Multiplicative Cascade Processes Research Articles

Introduction of k-means clustering into random cascade model for disaggregation of rainfall from daily to 1-hour resolution with improved preservation of extreme rainfall

Temporal disaggregation of rainfall has been of particular focus because of the non-availability of higher-resolution rainfall data for a long-duration period. Fine temporal resolution rainfall is used in a multitude of hydrological applications. Researchers have proposed various disaggregation models to disaggregate coarse temporal resolution rainfall. In this paper, firstly, the microcanonical multiplicative random cascade (MMRC) model is applied for disaggregation from daily rainfall to a one-hour scale. The model is applied in four different rainfall stations for disaggregation having varying rainfall patterns and characteristics. It is observed that the MMRC model can generate statistically reliable rainfall time series; however, the extreme rainfall characteristics are not well conserved by the model for all the stations.This paper describes a new model based on a random multiplicative cascade process where classification and parameter generation is done by k-means clustering such that it can better conserve extreme rainfall conditions and generate a reliable rainfall time series (MMRC-K). K-means clustering is a vector quantization method that divides the observations into a particular number of clusters based on the nearest mean called cluster centroid. The novel approach is tested with the same four Indian cities. The use of k-means clustering has made the classification and parameter generation of the model robust such that it can work with data sets of varying characteristics. It is found that MMRC-K provides improved conservation of extreme rainfall characteristics compared to the MMRC model for all four stations. The MMRC-K model reproduces the IDF curves of Delhi and Mumbai stations quite well; however, a little discrepancy was observed at higher resolution and larger return periods in Kolkata and Chennai stations. Extreme rainfall at finer resolution is used in various hydrological analyses and design problems like urban drainage design, stormwater management, etc. The overall superior conservation of the extreme rainfall characteristics in the model-generated rainfall time series by the MMRC-K model compared to the MMRC model supports the potential applicability of the model for temporal disaggregation.

A smart productivity evaluation method for shale gas wells based on 3D fractal fracture network model

The generation method of three-dimensional fractal discrete fracture network (FDFN) based on multiplicative cascade process was developed. The complex multi-scale fracture system in shale after fracturing was characterized by coupling the artificial fracture model and the natural fracture model. Based on an assisted history matching (AHM) using multiple-proxy-based Markov chain Monte Carlo algorithm (MCMC), an embedded discrete fracture modeling (EDFM) incorporated with reservoir simulator was used to predict productivity of shale gas well. When using the natural fracture generation method, the distribution of natural fracture network can be controlled by fractal parameters, and the natural fracture network generated coupling with artificial fractures can characterize the complex system of different- scale fractures in shale after fracturing. The EDFM, with fewer grids and less computation time consumption, can characterize the attributes of natural fractures and artificial fractures flexibly, and simulate the details of mass transfer between matrix cells and fractures while reducing computation significantly. The combination of AMH and EDFM can lower the uncertainty of reservoir and fracture parameters, and realize effective inversion of key reservoir and fracture parameters and the productivity forecast of shale gas wells. Application demonstrates the results from the proposed productivity prediction model integrating FDFN, EDFM and AHM have high credibility.

Infilling missing data of binary geophysical fields using scale invariant properties through an application to imperviousness in urban areas

ABSTRACT High-resolution modelling is needed to improve the understanding and management of storm water in cities. It requires data, which is not always available; hence the growing importance of handling missing data. Here, we use impervious areas in cities as case study. They are responsible for rapid runoff that can generate surface flooding. A methodology to handle such binary missing data relying on scale-invariant properties is presented. It uses a previous study, which showed in 10 peri-urban areas that imperviousness exhibits scale-invariant features from metres to kilometres, to generate realistic scenarios for the missing impervious data. More precisely, fractal fields are commonly simulated thanks to a simple binary multiplicative cascade process (β-model). Here we condition it to the available data. Numerical simulations are used to confirm theoretical expectations. They are then implemented to infill missing impervious data on a 3 km2 catchment and the corresponding uncertainty is quantified.

