The time-singularity multifractal spectrum distribution

Although the multifractal singularity spectrum revealed the distribution of singularity exponent, it failed to consider the temporal information, therefore it is hard to describe the dynamic evolving process of non-stationary and nonlinear systems. In this paper, we aim for a multifractal analysis and propose a time-singularity multifractal spectrum distribution (TS-MFSD), which will hopefully reveal the spatial dynamic character of fractal systems. Similar to the Wigner–Ville time-frequency distribution, the time-delayed conjugation of fractal signals is selected as the windows function. Furthermore, the time-varying Holder exponent and the time-varying wavelet singularity exponent are deduced based on the instantaneous self-correlation fractal signal. The time-singularity exponent distribution i.e. TS-MFSD is proposed, which involves time-varying Hausdorff singularity spectrum distribution, time-varying large deviation multifractal spectrum and time-varying Legendre spectrum distribution, which exhibit the singularity exponent distribution of fractal signal at arbitrary time. Finally, we studied the algorithm of the TS-MFSD based on the wavelet transform module maxima method, analyzed and discussed the characteristic of TS-MFSD based on Devil Staircase signal, stochastic fractional motion and real sea clutter.     

基于多尺度的人体ECG信号质量指数谱分析

提出一种多重分形质量指数谱曲率和面积的概念,Cantor集的验证结果证明该两个参数在探测混沌时间序列复杂性方面是完全有效的.引入多尺度的分析方法,详细研究了不同采样频率和数据点下健康人、心肌缺血患者和心梗患者心电图(ECG)信号的质量指数谱,并与其他非线性参数进行了比较,期望从不同个体间的这种差异,获得区分健康人及有疾病人的非线性特征值.分类指标和单因素方差分析(ANOVA)检验结果表明,该方法达到了最好的综合区分效果.这一结论在早期诊断和临床应用方面具有一定的价值. 关键词： ECG 多尺度 多重分形 质量指数谱     

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.     

脑电信号的标度分析及其在睡眠状态区分中的应用

脑电信号具有长程幂律相关性及多重分形的标度特性,并随生理病理状态改变.本文首次针对睡眠脑电信号应用单重分形去趋势波动分析(detrended fluctuation analysis,简记为DFA)方法与多重分形奇异谱对睡眠脑电信号的标度特征进行系统的对比研究.发现DFA标度指数α对于不同导联和样本组间的差异较为敏感,随睡眠状态的变化不规律;而多重分形奇异强度区间Δα随睡眠状态的变化更为规律,睡眠Ⅰ期至Ⅳ期不断增大,并且导联间差异和样本组间差异均较小.多重分形Δα参数更适合作为判定睡眠状态的定量参数.     

Mass exponent spectrum analysis of human ECG signals and its application to complexity detection

The complexity of electrocardiogram (ECG) signal may reflect the physiological function and healthy status of the heart. In this paper, we introduced two novel intermediate parameters of multifractality, the mass exponent spectrum curvature and area, to characterize the nonlinear complexity of ECG signal. These indicators express the nonlinear superposition of the discrepancies of singularity strengths from all the adjacent points of the spectrum curve and thus overall subsets of original fractal structure. The evaluation of binomial multifractal sets validated these two variables were entirely effective in exploring the complexity of this time series. We then studied the ECG mass exponent spectra taken from the cohorts of healthy, ischemia and myocardial infarction (MI) sufferer based on a large sets of 12 leads’ recordings, and took the statistical averages among each crowd. Experimental results suggest the two values from healthy ECG are apparently larger than those from the heart diseased. While the values from ECG of MI sufferer are much smaller than those from the other two groups. As for the ischemia sufferer, they are almost of moderate magnitude. Afterward, we compared these new indicators with the nonlinear parameters of singularity spectrum. The classification indexes and results of total separating ratios (TSR, defined in the paper) both indicated that our method could achieve a better effect. These conclusions may be of some values in early diagnoses and clinical applications.     

