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At least 793 records · Page 44Linked to original sources
PubMed2004

Average trajectory of returning walks.

We compute the average shape of trajectories of some one-dimensional stochastic processes x(t) in the (t,x) plane during an excursion, i.e., between two successive returns to a reference value, finding that it obeys a scaling form. For uncorrelated random walks the average shape is semicircular, independent from the single increments distribution, as long as it is symmetric. Such universality extends to biased random walks and Levy flights, with the exception of a particular class of biased Levy flights. Adding a linear damping term destroys scaling and leads asymptotically to flat excursions. The introduction of short and long ranged noise correlations induces nontrivial asymmetric shapes, which are studied numerically.

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PubMed2004

Probing universality classes in solid-on-solid deposition.

We consider several stochastic processes corresponding to the same physical solid-on-solid deposition problem. Simplified models presenting the same (conditional) mean and variance for each process are also introduced as well as generalizations in terms of the deposition of blobs and probabilistic deposition rules. We compare the evolution of the roughness as a function of time for a three-parameter family that includes as limit cases the Family model and the Edwards-Wilkinson equation, showing that in all cases the derived models with the same mean and variance are indistinguishable from the originating models in terms of the evolution of the roughness. Finally, we show that although all the models studied belong to the same universality class, some relevant features such as the final surface roughness are reproduced only for models within a restricted class determined by sharing the same (conditional) mean and variance.

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PubMed2004

Best linear forecast of volatility in financial time series.

The autocorrelation function of volatility in financial time series is fitted well by a superposition of several exponents. This case admits an explicit analytical solution of the problem of constructing the best linear forecast of a stationary stochastic process. We describe and apply the proposed analytical method for forecasting volatility. The leverage effect and volatility clustering are taken into account. Parameters of the predictor function are determined numerically for the Dow Jones 30 Industrial Average. Connection of the proposed method to the popular autoregressive conditional heteroskedasticity models is discussed.

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PubMed2004

Functional characterization of linear delay Langevin equations.

We present an exact functional characterization of linear delay Langevin equations driven by any noise structure defined through its characteristic functional. This method relies on the possibility of finding an explicitly analytical expression for each realization of the delayed stochastic process in terms of those of the driving noise. General properties of the transient dissipative dynamics are analyzed. The corresponding interplay with a color Gaussian noise is presented. As a full application of our functional method we study a model for population growth with non-Gaussian fluctuations: the Gompertz model driven by multiplicative white shot noise.

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PubMed2004

Dynamical epidemic suppression using stochastic prediction and control.

We consider the effects of noise on a model of epidemic outbreaks, where the outbreaks appear randomly. Using a constructive transition approach that predicts large outbreaks prior to their occurrence, we derive an adaptive control scheme that prevents large outbreaks from occurring. The theory is applicable to a wide range of stochastic processes with underlying deterministic structure.

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PubMed2004

Fluctuation theorem for a deterministic one-particle system.

A Duffing oscillator is driven by a sum of N chaotic time series. These time series are solutions of the undriven Duffing equation. It is shown that N=1 is sufficient to render the fluctuation theorem [Phys. Rev. Lett. 74, 2694 (1995)]; J. Math. Phys. 41, 4061 (2000)]; Adv. Phys. 51, 1529 (2002)]] for the power J(tau) averaged within intervals of length tau . In particular, the probabilities p( J(tau) ) follow a nearly Gaussian distribution. Also, ln[p( J(tau) )/p(- J(tau) )] versus J(tau) can be fitted by strikingly linear functions, the slopes being proportional to tau for large tau . These results indicate that validity of the fluctuation theorem requires neither a many-particle system nor a stochastic process, which are requirements used in previous works.

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PubMed2005

Persistence of randomly coupled fluctuating interfaces.