Enhancing fracture-network characterization and discrete-fracture-network simulation with high-resolution surveys using unmanned aerial vehicles

A workflow is presented that integrates unmanned aerial vehicle (UAV) imagery with discrete fracture network (DFN) geometric characterization and quantification of fluid flow. The DFN analysis allows for reliable characterization and reproduction of the most relevant features of fracture networks, including: identification of orientation sets and their characteristics (mean orientation, dispersion, and prior probability); scale invariance in distributions of fracture length and spatial location/clustering; and the distribution of aperture values used to compute network-scale equivalent permeability. A two-dimensional DFN-generation approach honors field data by explicitly reproducing observed multi-scale fracture clustering using a multiplicative cascade process and power law distribution of fracture length. The influence of aperture on network-scale equivalent permeability is investigated using comparisons between a sublinear aperture-to-length relationship and constant aperture. To assess the applicability of the developed methodology, DFN flow simulations are calibrated to pumping test data. Results suggest that even at small scales, UAV surveys capture the essential geometrical properties required for fluid flow characterization. Both the constant and sublinear aperture scaling approaches provide good matches to the pumping test results with only minimal calibration, indicating that the reproduced networks sufficiently capture the geometric and connectivity properties characteristic of the granitic rocks at the study site. The sublinear aperture scaling case honors the directions of dominant fractures that play a critical role in connecting fracture clusters and provides a realistic representation of network permeability.

Approximate multifractal correlation and products of universal multifractal fields, with application to rainfall data

Abstract. Universal multifractals (UMs) have been widely used to simulate and characterize, with the help of only two physically meaningful parameters, geophysical fields that are extremely variable across a wide range of scales. Such a framework relies on the assumption that the underlying field is generated through a multiplicative cascade process. Derived analysis techniques have been extended to study correlations between two fields not only at a single scale and for a single statistical moment as with the covariance, but across scales and for all moments. Such a framework of joint multifractal analysis is used here as a starting point to develop and test an approach enabling correlations between UM fields to be analysed and approximately simulated. First, the behaviour of two fields consisting of renormalized multiplicative power law combinations of two UM fields is studied. It appears that in the general case the resulting fields can be well approximated by UM fields with known parameters. Limits of this approximation will be quantified and discussed. Techniques to retrieve the UM parameters of the underlying fields as well as the exponents of the combination have been developed and successfully tested on numerical simulations. In a second step tentative correlation indicators are suggested. Finally the suggested approach is implemented to study correlation across scales of detailed rainfall data collected with the help of disdrometers of the Fresnel platform of Ecole des Ponts ParisTech (see available data at https://hmco.enpc.fr/portfolio-archive/taranis-observatory/, last access: 12 March 2020). More precisely, four quantities are used: the rain rate (R), the liquid water content (LWC) and the total drop concentration (Nt) along with the mass weighed diameter (Dm), which are commonly used to characterize the drop size distribution. Correlations across scales are quantified. Their relative strength (very strong between R and LWC, strong between DSD features and R or LWC, almost null between Nt and Dm) is discussed.

Blunt extension of discrete universal multifractal cascades: development and application to downscaling

ABSTRACT Scale issues are ubiquitous in geosciences. Because of their simplicity and intuitiveness, and despite strong limitations, notably its non-stationarity features, discrete random multiplicative cascade processes are very often used to address these scale issues. A novel approach based on the parsimonious framework of Universal Multifractals (UM) is introduced to tackle this issue while preserving the simple structure of discrete cascades. It basically consists in smoothing at each cascade step the random multiplicative increments with the help of a geometric interpolation over a moving window. The window size enables to introduce non-conservativeness in the simulated fields. It is established theoretically,] and numerically confirmed, that the simulated fields also exhibit a multifractal behaviour with expected features. It is shown that such an approach remains valid over a limited range of UM parameters. Finally, we test downscaling of rainfall fields with the help of this blunt discrete cascade process, and we discuss challenges for future developments.

