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.     

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.     

Multifractal analysis of streamflow records of the East River basin (Pearl River), China

Scaling behaviors of the long daily streamflow series of four hydrological stations (Longchuan (1952-2002), Heyuan (1951-2002), Lingxia (1953-2002) and Boluo (1953-2002)) in the mainstream East River, one of the tributaries of the Pearl River (Zhujiang River) basin, were analyzed using multifractal detrended fluctuation analysis (MF-DFA). The research results indicated that streamflow series of the East River basin are characterized by anti-persistence. MF-DFA technique showed similar scaling properties in the streamflow series of the East River basin on shorter time scales, indicating universal scaling properties over the East River basin. Different intercept values of the fitted lines of log-log curve of F_q(s) versus s implied hydrological regulation of water reservoirs. Based on the numerical results, we suggested that regulation activities by water reservoirs could not impact the scaling properties of the streamflow series. The regulation activities by water reservoir only influenced the fluctuation magnitude. Therefore, we concluded that the streamflow variations were mainly the results of climate changes, and precipitation variations in particular. Strong dependence of generalized Hurst exponent h(q) on q demonstrated multifractal behavior of streamflow series of the East River basin, showing ‘universal’ multifractal behavior of river runoffs. The results of this study may provide valuable information for prediction and assessment of water resources under impacts of climatic changes and human activities in the East River basin.     

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.     

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.     

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.     

Monofractal and multifractal analysis of simulated heat release fluctuations in a spark ignition heat engine

We study data from cycle-by-cycle variations in heat release for a simulated spark-ignited engine. Our analyses are based on nonlinear scaling properties of heat release fluctuations obtained from a turbulent combustion model. We apply monofractal and multifractal methods to characterize the fluctuations for several fuel-air ratio values, ?, from lean mixtures to stoichiometric situations. The monofractal approach reveals that, for lean and stoichiometric conditions, the fluctuations are characterized by the presence of weak anticorrelations, whereas for intermediate mixtures we observe complex dynamics characterized by a crossover in the scaling exponents: for short scales, the variations display positive correlations while for large scales the fluctuations are close to white noise. Moreover, a broad multifractal spectrum is observed for intermediate fuel ratio values, while for low and high ? the fluctuations lead to a narrow spectrum. Finally, we explore the origin of correlations by using the surrogate data method to compare the findings of multifractality and scaling exponents between original simulated and randomized data.     

Hierarchical structure of Azbel-Hofstadter problem: Strings and loose ends of Bethe ansatz

We present numerical evidence that solutions of the Bethe anstaz equations for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or AH model), consist of complexes—“strings”. String solutions are well known from integrable field theories. They become asymptotically exact in the thermodynamic limit. The string solutions for the AH model are exact in the incommensurate limit, where the flux through the unit cell is an irrational number in units of the elementary flux quantum.We introduce the notion of the integral spectral flow and conjecture a hierarchical tree for the problem. The hierarchical tree describes the topology of the singular continuous spectrum of the problem. We show that the string content of a state is determined uniquely by the rate of the spectral flow (Hall conductance) along the tree. We identify the Hall conductances with the set of Takahashi-Suzuki numbers (the set of dimensions of the irreducible representations of U_q(sl₂2) with definite parity).In this paper we consider the approximation of non-interacting strings. It provides the gap distribution function, the mean scaling dimension for the bandwidths and gives a very good approximation for some wave functions which even captures their multifractal properties. However, it misses the multifractal character of the spectrum. © 1998 Elsevier Science B.V     

Scaling in the distribution of intertrade durations of Chinese stocks

The distribution of intertrade durations, defined as the waiting times between two consecutive transactions, is investigated based upon the limit order book data of 23 liquid Chinese stocks listed on the Shenzhen Stock Exchange in the whole year 2003. A scaling pattern is observed in the distributions of intertrade durations, where the empirical density functions of the normalized intertrade durations of all 23 stocks collapse onto a single curve. The scaling pattern is also observed in the intertrade duration distributions for filled and partially filled trades and in the conditional distributions. The ensemble distributions for all stocks are modeled by the Weibull and the Tsallis q-exponential distributions. Maximum likelihood estimation shows that the Weibull distribution outperforms the q-exponential for not-too-large intertrade durations which account for more than 98.5% of the data. Alternatively, nonlinear least-squares estimation selects the q-exponential as a better model, in which the optimization is conducted on the distance between empirical and theoretical values of the logarithmic probability densities. The distribution of intertrade durations is Weibull followed by a power-law tail with an asymptotic tail exponent close to 3.     

