Detrended Fluctuation Analysis on Correlations of Complex Networks Under Attack and Repair Strategy

We analyze the correlation properties of the Erdos-Rényi random graph (RG) and the Barabási-Albert scale-free network (SF) under the attack and repair strategy with detrended fluctuation analysis (DFA). The maximum degree k representing the local property of the system, shows similar scaling behaviors for random graphs and scale-free networks. The fluctuations are quite random at short time scales but display strong anticorrelation at longer time scales under the same system size N and different repair probability pre. The average degree , revealing the statistical property of the system, exhibits completely different scaling behaviors for random graphs and scale-free networks. Random graphs display long-range power-law correlations. Scale-free networks are uncorrelated at short time scales; while anticorrelated at longer time scales and the anticorrelation becoming stronger with the increase of pre.     

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.     

Crossover Phenomena in Detrended Fluctuation Analysis Used in Financial Markets

A systematic analysis of Shanghai and Japan stock indices for the period of Jan. 1984 to Dec. 2005 is performed. After stationarity is verified by ADF （Augmented Dickey-Fuller） test, the power spectrum of the data exhibits a power law decay as a whole characterized by 1/f^β processes with possible long range correlations. Subsequently, by using the method of detrended fluctuation analysis （DFA） of the general volatility in the stock markets, we find that the long-range correlations are occurred among the return series and the crossover phenomena exhibit in the results obviously.Further, Shanghai stock market shows long-range correlations in short time scale and shows short-range correlations in long time scale. Whereas, for Japan stock market, the data behaves oppositely absolutely. Last, we compare the varying of scale exponent in large volatility between two stock markets. All results obtained may indicate the possibility of characteristic of multifractal scaling behavior of the financial markets.     

混合交通流时间序列的去趋势波动分析

应用去趋势波动分析法研究交通流时间序列的复杂性,探讨了混合交通流时间序列演变行为的标度指数.根据标度指数的变化特征,进而揭示交通流时间序列所具有的长程相关性和短程相关性.通过分析发现,存在一密度ρ,当ρ₁<ρ<ρ₂时,交通流时间序列具有长程相关性;而当ρ<ρ₁或ρ>ρ₂时,交通流时间序列具有短程相关性,即密度的变化影响着标度指数的变化.另外分析了在不同慢车比率条件下时间序列的标度指数,发现慢车比率的变化 关键词： 混合交通流 去趋势波动分析 时间序列 长程相关     

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.     

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.     

Chaoticity analysis of the current through pure, hydrogenated and hydrophobically modified PEG-Si thin films under varying relative humidity

Polyethylene Glycol has an irregular current characteristic under constant voltage and slowly varying relative humidity. The current through a thin film of Gamma-isocyanatopropyltriethoxysilane added Polyethylene glycol (PEG-Si), its hydrogenated and hydrophobically modified forms, as a function of increasing relative humidity at equal time steps is analyzed for chaoticity. We suggest that the irregular behavior of current through PEG-Si thin films as a function of increasing relative humidity could best be analyzed for chaoticity using both time series analysis and detrended uctuation analysis; the relative humidity is kept as a slowly varying parameter. The presence of more then one regime is suggested by the calculation of the maximal Lyapunov exponents. Furthermore, the maximal Lyapunov exponent in each of the regimes was positive, thus confirming the presence of low dimensional chaos. DFA also confirms the presence of at least two different regimes, in agreement with the behavior of the maximal Lyapunov exponent in the time series analysis. We also suggest that the irregular behavior of the current through PEG-Si can be reduced by hydrogenating and hydrophobically modifying PEG-Si and the improvement in stability can be confirmed by our study.      