A Modified Multifractal Detrended Fluctuation Analysis (MFDFA) Approach for Multifractal Analysis of Precipitation in Dongting Lake Basin, China

Multifractal detrended fluctuation analysis (MFDFA) method can examine higher-dimensional fractal and multifractal characteristics hidden in time series. However, removal of local trends in MFDFA is based on discontinuous polynomial fitting, resulting in pseudo-fluctuation errors. In this paper, we propose a two-stage modified MFDFA for multifractal analysis. First, an overlap moving window (OMW) algorithm is introduced to divide time series of the classic MFDFA method. Second, detrending by polynomial fitting local trend in traditional MFDFA is replaced by ensemble empirical mode decomposition (EEMD)-based local trends. The modified MFDFA is named OMW-EEMD-MFDFA. Then, the performance of the OMW-EEMD-MFDFA method is assessed by extensive numeric simulation experiments based on a p-model of multiplicative cascading process. The results show that the modified OMW-EEMD-MFDFA method performs better than conventional MFDFA and OMW-MFDFA methods. Lastly, the modified OMW-EEMD-MFDFA method is applied to explore multifractal characteristics and multifractal sources of daily precipitation time series data at the Mapoling and Zhijiang stations in Dongting Lake Basin. Our results showed that the scaling properties of the daily precipitation time series at the two stations presented a long-range correlation, showing a long-term persistence of the previous state. The strong q-dependence of H ( q ) and τ ( q ) indicated strong multifractal characteristics in daily precipitation time series data at the two stations. Positive Δ f values demonstrate that precipitation may have a local increasing trend. Comparing the generalized Hurst exponent and the multifractal strength of the original precipitation time series data with its shuffled and surrogate time series data, we found that the multifractal characteristics of the daily precipitation time series data were caused by both long-range correlations between small and large fluctuations and broad probability density function, but the broad probability density function was dominant. This study may be of practical and scientific importance in regional precipitation forecasting, extreme precipitation regulation, and water resource management in Dongting Lake Basin.

Stochastic explosion and non-uniqueness for α-Riccati equation

We consider the problem of global in time existence and uniqueness for the initial value problems u′(t)=−u(t)+u2(αt),u(0)=u0≥0 for scaling parameters α≥0 and a large class of initial data u0. Much of the focus is on the “super-critical” case α>1 where by using the multiplicative stochastic cascade techniques we prove global existence for small initial data and finite-time blow-up for large initial data. However, while uniqueness holds for solutions satisfying a growth condition, in general it fails even for arbitrary small initial data. We demonstrate that this lack of uniqueness is directly connected to the stochastic explosion of an associated multiplicative stochastic cascade process. The key tool in establishing the above-mentioned results is an iterative algorithm that allows one to exploit the stochastic explosion of the underlying multiplicative cascade to establish both existence and lack of uniqueness of solutions.

<b>A Brief multifractal analysis of rainfall dynamics in Piracicaba, São Paulo, Brazil

The aim of this study was to evaluate the statistical properties of a daily rainfall time series recorded in Piracicaba river basin, state of Sao Paulo, Brazil in the period 1917-2016. We apply Multifractal Detrended Fluctuation Analysis (MF-DFA) on deseasonalized data and calculate generalized scaling exponents, h ( q ). We find that h ( q ) is a decreasing function of q , indicating multifractal behavior of precipitation dynamics. We also perform calculations on two subseries corresponding to the periods before and after 1970, where the second period is characterised by a significant increase of positive trend in rainfall intensity. We find no significant change in behaviour between these two subseries and the entire series, indicating that the dynamics of the underlying process did not change. Our results agree with previous studies on multifractal properties of rainfall time series in other regions of the world supporting the hypotheses of the existence of multiplicative cascade processes in the atmosphere.

