Multifractal properties of the Indian financial market

We investigate the multifractal properties of the logarithmic returns of the Indian financial indices (BSE & NSE) by applying the multifractal detrended fluctuation analysis. The results are compared with that of the US S&P 500 index. Numerically we find that qth-order generalized Hurst exponents h(q) and τ(q) change with the moments q. The nonlinear dependence of these scaling exponents and the singularity spectrum f(α) show that the returns possess multifractality. By comparing the MF-DFA results of the original series to those for the shuffled series, we find that the multifractality is due to the contributions of long-range correlations as well as the broad probability density function. The financial markets studied here are compared with the Binomial Multifractal Model (BMFM) and have a smaller multifractal strength than the BMFM.     

Multifractal analysis of polyalanines time series

Multifractal properties of the energy time series of short α-helix structures, specifically from a polyalanine family, are investigated through the MF-DFA technique (multifractal detrended fluctuation analysis). Estimates for the generalized Hurst exponent h(q) and its associated multifractal exponents τ(q) are obtained for several series generated by numerical simulations of molecular dynamics in different systems from distinct initial conformations. All simulations were performed using the GROMOS force field, implemented in the program THOR. The main results have shown that all series exhibit multifractal behavior depending on the number of residues and temperature. Moreover, the multifractal spectra reveal important aspects of the time evolution of the system and suggest that the nucleation process of the secondary structures during the visits on the energy hyper-surface is an essential feature of the folding process.     

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 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.     

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.     

True and apparent scaling: The proximity of the Markov-switching multifractal model to long-range dependence

In this paper, we consider daily financial data of a collection of different stock market indices, exchange rates, and interest rates, and we analyze their multi-scaling properties by estimating a simple specification of the Markov-switching multifractal (MSM) model. In order to see how well the estimated model captures the temporal dependence of the data, we estimate and compare the scaling exponents H(q) (for q=1,2) for both empirical data and simulated data of the MSM model. In most cases the multifractal model appears to generate ‘apparent’ long memory in agreement with the empirical scaling laws.     

Fractal analysis of river flow fluctuations

We use some fractal analysis methods to study river flow fluctuations. The result of the Multifractal Detrended Fluctuation Analysis (MF-DFA) shows that there are two crossover timescales at s_1×∼12 and s_2×∼130 months in the fluctuation function. We discuss how the existence of the crossover timescales are related to a sinusoidal trend. The first crossover is due to the seasonal trend and the value of second one is approximately equal to the well-known cycle of sun activity. Using Fourier Detrended Fluctuation Analysis, the sinusoidal trend is eliminated. The values of Hurst exponents of the runoff water of rivers without the sinusoidal trend show a long-range correlation behavior. For the Daugava river, the value of Hurst exponent is 0.52±0.01 and also we find that these fluctuations have multifractal nature. Comparing the MF-DFA results for the remaining data set of Daugava river to those for shuffled and surrogate series, we conclude that its multifractal nature is almost entirely due to the broadness of probability density function.     

Multifractal behavior of wild-land and forest fire time series in Brazil

We examine statistical properties of a daily hot pixel time series recorded in Brazil during the period 1998–2006, using Multifractal Detrended Fluctuation Analysis (MF-DFA). We find that generalized scaling exponent h(q)$h\left(q\right)$ is a decreasing function of q$q$, indicating multifractal behavior of hot pixel dynamics. We also calculate multifractal spectra f(α)$f\left(\alpha \right)$ and use fourth-degree polynomial regression to estimate complexity parameters that describe the degree of multifractality of the underlying process. After July 2002, when a significant increase of the number of hot pixel observations is recorded, the complexity of the series is reduced (manifested by the reduction of width of the f(α)$f\left(\alpha \right)$ spectrum), while small fluctuations increase their dominance over large scale fluctuations (manifested by the increase of skew parameter r$r$). These results should be taken into account when devising ecological and climatic models for Brazil, that contemplate the phenomena of wild-land and forest fires.     

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.     

Some observations on Levy stability and intermittency in nucleus-nucleus interactions at SPS energies

The power law relation between higher order and second order scaled factorial moments is studied in one dimensional pseudo-rapidity phase (η) space in the interactions of ³²S beam with CNO, AgBr and Emulsion at incident energy of 200 AGeV. Observation for such a power law may indicate a self similar cascade mechanism in multiparticle production process. The values of slope, β_q are found to be independent of target size. The value of the scaling exponent υ = 1.412 obtained is higher than the critical value υ = 1.304, indicating that no second order phase transition exists in our data. The ratio of anomalous fractal dimensions, d_q/d₂ is found to increase with increase in the order of moments, q. The dependence of d_q/d₂ on q indicates a multifractal structure and the presence of self-similar cascading mechanism in our data. The d_q/d₂ values are well described by the Levy-stable distribution with Levy index μ = 1.562 which is consistent with and lies within the Levy stable region (0 ≤ μ ≤ 2). The multifractal spectrum is concave downward with a maximum at q = 0. The decrease in D_q with increasing q shows that there is a self affine multifractal behaviour in multiparticle production in our data.     