基于复杂网络的时间序列双变量联动波动

选择伦敦金与Au9999下午收盘价格作为样本数据研究时间序列双变量之间的联动波动规律.依据粗粒化方法,将伦敦金与Au9999价格的联动波动状态转化为由5个{P,N,M}字符组成的字符串,每个字符串代表5天的价格联动波动模态.将模态作为节点,模态之间的转化为边,构建价格联动波动复杂网络.运用复杂网络理论对时间序列双变量联动波动模态的统计、变化规律和演化机制进行分析.结果表明:时间序列双变量联动波动模态分布具有幂律性、群簇性和周期性,其联动波动模态主要通过少数几种模态进行转换与演化.本方法不仅可以研究不同类型时间序列双变量联动波动,同时可为多变量联动波动研究提供思路.     

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.     

On Time Scales of Intrinsic Oscillations in the Climate System

Proxy temperature data records featuring local time series, regional averages from areas all around the globe, as well as global averages, are analyzed using the Slow Feature Analysis (SFA) method. As explained in the paper, SFA is much more effective than the traditional Fourier analysis in identifying slow-varying (low-frequency) signals in data sets of a limited length. We find the existence of a striking gap from ~1000 to about ~20,000 years, which separates intrinsic climatic oscillations with periods ranging from ~60 years to ~1000 years, from the longer time-scale periodicities (20,000 year+) involving external forcing associated with Milankovitch cycles. The absence of natural oscillations with periods within the gap is consistent with cumulative evidence based on past data analyses, as well as with earlier theoretical and modeling studies.     

Correlation between detrended fluctuation analysis and the Lempel–Ziv complexity in nonlinear time series analysis

We focus on the study of the correlation between the detrended fluctuation analysis (DFA) and the Lempel-Ziv complexity (LZC) in nonlinear time series analysis in this paper. Typical dynamical systems including logistic map and Duffing model are investigated. Moreover, the influences of the Gaussian random noise on both DFA and LZC are analyzed. The results show a high correlation between DFA and LZC, which can quantify the non-stationarity and the nonlinearity of the time series, respectively. With the enhancement of the random component, the exponent α and the normalized complexity index C show increasing trends. In addition, C is found to be more sensitive to the fluctuation in the nonlinear time series than α. Finally, the correlation between DFA and LZC is applied to the feature extraction of vibration signals for a reciprocating compressor gas valve, and an effective fault diagnosis result is obtained.     

Analysis of the efficiency and multifractality of gold markets based on multifractal detrended fluctuation analysis

In this paper, we investigate the efficiency and multifractality of a gold market based on multifractal detrended fluctuation analysis. Our evidence shows that the gold return series are multifractal both for time scales smaller than a month and for time scales larger than a month. For time scales smaller than a month, the main contribution of multifractality is fat-tail distribution. For time scales larger than a month, both long-range correlations and fat-tail distribution play important roles in the contribution of multifractality. Using the method of rolling windows, we find that the gold market became more and more efficient over time, especially after 2001. The abnormal points of scaling exponents can also be related to some occasional events. By defining a new inefficiency measure related to the multifractality, we find that the gold market is more efficient during the upward periods than during the downward periods.     

基于去趋势波动分析方法确定极端事件阈值

极端事件或者极值事件脱离了自身的正常演化状态,是系统演化的极端状态或系统受到外界扰动而导致的异常状态.去趋势波动分析法得到的指数是衡量系统在某一时间尺度内演化的长程相关性的参数,系统的长程相关性不受极端事件的影响或影响很小.基于这一思想,提出了利用去趋势波动分析法确定极端事件的阈值方法,并验证了该方法的有效性.使用该方法对北京极端高温事件、极端低温事件和极端降水事件进行了分析和讨论,确定北京1951—2004年极端高温、极端低温事件和极端降水事件的阈值.50余年来,北京极端高温和低温事件在20世纪70年代 关键词： 去趋势波动分析 极端事件 阈值     

Scaling Analysis of an Image Encryption Scheme Based on Chaotic Dynamical Systems