Singularity analysis of global zircon U-Pb age series and implication of continental crust evolution

Recently compiled global databases of igneous and detrital zircon U-Pb ages have been integrated with other types of isotope data (e.g., neodymium, hafnium, and oxygen) to characterize the episodic growth of the continental crust and the development of supercontinents. However, to what extent do the age peaks reflect the rate of continent crust growth or the differential preservation due to supercontinent settings is a long-standing question. Instead of analyzing amplitudes and periodicity of the age series, here, the analysis focuses on shapes of the individual age peaks described by a power law model for measuring the scale invariant fractality and singularity of time-series records. The results indicate that zircon age distributions around peaks follow power law distribution, regardless of the bin-size used to measure the age distribution. Based on the commonly accepted mechanisms (phase transition, self-organized criticality and multiplicative cascade processes) for generation of power law distributions one can relate the nonlinearity of the age peaks to short spurts of accelerated magmatic activities due to “avalanches” (superplumes, slab breakoff etc.) occurred during episodic convection of the mantle. The exponents of the power law age distributions estimated from the age peaks can be used as an index to quantify the intensity of a singularity. The values of exponents calculated at all major age peaks from five global databases exhibit an episodic nature, with a mean duration of approximately 600–800Myr. Both intensity of the zircon episodes and their duration for the interval from 3Ga to 0.5Ga depict a descending trend, which may signify mantle cooling.

Multifractal characterization of mechanical vibration signals through improved empirical mode decomposition-based detrended fluctuation analysis

A novel multifractal detrended fluctuation analysis based on improved empirical mode decomposition for the non-linear and non-stationary vibration signal of machinery is proposed. As the intrinsic mode functions selection and Kolmogorov–Smirnov test are utilized in the detrending procedure, the present approach is quite available for contaminated data sets. The intrinsic mode functions selection is employed to deal with the undesired intrinsic mode functions named pseudocomponents, and the two-sample Kolmogorov–Smirnov test works on each intrinsic mode function and Gaussian noise to detect the noise-like intrinsic mode functions. The proposed method is adaptive to the signal and weakens the effect of noise, which makes this approach work well for vibration signals collected from poor working conditions. We assess the performance of the proposed procedure through the classic multiplicative cascading process. For the pure simulation signal, our results agree with the theoretical results, and for the contaminated time series, the proposed method outperforms the traditional multifractal detrended fluctuation analysis methods. In addition, we analyze the vibration signals of rolling bearing with different fault types, and the presence of multifractality is confirmed.

Hydrothermal mineralising systems as chemical reactors: Wavelet analysis, multifractals and correlations

Hydrothermal mineralising systems are discussed as large, open flow chemical reactors held far from equilibrium during their life-time by deformation and the influx of heat, fluid and dissolved chemical species. As such they are nonlinear dynamical systems and need to be analysed using the tools that have been developed for such systems. Hydrothermal systems undergo a number of phase transitions during their evolution and this paper focuses on methods for characterising these transitions in a quantitative manner and establishing whether they resemble either abrupt or continuous (critical) phase transitions or whether they have some other kind of nature. Critical phase transitions are characterised by long range correlations for some parameter characteristic of the system, power-law probability distributions, so that there is no characteristic length scale, and a high sensitivity to perturbations. The transitions undergone in mineralised hydrothermal systems are: (i) widespread, non-localised mineral alteration involving exothermic mineral reactions that produce hydrous silicate phases, carbonates and iron-oxides, (ii) strongly localised veining, brecciation and/or stock-work formation, (iii) a series of localised endothermic mineral reactions involving the formation of non-hydrous silicates, sulphides and metals such as gold, (iv) multiple overprinting repetitions of transitions (ii) and (iii). We quantify aspects of these transitions in some gold deposits from the Yilgarn craton of Western Australia using wavelet transforms. This technique is convenient and fast. It enables one to establish if the transition is multifractal (and if so, quantify the multifractal, or singularity, spectrum) and to determine the scale dependence of long range correlations or anti-correlations. Other aspects of the spectrum enable quantitative distinctions between sub-critical, critical and super-critical systems. The availability of long drill holes with detailed chemical analyses and mineral abundances derived from hyperspectral data enables individual ore bodies to be characterised rapidly in a quantitative manner and constraints placed on whether the various transitions are possibly critical or of some other form. We also present some simple nonlinear models, including numerical simulation, self-organised branching and multiplicative cascade processes that produce the multifractal character and correlation scaling relations observed in these data sets. Distinctions between systems that are strongly and weakly mineralised are discussed.