We study the persistence properties in a simple model of two coupled interfaces characterized by heights h(1) and h(2) , respectively, each growing over a d -dimensional substrate. The first interface evolves independently of the second and can correspond to any generic growing interface, e.g., of the Edwards-Wilkinson or of the Kardar-Parisi-Zhang variety. The evolution of h(2) , however, is coupled to h(1) via a quenched random velocity field. In the limit d-->0 , our model reduces to the Matheron-de Marsily model in two dimensions. For d=1, our model describes a Rouse polymer chain in two dimensions advected by a transverse velocity field. We show analytically that after a long waiting time t(0) -->infinity , the stochastic process h(2) , at a fixed point in space but as a function of time, becomes a fractional Brownian motion with a Hurst exponent, H2 =1- beta(1) /2 , where beta(1) is the growth exponent characterizing the first interface. The associated persistence exponent is shown to be theta(2)(s) =1- H2 = beta(1) /2 . These analytical results are verified by numerical simulations.

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PubMed2005

Mean-field results for the two-component model.

In previous papers we have introduced a new dynamical model of Ising spins: the two-component (TC) model. In this paper we formulate a mean-field version of the TC model by putting it on a complete graph. With such an approach we are able to describe the kinetics in terms of a one-dimensional stochastic process with hopping probabilities depending on the magnetization. This allows us to understand the differences in relaxation between phases observed previously in computer simulations for the TC model on the square lattice.

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PubMed2005

Lattice model for calcium dynamics.

We present a simplified lattice model to study calcium dynamics in the endoplasmic reticulum membrane. Calcium channels and calcium ions are placed in two interpenetrating square lattices which are connected in two ways: (i) via calcium release and (ii) because transitions between channel states are calcium dependent. The opening or closing of a channel is a stochastic process controlled by two functions which depend on the calcium density on the channel neighborhood. The model is studied through mean field calculations and simulations. We show that the critical behavior of the model changes drastically depending on the opening/closing functions. For certain choices of these functions, all channels are closed at very low and high calcium densities and the model presents one absorbing state.

Animals↗
PubMed2005

Natural cutoff in Lévy flights caused by dissipative nonlinearity.

Lévy flight models are often used to describe stochastic processes in complex systems. However, due to the occurrence of diverging position and/or velocity fluctuations Lévy flights are physically problematic if describing the dynamics of a particle of finite mass. Here we show that the velocity distribution of a random walker subject to Lévy noise can be regularized by nonlinear friction, leading to a natural cutoff in the velocity distribution and finite velocity variance.

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PubMed2005

Detecting a currency's dominance or dependence using foreign exchange network trees.

In a system containing a large number of interacting stochastic processes, there will typically be many nonzero correlation coefficients. This makes it difficult to either visualize the system's interdependencies, or identify its dominant elements. Such a situation arises in foreign exchange (FX), which is the world's biggest market. Here we develop a network analysis of these correlations using minimum spanning trees (MSTs). We show that not only do the MSTs provide a meaningful representation of the global FX dynamics, but they also enable one to determine momentarily dominant and dependent currencies. We find that information about a country's geographical ties emerges from the raw exchange-rate data. Most importantly from a trading perspective, we discuss how to infer which currencies are "in play" during a particular period of time.

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PubMed2005

Complexity of time series associated to dynamical systems inferred from independent component analysis.

A not trivial problem for every experimental time series associated to a natural system is to individuate the significant variables to describe the dynamics, i.e., the effective degrees of freedom. The application of independent component analysis (ICA) has provided interesting results in this direction, e.g., in the seismological and atmospheric field. Since all natural phenomena can be represented by dynamical systems, our aim is to check the performance of ICA in this general context to avoid ambiguities when investigating an unknown experimental system. We show many examples, representing linear, nonlinear, and stochastic processes, in which ICA seems to be an efficacious preanalysis able to give information about the complexity of the dynamics.

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PubMed2005

Volatility, persistence, and survival in financial markets.

