Measuring multifractality of stock price fluctuation using multifractal detrended fluctuation analysis

Analyzing the Shanghai stock price index daily returns using MF-DFA method, it is found that there are two different types of sources for multifractality in time series, namely, fat-tailed probability distributions and non-linear temporal correlations. Based on that, a sliding window of 240 frequency data in 5 trading days was used to study stock price index fluctuation. It is found that when the stock price index fluctuates sharply, a strong variability is clearly characterized by the generalized Hurst exponents h(q). Therefore, two measures, and σ, based on generalized Hurst exponents were proposed to compare financial risks before and after Price Limits and Reform of Non-tradable Shares. The empirical results verify the validity of the measures, and this has led to a better understanding of complex stock markets.     

Detection of changes in the fractal scaling of heart rate and speed in a marathon race

The aim of this study was to detect changes in the fractal scaling behavior of heart rate and speed fluctuations when the average runner’s speed decreased with fatigue. Scaling analysis in heart rate (HR) and speed (S) dynamics of marathon runners was performed using the detrended fluctuation analysis (DFA) and the wavelet based structure function. We considered both: the short-range (α₁) and the long-range (α₂) scaling exponents for the DFA method separated by a change-point, (box length), the same for all the races. The variability of HR and S decreased in the second part of the marathon race, while the cardiac cost time series (i.e. the number of cardiac beats per meter) increased due to the decreasing speed behavior. The scaling exponents α₁ and α₂ of HR and α₁ of S, increased during the race () as did the HR wavelet scaling exponent (τ). These findings provide evidence of the significant effect of fatigue induced by long exercise on the heart rate and speed variability.     

Blocks adjustment—reduction of bias and variance of detrended fluctuation analysis using Monte Carlo simulation

The length of minimal and maximal blocks equally distant on log-log scale versus fluctuation function considerably influences bias and variance of DFA. Through a number of extensive Monte Carlo simulations and different fractional Brownian motion/fractional Gaussian noise generators, we found the pair of minimal and maximal blocks that minimizes the sum of mean-squared error of estimated Hurst exponents for the series of length . Sensitivity of DFA to sort-range correlations was examined using ARFIMA(p,d,q) generator. Due to the bias of the estimator for anti-persistent processes, we narrowed down the range of Hurst exponent to      

Multifractal Detrended Cross-Correlation Analysis of sunspot numbers and river flow fluctuations

We use the Detrended Cross-Correlation Analysis (DCCA) to investigate the influence of sun activity represented by sunspot numbers on one of the climate indicators, specifically rivers, represented by river flow fluctuation for Daugava, Holston, Nolichucky and French Broad rivers. The Multifractal Detrended Cross-Correlation Analysis (MF-DXA) shows that there exist some crossovers in the cross-correlation fluctuation function versus time scale of the river flow and sunspot series. One of these crossovers corresponds to the well-known cycle of solar activity demonstrating a universal property of the mentioned rivers. The scaling exponent given by DCCA for original series at intermediate time scale, , is λ=1.17±0.04 which is almost similar for all underlying rivers at 1σ confidence interval showing the second universal behavior of river runoffs. To remove the sinusoidal trends embedded in data sets, we apply the Singular Value Decomposition (SVD) method. Our results show that there exists a long-range cross-correlation between the sunspot numbers and the underlying streamflow records. The magnitude of the scaling exponent and the corresponding cross-correlation exponent are λ∈(0.76,0.85) and γ_×∈(0.30,0.48), respectively. Different values for scaling and cross-correlation exponents may be related to local and external factors such as topography, drainage network morphology, human activity and so on. Multifractal cross-correlation analysis demonstrates that all underlying fluctuations have almost weak multifractal nature which is also a universal property for data series. In addition the empirical relation between scaling exponent derived by DCCA and Detrended Fluctuation Analysis (DFA), is confirmed.     

A generalized Kolmogorov-Sinai-like entropy under Markov shifts in symbolic dynamics

In this paper, a generalized Kolmogorov-Sinai-like entropy ( entropy) in the sense of Tsallis is proposed with a nonextensive parameter q under Markov shifts, which contains the classical Kolmogorov-Sinai (KS) entropy and the Rényi entropy as well as Bernoulli shifts as special cases. To verify the formula of this entropy, a one-dimensional iterative system is chosen as an example of Markov shifts, and its entropy is evaluated by a new refinement method of symbolic dynamics called symbolic refinement which differs from the conventional numerical method. The numerical results show that this entropy is monotonically decreasing as q increases.     

Fluctuation scaling of quotation activities in the foreign exchange market

We study the scaling behavior of quotation activities for various currency pairs in the foreign exchange market. The components’ centrality is estimated from multiple time series and visualized as a currency pair network. The power-law relationship between a mean of quotation activity and its standard deviation for each currency pair is found. The scaling exponent α and the ratio between common and specific fluctuations η increase with the length of the observation time window . The result means that although for , the market dynamics are governed by specific processes, and at a longer time scale the common information flow becomes more important. We point out that quotation activities are not independently Poissonian for , and temporally or mutually correlated activities of quotations can happen even at this time scale. A stochastic model for the foreign exchange market based on a bipartite graph representation is proposed.     

