多重分形降趋波动分析法和移动平均法的分形谱算法对比分析

多重分形降趋波动分析法(MFDFA)和多重分形降趋移动平均法(MFDMA)是用来估算一维随机分形信号多重分形谱的两种算法, 已被拓展应用于二维和高维分形信号的分析. 本文简要介绍了MFDFA和MFDMA算法及其在一维时间序列中的应用. 首次系统地从算法模型、计算统计精度、样本量的敏感性、无标度区选取的敏感性、矩选择的敏感性和计算量这六个方面对两种算法进行了对比分析, 以典型多重分形信号BMC信号为例, 分析两种算法的适用性和优劣性. 为实际应用中, 针对具体信号如何选用MFDFA或MFDMA算法, 以及两种算法的参数设置提供了有价值的参考.     

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.     

薄膜生长的多重分形谱的计算

规则分形具有理想的标度不变性,而随机分形(如薄膜生长)概率分布曲线及标度不变性的范围与计算方法有关:以偏离平均高度的方差值求概率,标度不变性不好,多重分形谱也不光滑;而以薄膜平均底面为基准面求概率得到的标度不变性可延伸到3个数量级,多重分形谱光滑.采用外延的方法修正灰度设置的偏移可以对多重分形谱有所改善.另外,讨论了权重因子q取值范围的影响,确定了q的取值范围.     

多重分形分析图象边缘提取算法

提出了一种基于多重分形分析的图象边缘提取算法,通过计算每个象素点的奇异值和多重分形谱,并根据多重分形谱的各种测度修正,提取出图象的边缘信息.在分析图象的各种象素点的多重分形谱特性的基础上,着重分析了多重分形谱常用的若干测度以及选取标准.该算法利用奇异性Hölder指数和多重分形谱以及它们组成的判据来进行图象边缘提取的思路不同于传统的基于梯度的局部极值点来进行图象边缘提取的方法.实验结果表明:该算法可以在保留重要边缘信息的情况下去除不重要细节,更能符合人的视觉心理.     

400GeV/c pp碰撞自仿射多重分形分析

利用分形来研究高能碰撞多粒子产生机制是新兴的交叉科学,而获取多粒子产生的多重分形谱对了解高能碰撞机制很重要.本文将改进的连续阶数阶乘矩方法应用于40 0GeV/cpp碰撞多重产生的自仿射分析中,首次得到了自仿射多重分形谱.     

基于多重分形谱的光纤周界振动信号识别

为了有效识别光纤周界系统的振动信号,提出一种多重分形谱参数和改进概率神经网络相结合的光纤振动信号识别方法.该方法能够避免特征提取过程中需要选择经验阈值和模式识别过程中需要确定平滑因子的不足.首先,检验分析光纤振动信号多重分形的存在性和有效性.然后,计算和提取光纤振动信号的多重分形谱参数,构成能够准确描述信号非线性和复杂性特性的特征向量.最后,采用改进的概率神经网络算法进行自适应地学习和分类,实现对不同光纤振动信号的识别.采用现场实验采集的四种振动信号对该方法进行验证,结果表明,平均识别率达到96.25%,识别时间为1.63s.该方法在正确识别率方面优于传统的概率神经网络算法.     

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.     

自仿射分形在高能碰撞多重产生中的应用

当分形体在不同方向的标度规律不同时，它就是自仿射而不是自相似．本文指出，高能多重产生中的分形属于这种情况，并建议了一种用实验检验自仿射性和测量自仿射的特征量──赫斯特指数的方法     

气液两相流波动信号多重分形去趋势波动分析

为了研究不同流型压差波动信号的分形特征,本文首先讨论了多重分形去趋势波动分析对周期信号、随机信号和混沌信号的识别能力,然后运用多重分形去趋势波动分析方法对水平管内气-液两相流压差波动信号进行分析.通过计算广义Hurst指数、尺度函数,多重分形谱,细致量化了压差波动信号的局部及不同层次的波动奇异性.研究结果表明:压差波动...     

多重分形及其描述

近年来在分形学及其应用的研究中逐渐发现,要想准确反映千姿百态的分形及其丰富多彩的特征,仅仅用单一取决于整体特征的标度指数即分维是远远不够的。于是研究工作者们提出了多重分形(multifractals)即分形测度(多分形,复分形,多标度     

Application of long-range correlation and multi-fractal analysis for the depiction of drought risk

By using the multi-fractal detrended fluctuation analysis method, we analyze the nonlinear property of drought in southwestern China. The results indicate that the occurrence of drought in southwestern China is multi-fractal and longrange correlated, and these properties are indifferent to timescales. A power-law decay distribution well describes the return interval of drought events and the auto-correlation. Furthermore, a drought risk exponent based on the multi-fractal property and the long-range correlation is presented. This risk exponent can give useful information about whether the drought may or may not occur in future, and provide a guidance function for preventing disasters and reducing damage.     