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.     

Electroproduction and annihilation scaling laws in a dual model

Scaling laws for large virtual photon mass (q²) in electroproduction and annihilation are studied in the framework of a simple planar dual model. We find, as has recently been conjectured, that the scaling behaviour depends on the number of space-time dimensions spanned by large momenta. In particular, for a certain range of parameters in the model, we find that the annihilation cross section is dominated by the one-dimensional configuration and increases with q² relative to its canonical behaviour while the electroproduction total cross section is dominated by the two-dimensional configuration and has the canonical Bjorken scaling behavior. In general the scaling laws and therefore the structure of events in the model are distinctively different from the conventional parton model. The problem of consistency of planar dual tree diagrams with unitarity sum rules is discussed.     

Multifractal description of wind power fluctuations using arbitrary order Hilbert spectral analysis

The objectives are to study and model the aggregate wind power fluctuations dynamics in the multifractal framework. We present here the analysis of aggregate power output sampled at 1 Hz during three years. We decompose the data into several Intrinsic Mode Functions (IMFs) using Empirical Mode Decomposition (EMD). We use a new approach, arbitrary order Hilbert spectral analysis, a combination of the EMD approach with Hilbert spectral analysis (or Hilbert–Huang Transform) and the classical structure-function analysis to extract the scaling exponents or multifractal spectrum ζ(q)$\zeta \left(q\right)$: this function provides a full characterization of a process at all intensities and all scales. The application of both methods, i.e. structure-function and arbitrary-order Hilbert spectral analyses, gives similar results indicating that the aggregate power output from a wind farm, possesses intermittent and multifractal properties. In order to check this result, we generate stochastic simulations of a Multifractal Random Walk (MRW) using a log-normal stochastic equation. We show that the simulation results are fully compatible with the experimental results.     

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.     

Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, “generic” interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in reference [23], naturally emerge. We then propose a simple solvable “stochastic volatility” model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided. Received 22 May 2000     

Three-dimensional 3-state Potts model revisited with new techniques

We report a fairly detailed finite-size scaling analysis of the first-order phase transition in the three-dimensional 3-state Potts model on cubic lattices with emphasis on recently introduced quantities whose infinite-volume extrapolations are governed only by exponentially small terms. In these quantities no asymptotic power series in the inverse volume are involved which complicate the finite-size scaling behaviour of standard observables related to the specific-heat maxima or Binder-parameter minima. Introduced initially for strong first-order phase transitions in q-state Potts models with “large enough” q, the new techniques prove to be surprisingly accurate for a q value as small as 3. On the basis of the high-precision Monte Carlo data of Alves et al. [Phys. Rev. B 43 (1991) 5846], this leads to a refined estimate of β_t = 0.550 565(10) for the infinite-volume transition point.     

Spin susceptibility of cuprates in the model of a 2D frustrated antiferromagnet: Role of renormalization of spin fluctuations in describing neutron experiments

Experimental data on the spin susceptibility of HTSC cuprates are reproduced on the basis of a spherically symmetric approach in the frustrated Heisenberg model. The inclusion of real and imaginary renormalizations in spin Green’s functions makes it possible to explain the evolution of spin excitation spectrum ω(q) and susceptibility spectrum χ(q, ω) in the range from insulator to optimal doping. In the low-frustration limit corresponding to the weakly doped mode, the saddle singularity of ω(q) and scaling of χ_2D(ω) =∫d q Im χ(q, ω) are reproduced and an analytic expression is derived for the scaling function. In the strong frustration (optimal doping) mode, the stripe scenario is demonstrated; this leads to a peak of χ_2D (ω) in the region of ω～60 meV.     

Strict parabolicity of the multifractal spectrum at the Anderson transition

Using the well-known “algebra of multifractality,” we derive the functional equation for anomalous dimensions Δ _q , whose solution Δ = χq(q–1) corresponds to strict parabolicity of the multifractal spectrum. This result demonstrates clearly that a correspondence of the nonlinear σ-models with the initial disordered systems is not exact.     

Multifractal Formalism for Almost all Self-Affine Measures

We conduct the multifractal analysis of self-affine measures for “almost all” family of affine maps. Besides partially extending Falconer’s formula of L ^q -spectrum outside the range 1 < q ≤ 2, the multifractal formalism is also partially verified.     

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.