Multifractal detrended cross-correlation analysis in the MENA area

In this paper, we investigated multifractal cross-correlations qualitatively and quantitatively using a cross-correlation test and the Multifractal detrended cross-correlation analysis method (MF-DCCA) for markets in the MENA area. We used cross-correlation coefficients to measure the level of this correlation. The analysis concerns four stock market indices of Morocco, Tunisia, Egypt and Jordan. The countries chosen are signatory of the Agadir agreement concerning the establishment of a free trade area comprising Arab Mediterranean countries. We computed the bivariate generalized Hurst exponent, Rényi exponent and spectrum of singularity for each pair of indices to measure quantitatively the cross-correlations. By analyzing the results, we found the existence of multifractal cross-correlations between all of these markets. We compared the spectrum width of these indices; we also found which pair of indices has a strong multifractal cross-correlation.     

The fractal energy measurement and the singularity energy spectrum analysis

The singularity exponent (SE) is the characteristic parameter of fractal and multifractal signals. Based on SE, the fractal dimension reflecting the global self-similar character, the instantaneous SE reflecting the local self-similar character, the multifractal spectrum (MFS) reflecting the distribution of SE, and the time-varying MFS reflecting pointwise multifractal spectrum were proposed. However, all the studies were based on the depiction of spatial or differentiability characters of fractal signals. Taking the SE as the independent dimension, this paper investigates the fractal energy measurement (FEM) and the singularity energy spectrum (SES) theory. Firstly, we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain, propose the conception of FEM and SES of multifractal signals, and investigate the Hausdorff measure and the local direction angle of the fractal energy element. Then, we prove the compatibility between FEM and traditional energy, and point out that SES can be measured in the fractal space. Finally, we study the algorithm of SES under the condition of a continuous signal and a discrete signal, and give the approximation algorithm of the latter, and the estimations of FEM and SES of the Gaussian white noise, Fractal Brownian motion and the multifractal Brownian motion show the theoretical significance and application value of FEM and SES.     

Harmonic Measure and SLE

In this paper we study the multifractal structure of Schramm’s SLE curves. We derive the values of the (average) spectrum of harmonic measure and prove Duplantier’s prediction for the multifractal spectrum of SLE curves. The spectrum can also be used to derive estimates of the dimension, Hölder exponent and other geometrical quantities. The SLE curves provide perhaps the only example of sets where the spectrum is non-trivial yet exactly computable.     

油菜光谱的多重分形分析及叶绿素诊断建模

作物信息科学的重要内容是如何利用作物的信息对其进行无损营养诊断,光谱分析是一种有效可行的途径。对于油菜而言,冠层光谱的特征是描述其营养状况的重要指标。但由于原始光谱总是受到一些如环境、气候等外在因素的影响,其巨大的波动导致难以直接用于油菜生物量的诊断。然而,光谱的多重分形特征将保持相对稳定。为研究油菜冠层光谱与叶绿素含量的关系,基于多重分形理论,提出了基于油菜冠层光谱特征的叶绿素定量预测模型和定性识别模型。以24个移栽种植小区和24个直播种植小区的高油酸油菜苗期样本为试验对象。首先,利用流行的多重分形去趋势波动分析提取了6个不同波段范围内光谱的广义Hurst指数和质量指数及其他相关的特征参数,发现它们都呈现典型的多重分形特性。但两种不同种植方式下的光谱特征也存在差异。接着,通过多重分形特征参数与SPAD值的相关分析发现不同波段的光谱所含的有效信息不同。以多重分形特征参数建立单变量油菜叶片SPAD值预测模型,移栽方式、直播方式及混合样本的预测模型相对均方根误差均小于5%。最后,以多重分形特征组合建立识别模型,以Fisher线性判别法识别移栽和直播两种种植方式的最大约登指数为0.902 5,对应最敏感波段为350~1 350 nm。这项有意义的工作为预测油菜叶绿素提供了理论基础和实践方法,也为寻找敏感波段进行识别诊断提供了有效的途径。     

