Statistics of biophysical signal characteristics and state specificity of the human EEG

The arguments are given that local exponents obtained in multifractal analysis by two methods: wavelet transform modulus maxima (WTMM) and multifractal detrended fluctuation analysis (MDFA) allow to separate statistically hearts of healthy people and subjects suffering from reduced left ventricle systolic function (NYHA I–III class). Proposed indices of fractality suggest that a signal of human heart rate is a mixture of two processes: monofractal and multifractal ones.     

A new universality class in corpus of texts; A statistical physics study

Text can be regarded as a complex system. There are some methods in statistical physics which can be used to study this system. In this work, by means of statistical physics methods, we reveal new universal behaviors of texts associating with the fractality values of words in a text. The fractality measure indicates the importance of words in a text by considering distribution pattern of words throughout the text. We observed a power law relation between fractality of text and vocabulary size for texts and corpora. We also observed this behavior in studying biological data.     

Extremal coupled map lattices

Multifractal critical phenomena with infinite-temperature critical point and with complex coexistence of the infinite and finite temperature critical points are considered and it is shown that strange attractors generated by cascades of period-doubling bifurcations (Feigenbaum scenario) as well as fields of velocity differences in fluid turbulence belong to the former subclass of the multifractal critical phenomena, while the real traffic processes and real currency exchange processes belong to the last (complex) subclass of the multifractal critical phenomena. Data obtained by different authors are used for this purpose. Received 5 February 1999     

Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.     

Effect of uniaxial tension and hydrostatic compression on the geometry and morphology of amorphous Fe<Subscript>77</Subscript>Ni<Subscript>1</Subscript>Si<Subscript>9</Subscript>B<Subscript>13</Subscript> alloy ribbon surface

The effect of hydrostatic pressure and uniaxial compression on the relief of an amorphous Fe₇₇Ni₁Si₉B₁₃ alloy ribbon surface was studied using scanning tunneling and atomic-force microscopy. The fracture surfaces of samples were also studied. It is found that both the initial surfaces and the surfaces of samples subjected to hydrostatic compression or tension, as well as fracture surfaces, are fractal or multifractal, but their fractality parameters are different. Hydrostatic pressure decreases the surface roughness and the average fractal dimension of the surface on both sides of the ribbons. The dependence of the surface fractal characteristics on tension is more complex. Prior to the occurrence of a “critical event” on the surface (formation of a deformation band or a through crack), the Hölder index and the half-width of the singularity spectrum decrease. The correlation is discussed between the fractal characteristics of the ribbon surface and those of a fracture surface, and the role of an excess free volume in the initiation of fracture of amorphous alloys is analyzed.     

400GeV/cpp碰撞多粒子产生的自仿射分形

利用４００ＧｅＶ／ｃｐｐ碰撞多粒子产生的实验数据进行了自仿射分形分析，并与自相似分析相比较，结果表明在实验分辨能力范围内，自仿射分形分析具有嵔虾玫谋甓刃形     

Self-affine Fractality of Multiparticle Productionin pp Collisions at 400 GeV/c

A Self-affine fractality analysis was performed by using the experimental data of multiparticle production in pp collisions at 400GeV/c. Compared with the results obtained from the selfsimilar analysis, the self-affine fractality analysis shows a better scaling behavior within the limits of the experimenal resolution.     

Improved visibility graph fractality with application for the diagnosis of Autism Spectrum Disorder

Recently, the visibility graph (VG) algorithm was proposed for mapping a time series to a graph to study complexity and fractality of the time series through investigation of the complexity of its graph. The visibility graph algorithm converts a fractal time series to a scale-free graph. VG has been used for the investigation of fractality in the dynamic behavior of both artificial and natural complex systems. However, robustness and performance of the power of scale-freeness of VG (PSVG) as an effective method for measuring fractality has not been investigated. Since noise is unavoidable in real life time series, the robustness of a fractality measure is of paramount importance. To improve the accuracy and robustness of PSVG to noise for measurement of fractality of time series in biological time-series, an improved PSVG is presented in this paper. The proposed method is evaluated using two examples: a synthetic benchmark time series and a complicated real life Electroencephalograms (EEG)-based diagnostic problem, that is distinguishing autistic children from non-autistic children. It is shown that the proposed improved PSVG is less sensitive to noise and therefore more robust compared with PSVG. Further, it is shown that using improved PSVG in the wavelet-chaos neural network model of Adeli and c-workers in place of the Katz fractality dimension results in a more accurate diagnosis of autism, a complicated neurological and psychiatric disorder.     

Entropy and Multifractal-Multiscale Indices of Heart Rate Time Series to Evaluate Intricate Cognitive-Autonomic Interactions

Recent research has clarified the existence of a networked system involving a cortical and subcortical circuitry regulating both cognition and cardiac autonomic control, which is dynamically organized as a function of cognitive demand. The main interactions span multiple temporal and spatial scales and are extensively governed by nonlinear processes. Hence, entropy and (multi)fractality in heart period time series are suitable to capture emergent behavior of the cognitive-autonomic network coordination. This study investigated how entropy and multifractal-multiscale analyses could depict specific cognitive-autonomic architectures reflected in the heart rate dynamics when students performed selective inhibition tasks. The participants (N=37) completed cognitive interference (Stroop color and word task), action cancellation (stop-signal) and action restraint (go/no-go) tasks, compared to watching a neutral movie as baseline. Entropy and fractal markers (respectively, the refined composite multiscale entropy and multifractal-multiscale detrended fluctuation analysis) outperformed other time-domain and frequency-domain markers of the heart rate variability in distinguishing cognitive tasks. Crucially, the entropy increased selectively during cognitive interference and the multifractality increased during action cancellation. An interpretative hypothesis is that cognitive interference elicited a greater richness in interactive processes that form the central autonomic network while action cancellation, which is achieved via biasing a sensorimotor network, could lead to a scale-specific heightening of multifractal behavior.     