New nonlinear mechanisms of midlatitude atmospheric low-frequency variability

This paper studies the dynamical mechanisms potentially involved in the so-called atmospheric low-frequency variability, occurring at midlatitudes in the Northern Hemisphere. This phenomenon is characterised by recurrent non-propagating and temporally persistent flow patterns, with typical spatial and temporal scales of 6000-10 000 km and 10-50 days, respectively.We study a low-order model derived from the 2-layer shallow-water equations on a β-plane channel. The main ingredients of the low-order model are a zonal flow, a planetary scale wave, orography, and a baroclinic-like forcing.A systematic analysis of the dynamics of the low-order model is performed using techniques and concepts from dynamical systems theory. Orography height (h₀) and magnitude of zonal wind forcing (U₀) are used as control parameters to study the bifurcations of equilibria and periodic orbits. Along two curves of Hopf bifurcations an equilibrium loses stability () and gives birth to two distinct families of periodic orbits. These periodic orbits bifurcate into strange attractors along three routes to chaos: period doubling cascades, breakdown of 2-tori by homo- and heteroclinic bifurcations, or intermittency ( and ).The observed attractors exhibit spatial and temporal low-frequency patterns comparing well with those observed in the atmosphere. For the periodic orbits have a period of about 10 days and patterns in the vorticity field propagate eastward. For , the period is longer (30-60 days) and patterns in the vorticity field are non-propagating. The dynamics on the strange attractors are associated with low-frequency variability: the vorticity fields show weakening and strengthening of non-propagating planetary waves on time scales of 10-200 days. The spatio-temporal characteristics are “inherited” (by intermittency) from the two families of periodic orbits and are detected in a relatively large region of the parameter plane. This scenario provides a characterisation of low-frequency variability in terms of intermittency due to bifurcations of waves.     

Scaling instability in self-potential earthquake-related signals

Scaling behavior of the temporal variability in self-potential data recorded in the frequency range during 1995 at Acapulco station (Mexico) was identified and analyzed. In this period, a strong earthquake (Ms=7.4) struck the monitored area on September 14, 1995. Using the detrended fluctuation analysis (DFA), which is a powerful method to detect scaling in nonstationary time series, deviations from stable power-law scaling were quantified. Our findings point to an evident unstable scaling behavior in self-potential data before the occurrence of the seismic event. These first results could be useful in the framework of earthquake prediction studies.     

Geometrical properties of local dynamics in Hamiltonian systems: The Generalized Alignment Index (GALI) method

We investigate the detailed dynamics of multi-dimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on N-dimensional tori. More specifically we introduce the Generalized Alignment Index of order k () as the volume of a generalized parallelepiped, whose edges are k initially linearly independent unit deviation vectors with respect to the orbit studied whose magnitude is normalized to unity at every time step. We show analytically and verify numerically on particular examples of N-degree-of-freedom Hamiltonian systems that, for chaotic orbits, tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, fluctuates around non-zero values for 2≤k≤N and goes to zero for N<k≤2N following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus: ∝t^{−2(k−N)+m} if 0≤m<k−N, and ∝t^−(k−N) if m≥k−N. The is a generalization of the Smaller Alignment Index (SALI) (). However, provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.     

A simple mathematical model for anomalous diffusion via Fisher's information theory

Starting with the relative entropy based on a previously proposed entropy function , we find the corresponding Fisher's information measure. After function redefinition we then maximize the Fisher information measure with respect to the new function and obtain a differential operator that reduces to a space coordinate second derivative in the q→1 limit. We then propose a simple differential equation for anomalous diffusion and show that its solutions are a generalization of the functions in the Barenblatt-Pattle solution. We find that the mean squared displacement, up to a q-dependent constant, has a time dependence according to 〈x²〉∼K1/qt1/q, where the parameter q takes values (superdiffusion) and (subdiffusion), ∀n?1.     

Roundoff-induced attractors and reversibility in conservative two-dimensional maps

We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov exponent is known to be positive), and a typical one (exhibiting a vanishing Lyapunov exponent) chosen from the generalized shift family of maps introduced by C. Moore [Phys. Rev. Lett. 64 (1990) 2354] in the context of undecidability. We calculate the time evolution of the entropy (). We exhibit the dramatic effect introduced by numerical precision. Indeed, in spite of being area-preserving maps, they present, well after the initially concentrated ensemble has spread virtually all over the phase space, unexpected pseudo-attractors (fixed-point like for the baker map, and more complex structures for the Moore map). These pseudo-attractors, and the apparent time (partial) reversibility they provoke, gradually disappear for increasingly large precision. In the case of the Moore map, they are related to zero Lebesgue-measure effects associated with the frontiers existing in the definition of the map. In addition to the above, and consistent with the results by V. Latora and M. Baranger [Phys. Rev. Lett. 82 (1999) 520], we find that the rate of the far-from-equilibrium entropy production of baker map numerically coincides with the standard Kolmogorov-Sinai entropy of this strongly chaotic system.     