A groundwater separation study in boreal wetland terrain: the WATFLOOD hydrological model compared with stable isotope tracers

Monitoring of stable water isotopes (18O and 2H) in precipitation and surface waters in the Mackenzie River basin of northern Canada has created new opportunities for researchers to study the complex hydrology and hydroclimatology of this remote region. A number of prior studies have used stable isotope data to investigate aspects of the hydrological regime of the wetland-dominated terrain near Fort Simpson, Northwest Territories, Canada. The present paper compares estimates of groundwater contributions to streamflow derived using the WATFLOOD distributed hydrological model, equipped with a new water isotope tracer module, with the results of conventional isotope hydrograph separation for five wetland-dominated catchments along the lower Liard River. The comparison reveals highly promising agreement, verifying that the hydrological model is simulating groundwater flow contributions to total streamflow with reasonable fidelity, especially during the crucial snowmelt period. Sensitivity analysis of the WATFLOOD simulations also reveals intriguing features about runoff generation from channelized fens, which may contribute less to streamflow than previously thought.     

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.     

Scaling characteristics in aftershock sequence of earthquake

The possible scale-invariant behavior and the clustering characteristics in aftershock sequence of Chi-Chi (Taiwan) main earthquake (ASCCME) that occurred in 1999/9/20/17/47 were investigated by means of some statistical tools: histogram, spectral analysis, and fractal theory. The examined data were constructed from the aftershocks that occurred at the locations defined at longitude 120.1–121.3 and latitude 23.3–24.5 during the 1999/9/20/17/47–1999/9/24/08/13 period. It was found that the aftershock sequence exhibited the characteristic right-skewed frequency distribution and could be well described with the lognormal distribution. Long-term memory and the possibility of scale invariance were first roughly identified through the analysis of autocorrelation and power spectrum, respectively. Scale invariance was clearly revealed with the aid of box-counting method and the box dimension was shown to be a decreasing function of the threshold magnitude level, i.e., the weak and intense regions scaled differently. To verify the existence of multifractal characteristics, the aftershock sequence was transferred into a useful compact form through the multifractal formalism, namely, the τ(q)–q and f(α)–α plots. The analysis confirmed the existence of multifractal characteristics in the examined aftershock sequence. The origin of both the pronounced right-skewness and multifractal phenomena in aftershock sequence might be interpreted in terms of the multiplicative cascade process of the stress in the Earth's crust. A simple two-scale Cantor set with unequal scales and weights was then used to fit the calculated τ(q)–q plot. This model fitted remarkably well the entire spectrum of scaling exponents of the examined ASCCME.     

Multifractal analysis of the fracture surfaces of foamed polypropylene/polyethylene blends

The two-dimensional multifractal detrended fluctuation analysis is applied to reveal the multifractal properties of the fracture surfaces of foamed polypropylene/polyethylene (PP/PE) blends at different temperatures. Nice power-law scaling relationship between the detrended fluctuation function F_q and the scale s is observed for different orders q and the scaling exponent h(q) is found to be a nonlinear function of q, confirming the presence of multifractality in the fracture surfaces. The multifractal spectra f(α) are obtained numerically through Legendre transform. The shape of the multifractal spectrum of singularities can be well captured by the width of spectrum and the difference of dimension . With the increase of the PE content, the fracture surface becomes more irregular and complex, as is manifested by the facts that increases and decreases from positive to negative. A qualitative interpretation is provided based on the foaming process.     

A brief description to different multi-fractal behaviors of daily wind speed records over China

Scaling behaviors of the long daily wind speed records of four selected weather stations over China were analyzed by using Multi-Fractal Detrended Fluctuation Analysis (MF-DFA). The results indicated that all these four stations are characterized by long-range power-law correlations, but MF-DFA results showed non-universal multi-fractal behaviors over China. We fitted generalized Hurst exponent h(q) via a modified generalized binomial multiplicative cascade model, and different widths of the multi-fractal spectrum are estimated.     

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 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.     

Linearization effect in multifractal analysis: Insights from the Random Energy Model

The analysis of the linearization effect in multifractal analysis, and hence of the estimation of moments for multifractal processes, is revisited borrowing concepts from the statistical physics of disordered systems, notably from the analysis of the so-called Random Energy Model. Considering a standard multifractal process (compound Poisson motion), chosen as a simple representative example, we show the following: (i) the existence of a critical order q^∗ beyond which moments, though finite, cannot be estimated through empirical averages, irrespective of the sample size of the observation; (ii) multifractal exponents necessarily behave linearly in q, for q>q^∗. Tailoring the analysis conducted for the Random Energy Model to that of Compound Poisson motion, we provide explicative and quantitative predictions for the values of q^∗ and for the slope controlling the linear behavior of the multifractal exponents. These quantities are shown to be related only to the definition of the multifractal process and not to depend on the sample size of the observation. Monte Carlo simulations, conducted over a large number of large sample size realizations of compound Poisson motion, comfort and extend these analyses.     

Use of isotopes to study floodplain wetland and river flow interaction in the White Volta River basin,Ghana

Floodplain wetlands influence the timing and magnitude of stream responses to rainfall. In managing and sustaining the level of water resource usage in any river catchment as well as when modelling hydrological processes, it is essential that the role of floodplain wetlands in stream flows is recognised and understood. Existing studies on hydrology within the Volta River basin have not adequately represented the variability of wetland hydrological processes and their contribution to the sustenance of river flow. In order to quantify the extent of floodwater storage within riparian wetlands and their contribution to subsequent river discharges, a series of complementary studies were conducted by utilising stable isotopes, physical monitoring of groundwater levels and numerical modelling. The water samples were collected near Pwalugu on the White Volta River and at three wetland sites adjacent to the river using the grab sampling technique. These were analysed for ¹⁸O and ²H. The analysis provided an estimate of the contribution of pre-event water to overall stream flow. In addition, the variation in the isotopic composition in the river and wetland water samples, respectively, revealed the pattern of flow and exchange of water between the wetlands and the main river system.     

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.