Comment on ‘multifractal diffusion entropy analysis on stock volatility in financial markets’ [Physica A 391 (2012) 5739–5745]

In their recent article ‘multifractal diffusion entropy analysis on stock volatility in financial markets’ Huang, Shang and Zhao (2012) [6] suggested a generalization of the diffusion entropy analysis method with the main goal of being able to reveal scaling exponents for multifractal times series. The main idea seems to be replacing the Shannon entropy by the Rényi entropy, which is a one-parametric family of entropies. The authors claim that based on their method they are able to separate long range and short correlations of financial market multifractal time series. In this comment I show that the suggested new method does not bring much valuable information in obtaining the correct scaling for a multifractal/mono-fractal process beyond the original diffusion entropy analysis method. I also argue that the mathematical properties of the multifractal diffusion entropy analysis should be carefully explored to avoid possible numerical artefacts when implementing the method in analysis of real sequences of data.     

Multifractality through Non-Markovian Stochastic Processes in the Scale Relativity Theory. Acute Arterial Occlusions as Scale Transitions

By assimilating biological systems, both structural and functional, into multifractal objects, their behavior can be described in the framework of the scale relativity theory, in any of its forms (standard form in Nottale’s sense and/or the form of the multifractal theory of motion). By operating in the context of the multifractal theory of motion, based on multifractalization through non-Markovian stochastic processes, the main results of Nottale’s theory can be generalized (specific momentum conservation laws, both at differentiable and non-differentiable resolution scales, specific momentum conservation law associated with the differentiable–non-differentiable scale transition, etc.). In such a context, all results are explicated through analyzing biological processes, such as acute arterial occlusions as scale transitions. Thus, we show through a biophysical multifractal model that the blocking of the lumen of a healthy artery can happen as a result of the “stopping effect” associated with the differentiable-non-differentiable scale transition. We consider that blood entities move on continuous but non-differentiable (multifractal) curves. We determine the biophysical parameters that characterize the blood flow as a Bingham-type rheological fluid through a normal arterial structure assimilated with a horizontal “pipe” with circular symmetry. Our model has been validated based on experimental clinical data.     

Multifractal moving average analysis and test of multifractal model with tuned correlations

We address two common major problems in the study of time series characterizing fluctuations in complex systems: multifractal analysis and multifractal modeling. Specifically, we introduce a multi-fractal centered moving average (MF-CMA) analysis, which is computationally easier but equivalently performing compared with the well-established multi-fractal detrended fluctuation analysis (MF-DFA) with linear detrending. In addition, we study in detail a generalized binomial multi-fractal model (GB-MFM) to conveniently and reliably generate multifractal surrogate data with arbitrary singularity strengths and arbitrary long-term persistence. We use the data generated by this model as well as realistic, by construction monofractal data series with crossovers and trends to test and compare the multifractal analysis methods and discuss finite-size effects as well as limitations due to spurious multifractality.     

基于多个无标度区的多重分形分析方法

用计算机模拟生成了多重分形结构,通过对比分析结构的解析多重分形谱和配分函数法计算得到的多重分形谱,总结出多重分形谱可以描述结构在某一无标度区内生长规律的特性,发现结构的各个无标度区都具有研究价值,针对传统方法不能充分利用数据的缺陷,提出了基于多个无标度区的多重分形谱计算方法.     

Scaling, multifractality, and long-range correlations in well log data of large-scale porous media

Three distinct methods, namely, the spectral density, the multifractal random walk approach, and the multifractal detrended fluctuation analysis are utilized to study the properties of four distinct types of well logs from three oil and gas fields, namely, the natural gamma ray emission, neutron porosity, bulk density, and the sonic transient time logs. Such well logs have never been analyzed by the methods that we utilize in the present study. The results indicate that the well logs exhibit multifractal characteristics, and the estimated Hurst exponents by the three methods are close to each other. Using multifractal detrended fluctuation analysis and the shuffled and surrogated data, we find that the source of multifractality is due to both broad probability density functions of the data and long-range correlations in them. The correlations are persistent and are characterized by a Hurst exponent H>0.5. Despite very significant differences in the geology of the three reservoirs-ranging from shaly sands to fractured carbonate reservoirs-there is a rough universality in the log data in that, the Hurst exponents for all the logs vary in a very narrow range.     

Multifractal models via products of geometric OU-processes: Review and applications

This paper reviews a class of multifractal models obtained via products of exponential Ornstein–Uhlenbeck processes driven by Lévy motion. Given a self-decomposable distribution, conditions for constructing multifractal scenarios and general formulas for their Renyi functions are provided. Together with several examples, a model with multifractal activity time is discussed and an application to exchange data is presented.     

A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect

Stock markets can become inefficient due to calendar anomalies known as the day-of-the-week effect. Calendar anomalies are well known in the financial literature, but the phenomena remain to be explored in econophysics. This paper uses multifractal analysis to evaluate if the temporal dynamics of market returns also exhibit calendar anomalies such as day-of-the-week effects. We apply multifractal detrended fluctuation analysis (MF-DFA) to the daily returns of market indices worldwide for each day of the week. Our results indicate that distinct multifractal properties characterize individual days of the week. Monday returns tend to exhibit more persistent behavior and richer multifractal structures than other day-resolved returns. Shuffling the series reveals that multifractality arises from a broad probability density function and long-term correlations. The time-dependent multifractal analysis shows that the Monday returns’ multifractal spectra are much wider than those of other days. This behavior is especially persistent during financial crises. The presence of day-of-the-week effects in multifractal dynamics of market returns motivates further research on calendar anomalies for distinct market regimes.     

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.