The magnetocaloric effect and low temperature specific heat of SmNi

The magnetocaloric effect (MCE) in a magnetic SmNi sample was evaluated from magnetization and heat capacity measurements. The MCE phenomena in the vicinity of magnetic phase transitions in terms of magnetic entropy change, , and adiabatic temperature change, , are reported. Isothermal magnetization measurements at several temperatures around the transition were carried out and used for versusT calculations. A similar dependence of the magnetic entropy change was evaluated from heat capacity C_p(T) measurements under zero field and 5 T. The SmNi system provides magnetic refrigerants that induce an adiabatic cooling of about during the magnetization process with a field of 5 T in the temperature range of 35-45 K. The temperature dependence of C_p(T) is analyzed in terms of the magnetic and the lattice contributions.     

Absorbing phase transition in contact process on fractal lattices

We investigate the critical behavior of nonequilibrium phase transition from an active phase to an absorbing state on two selected fractal lattices, i.e., on a checkerboard fractal and on a Sierpinski carpet. The checkerboard fractal is finitely ramified with many dead ends, while the Sierpinski carpet is infinitely ramified. We measure various critical exponents of the contact process with a diffusion-reaction scheme A→AA and A→0, characterized by a spreading with a rate λ and an annihilation with a rate μ, and the results are confirmed by a crossover scaling and a finite-size scaling. The exponents, compared with the ?-expansion results assuming , being the fractal dimension of the underlying fractal lattice, exhibit significant deviations from the analytical results for both the checkerboard fractal and the Sierpinski carpet. On the other hand, the exponents on a checkerboard fractal agree well with the interpolated results of the regular lattice for the fractional dimensionality, while those on a Sierpinski carpet show marked deviations.     

What is a complex graph?

Many papers published in recent years show that real-world graphs G(n,m) (n nodes, m edges) are more or less “complex” in the sense that different topological features deviate from random graphs. Here we narrow the definition of graph complexity and argue that a complex graph contains many different subgraphs. We present different measures that quantify this complexity, for instance C_1e, the relative number of non-isomorphic one-edge-deleted subgraphs (i.e. DECK size). However, because these different subgraph measures are computationally demanding, we also study simpler complexity measures focussing on slightly different aspects of graph complexity. We consider heuristically defined “product measures”, the products of two quantities which are zero in the extreme cases of a path and clique, and “entropy measures” quantifying the diversity of different topological features. The previously defined network/graph complexity measures Medium Articulation and Offdiagonal complexity (OdC) belong to these two classes. We study OdC measures in some detail and compare it with our new measures. For all measures, the most complex graph has a medium number of edges, between the edge numbers of the minimum and the maximum connected graph . Interestingly, for some measures this number scales exactly with the geometric mean of the extremes: . All graph complexity measures are characterized with the help of different example graphs. For all measures the corresponding time complexity is given.Finally, we discuss the complexity of 33 real-world graphs of different biological, social and economic systems with the six computationally most simple measures (including OdC). The complexities of the real graphs are compared with average complexities of two different random graph versions: complete random graphs (just fixed n,m) and rewired graphs with fixed node degrees.     

Energy fluctuation and correlation in Tsallis statistics

We investigate the fluctuation of the energy in the framework of Tsallis statistics and find the correlation plays an important role in energy fluctuations. In Tsallis statistics, the correlation is induced by the nonextensivity of Tsallis entropy and exists between particles even if the particles are dynamically independent. By taking the generalized ideal gas as an example, we get that when the particle number N is large enough, the relative fluctuation of the energy is proportional to 1/N instead of in Boltzmann statistics. Thus, the relative energy fluctuation is much smaller in Tsallis statistics than that in Boltzmann statistics. Besides, we demonstrate that the introduction of correlation between particle energies leads to smaller energy fluctuations in Tsallis statistics.     

Detrending moving average algorithm: A closed-form approximation of the scaling law

The Hurst exponent H of long range correlated series can be estimated by means of the detrending moving average (DMA) method. The computational tool, on which the algorithm is based, is the generalized variance , with being the average over the moving window n and N the dimension of the stochastic series y(i). The ability to yield H relies on the property of to vary as n^2H over a wide range of scales [E. Alessio, A. Carbone, G. Castelli, V. Frappietro, Eur. J. Phys. B 27 (2002) 197]. Here, we give a closed form proof that is equivalent to C_Hn^2H and provide an explicit expression for C_H. We furthermore compare the values of C_H with those obtained by applying the DMA algorithm to artificial self-similar signals.