A comparative study of two-dimensional multifractal detrended fluctuation analysis and two-dimensional multifractal detrended moving average algorithm to estimate the multifractal spectrum

Multifractal detrended fluctuation analysis (MFDFA) and multifractal detrended moving average (MFDMA) algorithm have been established as two important methods to estimate the multifractal spectrum of the one-dimensional random fractal signal. They have been generalized to deal with two-dimensional and higher-dimensional fractal signals. This paper gives a brief introduction of the two-dimensional multifractal detrended fluctuation analysis (2D-MFDFA) and two-dimensional multifractal detrended moving average (2D-MFDMA) algorithm, and a detailed description of the application of the two-dimensional fractal signal processing by using the two methods. By applying the 2D-MFDFA and 2D-MFDMA to the series generated from the two-dimensional multiplicative cascading process, we systematically do the comparative analysis to get the advantages, disadvantages and the applicabilities of the two algorithms for the first time from six aspects such as the similarities and differences of the algorithm models, the statistical accuracy, the sensitivities of the sample size, the selection of scaling range, the choice of the q-orders and the calculation amount. The results provide a valuable reference on how to choose the algorithm from 2D-MFDFA and 2D-MFDMA, and how to make the schemes of the parameter settings of the two algorithms when dealing with specific signals in practical applications.

Dual-tree complex wavelet transform based multifractal detrended fluctuation analysis for nonstationary time series

Multifractal detrended fluctuation analysis is an effective tool for dealing with the non-uniformity and singularity of nonstationary time series. For the serious issues of the trend extraction and the inefficient computation in the traditional polynomial fitting based multifractal detrended fluctuation analysis, based on the dual-tree complex wavelet transform, a novel multifractal analysis is proposed. To begin with, as the dual-tree complex wavelet transform has the anti-aliasing and nearly shift-invariance, it is first utilized to decompose the signal through the pyramid algorithm, and the scale-dependent trends and the fluctuations are extracted from the wavelet coefficients. Then, using the wavelet coefficients, the length of the non-overlapping segment on a corresponding time scale is computed through the Hilbert transform, and each of the extracted fluctuations is divided into a series of non-overlapping segments whose sizes are identical. Next, on each scale, the detrended fluctuation function for each segment is calculated, and the overall fluctuation function can be obtained by averaging all segments with different orders. Finally, the generalized Hurst index and scaling exponent spectrum are determined from the logarithmic relations between the overall detrended fluctuation function and the time scale and the standard partition function, respectively, and then the multifractal singularity spectrum is calculated with the help of Legendre transform. We assess the performance of the dual-tree-complex wavelet transform based multifractal detrended fluctuation analysis (MFDFA) procedure through the classic multiplicative cascading process and the fractional Brownian motions, which have the theoretical fractal measures. For the multiplicative cascading process, compared with the traditional polynomial fitting based MFDFA methods, the proposed multifractal approach defines the trends and the length of non-overlapping segments adaptively and obtains a more precise result, while for the traditional MFDFA method, for the negative orders, no matter the generalized Hurst index, scaling exponents spectrum, or the multifractal singularity spectrum, the acquired results each have a significant deviation from the theoretical one. For the time series with different sizes, the proposed method can also give a stable result. Compared with the other adaptive method such as maximal overlap discrete wavelet transform based MFDFA and the discrete wavelet transfrom based MFDFA, the proposed approach obtains a very accurate result and has a fast calculation speed. For another time series of fractional Brownian motions with different Hurst indexes of 0.4, 0.5 and 0.6, which represent the anticorrelated, uncorrelated, correlated process, respectively, the results of the proposed method are consistent with those analytical results, while the results of the polynomial fitting based MFDFA methods are most greatly affected by the order of the fitting polynomial. The method in this article provides a valuable reference for how to use the dual-tree complex wavelet transform to realize the multifractal detrended fluctuation analysis, and we can benefit from the signal self-adaptive trend extraction and the high computation efficiency.