Despite that many image encryption systems based on chaotic or hyperchaotic systems have been proposed to protect different kinds of information, it has been crucial to achieve as much security as possible in such systems. In this sense, we numerically implement a known image encryption system with some variants, making special emphasis when two operations are considered in the scrambling stage. The variants of such an encryption system are based on some hyperchaotic systems, which generated some substitution boxes and the keys of the system. With the aim to have a more complete evaluation, some internal stages of the image encryption scheme have been evaluated by using common statistical tests, and also the scaling behavior of the encrypted images has been calculated by means of a two-dimensional detrended fluctuation analysis (2D-DFA). Our results show that the image encryption systems that include two operations or transformations in the scrambling stage present a better performance than those encryption systems that consider just one operation. In fact, the 2D-DFA approach was more sensitive than some common statistical tests to determine more clearly the impact of multiple operations in the scrambling process, confirming that this scaling method can be used as a perceptual security metric, and it may contribute to having better image encryption systems.     

Theoretical Analysis of the Fluctuation Theorem Applied to Electric Circuits

We carry out a theoretical analysis of the fluctuations in the Johnson-Nyquist noise in an RC electric circuit being driven by a non-ideal constant current source. Using the appropriate Langevin equation, we derive the fluctuation theorem for the system. This analysis exhibits some interesting features. Firstly, the average of the dissipation function is not necessarily just an instantaneous entropy production, i.e. the work being done on the circuit by the external power source divided by the ambient temperature. Secondly, by appropriate selection of the values for the circuit components, the transient response of the system can be made identical to the nonequilibrium steady state response. In the limit of an ideal current source, the average of the dissipation function reduces to the entropy production. Our analysis accurately reproduces published experimental results.     

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.     

The effects of observational correlated noises on multifractal detrended fluctuation analysis

We have numerically investigated the effects that observational correlated noises have on the generalized Hurst exponents, h(q)$h\left(q\right)$, estimated by using the multifractal generalization of detrended fluctuation analysis (MF-DFA). More precisely, artificially generated stochastic binomial multifractals with increased amount of colored noises were analyzed via MF-DFA. It has been recently shown that for moderate additions of white noise, the generalized Hurst exponents are significantly underestimated for q<2$q<2$ and they are nearly unchanged for q≥2$q\ge 2$ [J. Ludescher, M.I. Bogachev, J.W. Kantelhardt, A.Y. Schumann, A. Bunde, On spurious and corrupted multifractality: the effects of additive noise, short- term memory and periodic trends, Physica A 390 (2011) 2480–2490]. In this paper, we have found that h(q)$h\left(q\right)$ with q≥2$q\ge 2$ are also affected when correlated noises are considered. This is due to the fact that the spurious correlations influence the scaling behaviors associated to large fluctuations. The results obtained are significant for practical situations, where noises with different correlations are inherently present.     

Multifractal modeling of the production of concentrated sugar syrup crystal

High quality, concentrated sugar syrup crystal is produced in a critical step in cane sugar production: the clarification process. It is characterized by two variables: the color of the produced sugar and its clarity degree. We show that the temporal variations of these variables follow power-law distributions and can be well modeled by multiplicative cascade multifractal processes. These interesting properties suggest that the degradation in color and clarity degree has a systemwide cause. In particular, the cascade multifractal model suggests that the degradation in color and clarity degree can be equivalently accounted for by the initial "impurities" in the sugarcane. Hence, more effective cleaning of the sugarcane before the clarification stage may lead to substantial improvement in the effect of clarification.     

中能重离子碰撞中的涨落和中间质量碎片发射的机制

在ＱＭＤ模型的基础上，系统研究了４０Ｃａ＋４０Ｃａ中心碰撞系统的中间质量碎片随能量和时间的演化以及它们的动力学起源．集中讨论了反应机制的跃迁，动力学过程中密度涨落和平均中间质量碎片的关系以及碎裂的时间尺度．发现中间质量碎片随能量的上升和下降伴随着涨落的上升和下降．并预言对所研究的反应，当入射能量为６５ＭｅＶ／ｕ时，彻底解体可能发生．碎裂的时间尺度大约为１４０ｆｍ／ｃ，且随能量的增大，此时间尺度不再减小，达到饱和．