On the spectrum of multifractal diffusion process

A quasi-multifractal model of stochastic processes is considered. In contrast to the more widely known multifractal random walk model, it is free of such substantial drawbacks as infinite variance of the modeled processes and time-dependent increments. An analysis of a multifractal diffusion-type process is presented, including the moments of increments and local scaling exponents of the process. A quasi-multifractal spectrum of the process is computed. A focus is on two new concepts in the theory of multifractal processes: effective scaling exponent and quasi-multifractal spectrum of a process.     

Investigating the nanostructured gold thin films using the multifractal analysis

The atomic force microscopy images representing the surface morphology of the nanostructured gold thin films of thickness of 20, 50 and 200 nm, respectively, were investigated using the multifractal analysis. The interface width and growth exponent corresponding to films of different thicknesses were estimated. The surfaces having greater roughness give rise to larger nonlinearity and wider width of the multifractal spectrum. The statistical tests confirm that the gold thin film surfaces under investigation are multifractal in nature.     

Forecasting volatility in Shanghai and Shenzhen markets based on multifractal analysis

This paper analyzes the multifractality in Shanghai and Shenzhen stock markets using multifractal spectrum analysis and multifractal detrended fluctuation analysis. We find that the main source of multifractality is long-range correlations of large and small fluctuations. Then, we introduce a multifractal volatility measure (MV) and find that by taking MV as daily conditional volatility, the simulated series displayed similar “stylized facts” to the original daily return series. By capturing the dynamics of MV using the ARFIMA model, we find that the out-of-sample forecasting performance of the ARFIMA-MV model is better than some GARCH-class models and the ARFIMA-RV model under some criteria of loss function.     

基于双树复小波变换的非平稳时间序列去趋势波动分析方法

多重分形去趋势波动分析是研究非平稳时间序列非均匀性和奇异性的有效工具, 针对该方法中趋势项难以确定的问题, 提出一种基于双树复小波变换的方法, 实现了非平稳信号的多重分形自适应去趋势波动分析. 利用双树复小波变换提取信号的多尺度趋势和波动信息, 通过小波系数的希尔伯特变换确定每个时间尺度不重叠子区间的长度, 使多重分形分析具有信号自适应性及较高的计算效率. 以具有解析形式分形特征的倍增级联信号和分数布朗运动时间序列为例验证本文方法的有效性, 所得结果与解析解相吻合. 与传统的多项式去趋势多重分形方法相比, 本文方法根据信号自身特点自适应地确定信号的趋势和不重叠等长度子区间长度, 所得结果更加精确. 对倍增级联信号时间序列取不同的长度, 验证了算法的稳定性. 分别与基于极大重叠离散小波变换和离散小波变换多重分形方法进行比较, 表明本文方法具有更精确的结果和更快的运算速度.     

On fractal structure of large-scale electron-density inhomogeneities of traveling disturbances in the midlatitude ionosphere

We present the results of studies of the multifractal structure of slow (of duration τ ≈ 10 s) fluctuations of the received-signal amplitudes in special experiments on radio-raying of the midlatitude ionosphere by signals from orbital satellites in 2004–2006. It is shown, in particular, that the method of multifractal analysis of amplitude records of the received signals yields information on the spectrum of large-scale ionospheric inhomogeneities, which is inaccessible for the classical method of radio scintillations. From the results of measurements with the use of multifractal processing of experimental data, we found that large-scale (tens of kilometers) quasiregular electron-density inhomogeneities of traveling ionospheric disturbances (TIDs) have a power-law spectrum. It is exactly the power-law form of the spatial spectrum of large-scale inhomogeneities of TIDs that can be the reason for the observed multifractal structure of the intermittency of slow fluctuations of the received-signal amplitudes. However, under conditions of a developed small-scale turbulence of TIDs, the observed multifractal structure of the received signals is, as a rule, stipulated by the spatial inhomogeneity of the variance of the integral electron-density fluctuations of small-scale inhomogeneities on scales comparable with the sizes of large-scale inhomogeneities of TIDs. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 3, pp. 191–198, March 2008.     