Parameter estimation and forecasting for multiplicative log-normal cascades

We study the well-known multiplicative log-normal cascade process in which the multiplication of Gaussian and log normally distributed random variables yields time series with intermittent bursts of activity. Due to the nonstationarity of this process and the combinatorial nature of such a formalism, its parameters have been estimated mostly by fitting the numerical approximation of the associated non-Gaussian probability density function to empirical data, cf. Castaing et al. [Physica D 46, 177 (1990)]. More recently, alternative estimators based upon various moments have been proposed by Beck [Physica D 193, 195 (2004)] and Kiyono et al. [Phys. Rev. E 76, 041113 (2007)]. In this paper, we pursue this moment-based approach further and develop a more rigorous generalized method of moments (GMM) estimation procedure to cope with the documented difficulties of previous methodologies. We show that even under uncertainty about the actual number of cascade steps, our methodology yields very reliable results for the estimated intermittency parameter. Employing the Levinson-Durbin algorithm for best linear forecasts, we also show that estimated parameters can be used for forecasting the evolution of the turbulent flow. We compare forecasting results from the GMM and Kiyono et al.'s procedure via Monte Carlo simulations. We finally test the applicability of our approach by estimating the intermittency parameter and forecasting of volatility for a sample of financial data from stock and foreign exchange markets.

The temporal cascade structure of reanalyses and global circulation models

AbstractThe spatial stochastic structure of deterministic models of the atmosphere has been shown recently to be well modelled by multiplicative cascade processes; in this paper we extend this to the time domain. Using data from the European Centre for Medium Range Weather Forecasting's (ECMWF) reanalyses (ERA40) and two meteorological models (Global Forecast System and Global Environmental Multiscale), we investigate the temporal cascade structures of the temperature, humidity and zonal wind at various altitudes, latitudes and forecast times. First, we estimate turbulent fluxes from the absolute second‐order time differences, showing that the fluxes are generally very close to those estimated in space at the model dissipation scale; thus validating the flux estimates. We then show that temporal cascades with outer scales typically in the range 5–20 days can accurately account for the statistical properties over the range from 6 h up to 3–5 days. We quantify the (typically small) differences in the cascades from model to model, as functions of latitude, altitude and forecast time. By normalizing the moments by the theoretical predictions for universal multifractals, we investigate the ‘Levy collapse’ of the statistics somewhat beyond the outer cascade limit into the low‐frequency weather regime. Although due to finite size effects and the small outer temporal scale, the temporal scaling range is narrow (12–24 h up to 2–10 days), we compare the spatial and temporal statistics by constructing space‐time (Stommel) diagrams, finding space‐time transformation velocities of 450–1000 km day−1, comparable to those predicted based on the solar energy flux driving the system. This transition time‐scale corresponding to planetary size structures objectively defines the transition from usual weather to a low‐frequency weather regime with much lower variability. Finally, we discuss the implications for ensemble forecasting systems, stochastic parametrization and stochastic forecasting. Copyright © 2012 Royal Meteorological Society

Multiplicative cascade processes and information integration for predictive mapping