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.     

海杂波背景下小目标检测的分形方法

针对海杂波背景下小目标检测对海情依赖性强的问题, 本文采用分数布朗运动模型对实测海杂波建模, 结合多重分形去势波动分析法确定分形参数, 分析了海杂波的单尺度、多重分形特性. 在单尺度分形的基础上, 利用表征海杂波分形特征的分数维和Hurst指数构建了分形差量, 提出了基于分形差量的小目标检测方法;在多重分形基础上, 比较了两种海杂波的高尺度多重分形特性. 结果表明, 当尺度q > 10时, 纯海杂波的多重分形参数H(q) < 0, 而存在小目标的H(q) > 0, 此差异性为高尺度分形参数的海杂波背景小目标检测提供了判定依据. 所研究的两种方法均能实现不同海情下的小目标检测.     

Multifractal detrended fluctuation analysis for clustering structures of electricity price periods

A new model is proposed to investigate the structure of electricity price in different time periods. A popular method — the multifractal detrended fluctuation analysis (MF-DFA) method is employed to analyze the features achieved from three types of electricity price data after filtering some trends by Fourier detrended fluctuation function. Twelve multifractal parameters are calculated and selected as the characteristic indicators for comparison. Moreover, the minimum number of indicators is determined so that the discriminant accuracy reaches maximum based on Fisher’s linear discriminant algorithm (Fisher’s LDA) for each time period. These indicators form a multi-dimensional space, in which each point represents a price time series. This allows us to cluster the three price time periods, namely, the low price time periods, the average price time periods and the peak price time periods. Fisher’s LDA is employed to evaluate the discriminant accuracy on these three kinds of time periods. Our analysis is then applied to the data in California1999–2000 and PJM2001–2002 electricity markets to demonstrate the applicability of our methods.     

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.     

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.     

Multiresolution analysis of fluctuations in non-stationary time series through discrete wavelets

We illustrate the efficacy of a discrete wavelet based approach to characterize fluctuations in non-stationary time series. The present approach complements the multifractal detrended fluctuation analysis (MF-DFA) method and is quite accurate for small size data sets. As compared to polynomial fits in the MF-DFA, a single Daubechies wavelet is used here for detrending purposes. The natural, built-in variable window size in wavelet transforms makes this procedure well suited for non-stationary data. We illustrate the working of this method through the analysis of binomial multifractal model. For this model, our results compare well with those calculated analytically and obtained numerically through MF-DFA. To show the efficacy of this approach for finite data sets, we also do the above comparison for Gaussian white noise time series of different sizes. In addition, we analyze time series of three experimental data sets of tokamak plasma and also spin density fluctuations in 2D Ising model.     

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.     

海杂波FRFT域的分形特征分析及小目标检测方法

针对海杂波背景下海情对小目标检测的严重影响, 本文研究了实测海杂波在分数阶Fourier变换(FRFT)域的分形特征, 分别提出了单、高尺度下的分形检测方法. 由数学定义推得, FRFT 在不同阶数和尺度情况下, 不具有一致的自相似特性, 采用多重分形趋势波动分析法确定分形参数H(q), 分析了海杂波在不同海情、距离和极化条件下的分形特征. 在单尺度基础上结合FRFT的变阶优势, 提出了阶数自适应的小目标检测方法; 高尺度条件下, 比较了不同因素对海杂波FRFT域多重分形参数的影响. 结果表明:海杂波FRFT域可用变换阶数的方法检测到湮没在复杂海情中的小信号, 检测门限多数提高200%以上, 比采用时域信号提高26.3%. H(q) 在负高尺度上具有明显的多重分形特征差异, H(q)-q曲线满足反正切分布, 纯海杂波与含目标数据的拟合幅值比分别大于1.8(HH)和1.4(VV), 为海杂波背景小目标检测提供了判定依据.     

A multifractal detrended fluctuation analysis of trading behavior of individual and institutional traders in Tehran stock market

Employing the multifractal detrended fluctuation analysis (MF-DFA), the multifractal properties of trading behavior of individual and institutional traders in the Tehran Stock Exchange (TSE) are numerically investigated. Using daily trading volume time series of these two categories of traders, the scaling exponents, generalized Hurst exponents, generalized fractal dimensions and singularity spectrum are derived. Furthermore, two main sources of multifractality, i.e. temporal correlations and fat-tailed probability distributions are also examined. We also compare our results with data of S&P 500. Results of this paper suggest that for both classes of investors in TSE, multifractality is mainly due to long-range correlation while for S&P 500, the fat-tailed probability distribution is the main source of multifractality.