Multifractal analyses of row sum signals of elementary cellular automata

We first apply the WT-MFDFA, MFDFA, and WTMM multifractal methods to binomial multifractal time series of three different binomial parameters and find that the WTMM method indicates an enhanced difference between the fractal components than the known theoretical result. Next, we make use of the same methods for the time series of the row sum signals of the two complementary ECA pairs of rules (90,165) and (150,105) for ten initial conditions going from a single 1 in the central position up to a set of ten 1’s covering the ten central positions in the first row. Since the members of the pairs are actually similar from the statistical point of view, we can check which method is the most stable numerically by recording the differences provided by the methods between the two members of the pairs for various important quantities of the scaling analyses, such as the multifractal support, the most frequent Hölder exponent, and the Hurst exponent and considering as the better one the method that provides the minimum differences. According to this criterion, our results show that the MFDFA performs better than WT-MFDFA and WTMM in the case of the multifractal support, while for the other two scaling parameters the WT-MFDFA is the best. The employed set of initial conditions does not generate any specific trend in the values of the multifractal parameters.     

Wavelet-based multifractal analysis of DNA sequences by using chaos-game representation

Chaos game representation (CGR) is proposed as a scale-independent representation for DNA sequences and provides information about the statistical distribution of oligonucleotides in a DNA sequence. CGR images of DNA sequences represent some kinds of fractal patterns, but the common multifractal analysis based on the box counting method cannot deal with CGR images perfectly. Here, the wavelet transform modulus maxima (WTMM) method is applied to the multifractal analysis of CGR images. The results show that the scale-invariance range of CGR edge images can be extended to three orders of magnitude, and complete singularity spectra can be calculated. Spectrum parameters such as the singularity spectrum span are extracted to describe the statistical character of DNA sequences. Compared with the singularity spectrum span, exon sequences with a minimal spectrum span have the most uniform fractal structure. Also, the singularity spectrum parameters are related to oligonucleotide length, sequence component and species, thereby providing a method of studying the length polymorphism of repeat oligonucleotides.     

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.     

A multi-length-scale theory of the anomalous mixing-length growth for tracer flow in heterogeneous porous media

We develop a multi-length-scale (multifractal) theory for the effect of rock heterogeneity on the growth of the mixing layer of the flow of a passive tracer through porous media. The multifractal exponent of the size of the mixing layer is determined analytically from the statistical properties of a random velocity (permeability) field. The anomalous diffusion of the mixing layer can occur both on finite and on asymptotic length scales.     

Continuous-distribution puddle model for conduction in trilayer graphene

Influence of the weak electric field on the electronic structure of the Fibonacci superlattice is considered. The electric field produces a nonlinear dynamics of the energy spectrum of the aperiodic superlattice. Mechanism of the nonlinearity is explained in terms of energy levels anticrossings. The multifractal formalism is applied to investigate the effect of weak electric field on the statistical properties of electronic eigenfunctions. It is shown that the applied electric field does not remove the multifractal character of the electronic eigenfunctions, and that the singularity spectrum remains non-parabolic, however with a modified shape. Changes of the distances between energy levels of neighbouring eigenstates lead to the changes of the inverse participation ratio of the corresponding eigenfunctions in the weak electric field. It is demonstrated, that the local minima of the inverse participation ratio in the vicinity of the anticrossings correspond to discontinuity of the first derivative of the difference between marginal values of the singularity strength. Analysis of the generalized dimension as a function of the electric field shows that the electric field correlates spatial fluctuations of the neighbouring electronic eigenfunction amplitudes in the vicinity of anticrossings, and the nonlinear character of the scaling exponent confirms multifractality of the corresponding electronic eigenfunctions.     

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.