Abstract. This paper presents a new model proposed on the basis of multiplicative cascade process (MCP) theory for integrating spatial information to be used for mineral resources prediction and environmental impact assessment. Probability of a spatial point event is defined as the probability that a small map calculating unit (map unit) randomly selected from a study area contains one or more points. The probability that such unit randomly selected from a subarea with known spatial binary map patterns (evidential layers) contains one or more points is defined as the posterior point event probability. In this paper, processes of integrating multiple binary map patterns that divide the study area into smaller areas with updated posterior probabilities are viewed as multiplicative cascade processes resulting in a new log-linear model for calculating conditional probabilities from the multiple evidential input layers. The coefficients (weights) involved in this model measuring degree of spatial correlation between point event and the evidential layers are found to be associated with singularity indices involved in multifractal modeling. It is demonstrated that the model is simple and easy to be implemented in comparison with the existing weights of evidence model which is commonly applied in spatial decision modeling. In addition, the posterior probability as the end product of a multiplicative cascade process can be used to describe multifractality and singularity which are useful properties for characterizing spatial distribution of predicted point events. A case study of tin mineral potential mapping in the Gejiu mineral district in China is used to illustrate principles and use of the modeling process. Four binary layers: formation of limestone, buffer distance for intersections of three groups of faults, local and regional geochemical anomalies of elements As, Sn, Cu, Pb, Zn and Cd, were combined for mapping potential areas for occurrence of tin mineral deposits.

Dimension result and KPZ formula for two-dimensional multiplicative cascade processes

We prove a Hausdorff dimension result for the image of two-dimensional multiplicative cascade processes, and we obtain from this result a KPZ-type formula which normally has one point of phase transition.

Modified detrended fluctuation analysis based on empirical mode decomposition for the characterization of anti-persistent processes

Detrended fluctuation analysis (DFA) is a simple but very efficient method for investigating the power-law long-term correlations of non-stationary time series, in which a detrending step is necessary to obtain the local fluctuations at different timescales. We propose to determine the local trends through empirical mode decomposition (EMD) and perform the detrending operation by removing the EMD-based local trends, which gives an EMD-based DFA method. Similarly, we also propose a modified multifractal DFA algorithm, called an EMD-based MFDFA. The performance of the EMD-based DFA and MFDFA methods is assessed with extensive numerical experiments based on fractional Brownian motion and multiplicative cascading process. We find that the EMD-based DFA method performs better than the classic DFA method in the determination of the Hurst index when the time series is strongly anticorrelated and the EMD-based MFDFA method outperforms the traditional MFDFA method when the moment order q of the detrended fluctuations is positive. We apply the EMD-based MFDFA to the 1 min data of Shanghai Stock Exchange Composite index, and the presence of multifractality is confirmed. We also analyze the daily Austrian electricity prices and confirm its anti-persistence.

Near Scale-Free Dynamics in Neural Population Activity of Waking/Sleeping Rats Revealed by Multiscale Analysis

A neuron embedded in an intact brain, unlike an isolated neuron, participates in network activity at various spatial resolutions. Such multiple scale spatial dynamics is potentially reflected in multiple time scales of temporal dynamics. We identify such multiple dynamical time scales of the inter-spike interval (ISI) fluctuations of neurons of waking/sleeping rats by means of multiscale analysis. The time scale of large non-Gaussianity in the ISI fluctuations, measured with the Castaing method, ranges up to several minutes, markedly escaping the low-pass filtering characteristics of neurons. A comparison between neural activity during waking and sleeping reveals that non-Gaussianity is stronger during waking than sleeping throughout the entire range of scales observed. We find a remarkable property of near scale independence of the magnitude correlations as the primary cause of persistent non-Gaussianity. Such scale-invariance of correlations is characteristic of multiplicative cascade processes and raises the possibility of the existence of a scale independent memory preserving mechanism.