis. For a better perspective, the three-dimensional diagrams are used to present the analysis results. The empirical research in this paper indicates that the long-range dependence property and the multifractal phenomenon exist in the real returns and the proposed model. Therefore, the new agent-based financial model can recurrence some important features of real stock markets.

Multifractal Detrended Cross-Correlation Analysis of Geochemical Element Concentration

We use multifractal detrended cross -fluctuation analysis (MF-DXA) to investigate nonlinear behavior of geochemical element concentration, Au-Cu-Pb-Zn-Ag, in Shangzhuang Deposit, Shandong Province, China. We find that the generalized Hurst exponent h(q) and cross-correlation exponent hxy(q) decrease with the increase of q, which indicate that all element concentration series and their cross pairs exhibit multifractal phenomena. By comparing the variability of h(q) and hxy(q), we have found that the multifractal behavior is more obvious when q > 0 than q < 0 for the element Au- Cu-Pb-Zn and their cross pairs. These analyses, given quantitative information about the complexity of the element concentration, lead to a better understanding of the geochemical phenomena underlying mineralization process.

Nonlinear Fluctuation Behavior of Financial Time Series Model by Statistical Physics System

We develop a random financial time series model of stock market by one of statistical physics systems, the stochastic contact interacting system. Contact process is a continuous time Markov process; one interpretation of this model is as a model for the spread of an infection, where the epidemic spreading mimics the interplay of local infections and recovery of individuals. From this financial model, we study the statistical behaviors of return time series, and the corresponding behaviors of returns for Shanghai Stock Exchange Composite Index (SSECI) and Hang Seng Index (HSI) are also comparatively studied. Further, we investigate the Zipf distribution and multifractal phenomenon of returns and price changes. Zipf analysis and MF-DFA analysis are applied to investigate the natures of fluctuations for the stock market.

On the multifractal effects generated by monofractal signals

We study quantitatively the level of false multifractal signal one may encounter while analyzing multifractal phenomena in time series within multifractal detrended fluctuation analysis (MF-DFA). The investigated effect appears as a result of finite length of used data series and is additionally amplified by the long-term memory the data eventually may contain. We provide the detailed quantitative description of such apparent multifractal background signal as a threshold in spread of generalized Hurst exponent values Δh or a threshold in the width of multifractal spectrum Δα below which multifractal properties of the system are only apparent, i.e. do not exist, despite Δα≠0 or Δh≠0. We find this effect quite important for shorter or persistent series and we argue it is linear with respect to autocorrelation exponent γ. Its strength decays according to power law with respect to the length of time series. The influence of basic linear and nonlinear transformations applied to initial data in finite time series with various levels of long memory is also investigated. This provides additional set of semi-analytical results. The obtained formulas are significant in any interdisciplinary application of multifractality, including physics, financial data analysis or physiology, because they allow to separate the ‘true’ multifractal phenomena from the apparent (artificial) multifractal effects. They should be a helpful tool of the first choice to decide whether we do in particular case with the signal with real multiscaling properties or not.

Dependence phenomenon analysis of the stock market

A random financial stock price model is developed by the interacting contact process, which is one of the statistical-physics systems. The contact process is a continuous-time Markov process, one interpretation of this process is as a model for the spread of an infection, where the epidemic spreading mimics the interplay of local infections and the recovery of individuals. We investigate and analyze the long-term memory, the nonlinear correlations and the multifractal phenomenon of normalized returns of the price model by statistical analysis methods, which include autocorrelation analysis, the Gaussian copula method and the multifractal analysis method. Moreover, we consider the daily returns of the Shanghai Stock Exchange Composite Index and the Shenzhen Stock Exchange Composite Index, and the comparisons of statistical behaviors of returns between the actual data and the simulation data are presented.

Convergence of complex multiplicative cascades

The familiar cascade measures are sequences of random positive measures obtained on [0, 1] via b-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to nontrivial statistically self-similar limits. Is the limit function a monofractal function in multifractal time? General sufficient conditions are given under which such is the case, as well as examples for which no natural time change can be used. In most cases when the sufficient condition for convergence does not hold, we show that either the limit is 0 or the sequence diverges almost surely. In the later case, a functional central limit theorem holds, under some conditions. It provides a natural normalization making the sequence converge in law to a standard Brownian motion in multifractal time.

The L q -spectrum of a class of graph directed self-affine measures

The L q -spectrum of measure is a basic ingredient in the study of fractal geometry, particularly in the study of multifractal phenomena. We calculate the L q -spectrum for a class of graph directed self-affine measures. As an application, the Hausdorff dimensions of these measures and the box-counting dimensions of the support sets are obtained.

Numerical simulation of the realizations and spectra of a quasi-multifractal diffusion process

A discrete model has been developed for a quasi-multifractal diffusion process and a new method for calculating the quasi-multifractal spectrum has been proposed with the use of the statistical processing of the realizations of the process. This method makes it possible to obtain an almost continuous process spectrum, which is impossible by means of traditional analytical methods. The numerical experiments indicate three significantly different regions of the parameters: regions of “monofractality,” “tempered” multifractality, and “strong” multifractality. The model allows deeper insight into the mechanisms of multifractal phenomena in strong turbulence and in the stochastic behavior of financial markets.

Modeling network traffic using generalized Cauchy process

Processes with long-range dependence (LRD) have gained wide applications in many fields of science and technologies ranging from hydrology to network traffic. Two key properties of such processes are LRD that is characterized by the Hurst parameter H and self-similarity (SS) that is measured by the fractal dimension D . However, in the popular traffic model using fractional Gaussian noise (fGn), these two parameters are linearly related. This may be regarded as a limitation of fGn in traffic modeling from the point of view of either accurately fitting real traffic or appropriately explaining the particular multi-fractal phenomena of traffic. In this paper, we discuss recent results in traffic modeling from a view of the generalized Cauchy (GC) process. The GC process is indexed by two parameters D and H . The parameter D in the GC model is independent of H . Hence, it provides a more flexible way to describe the multi-fractal phenomena of traffic in addition to accurately modeling traffic for both short-term lags and long-term ones.

Multifractal spectra of in-homogenous self-similar measures

Let Si:Rd?Rd for i=1,...,N be contracting similarities. Also, let (p1,...,pN,p) be a probability vector and let ? be a probability measure on Rd with compact support. Then there exists a unique probability measure µ on Rd such that µ=?ipiµ°Si-1+p?. The measure µ is called an in-homogenous self-similar measure. In previous work we computed the Lq spectra of in-homogenous self-similar measures. In this paper we study the significantly more difficult problem of computing the multifratal spectra of in-homogenous self-similar measures satisfying the In-homogenous Open Set Condition. We prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the in-homogenous case. In particular, we show that the multifractal spectra of in-homogenous self-similar measures may be non-concave. This is in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Several applications are presented. Many of our applications are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. We show that our main results can be applied to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. Other applications to non-linear self-similar measures introduced by Glickenstein and Strichartz are also presented.

A universal model to characterize different multi-fractal behaviors of daily temperature records over China

The multi-fractal properties of surface air temperature over China are studied through Multi-fractal detrended fluctuation analysis (MF-DFA). It is shown that there indeed exists the multi-fractal phenomenon in daily surface air temperature time series, reflecting a great deal of fluctuations at various time scales. The multi-fractal properties can be characterized very well by a universal generalized binomial multiplicative cascade model with only two parameters a and b . For different stations, the width of singularity spectrum f ( α ) is different, indicating different strengths of temperature multi-fractal behavior from station to station.