We study the temporal fluctuations in time-dependent stock prices (both individual and composite) as a stochastic phenomenon using general techniques and methods of nonequilibrium statistical mechanics. In particular, we analyze stock price fluctuations as a non-Markovian stochastic process using the first-passage statistical concepts of persistence and survival. We report the results of empirical measurements of the normalized qth-order correlation functions fq(t), survival probability S(t), and persistence probability P(t) for several stock market dynamical sets. We analyze both minute-to-minute and higher-frequency stock market recordings (i.e., with the sampling time deltat of the order of days). We find that the fluctuating stock price is multifractal and the choice of deltat has no effect on the qualitative multifractal behavior displayed by the 1/q dependence of the generalized Hurst exponent Hq associated with the power-law evolution of the correlation function fq(t) approximately tHq. The probability S(t) of the stock price remaining above the average up to time t is very sensitive to the total measurement time tm and the sampling time. The probability P(t) of the stock not returning to the initial value within an interval t has a universal power-law behavior P(t) approximately t(-theta), with a persistence exponent theta close to 0.5 that agrees with the prediction theta=1-H2. The empirical financial stocks also present an interesting feature found in turbulent fluids, the extended self-similarity.

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PubMed2005

Exact numerical simulation of power-law noises.

Many simulations of stochastic processes require colored noises: I describe here an exact numerical method to simulate power-law noises: the method can be extended to more general colored noises, and is exact for all time steps, even when they are unevenly spaced [as may often happen for astronomical data; see, e.g., N. R. Lomb, Astrophys. Space Sci. 39, 447 (1976)]. The algorithm has a well-behaved computational complexity, it produces a nearly perfect Gaussian noise, and its computational efficiency depends on the required degree of noise Gaussianity.

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PubMed2006

Numerical method for solving stochastic differential equations with dichotomous noise.

We propose a numerical method for solving stochastic differential equations with dichotomous Markov noise. The numerical scheme is formulated such that (i) the stochastic formula used follows the Stratonovich-Taylor form over the entire range of noise correlation times, including the Gaussian white noise limit; and (ii) the method is readily applicable to dynamical systems driven by arbitrary types of noise, provided there exists a way to describe the random increment of the stochastic process expressed in the Stratonovich-Taylor form. We further propose a simplified Taylor scheme that significantly reduces the computation time, while still satisfying the moment properties up to the required order. The accuracies and efficiencies of the proposed algorithms are validated by applying the schemes to two prototypical model systems that possess analytical solutions.

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PubMed2006

Accuracy and efficiency of reduced stochastic models for chaotic Hamiltonian systems with time-scale separation.

The study of the long time behavior of systems with time-scale separation is considerably facilitated if the fast degrees of freedom can be eliminated without affecting the slow dynamics. We investigate a technique in which the fluctuations due to a fast chaotic subsystem are replaced by a suitable stochastic process so that a Fokker-Planck equation for the slow variables results. The accuracy and efficiency of this technique is verified by the detailed numerical investigation of several coupled systems. The asymptotic behavior as well as transients turn out to be well modeled by the reduced dynamics. We concentrate on low-dimensional problems and cover different types of coupling schemes as well as different chaotic subsystems. As a physical application we discuss the classical dynamics of a hydrogen atom in a strong magnetic field.

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PubMed2006

Scaling in a general class of critical random Boolean networks.

We derive analytically the scaling behavior in the thermodynamic limit of the number of nonfrozen and relevant nodes in the most general class of critical Kauffman networks for any number of inputs per node, and for any choice of the probability distribution for the Boolean functions. By defining and analyzing a stochastic process that determines the frozen core we can prove that the mean number of nonfrozen nodes in any critical network with more than one input per node scales with the network size N as N2/3, with only N1/3 nonfrozen nodes having two nonfrozen inputs and the number of nonfrozen nodes with more than two inputs being finite in the thermodynamic limit. Using these results we can conclude that the mean number of relevant nodes increases for large N as N1/3, with only a finite number of relevant nodes having two relevant inputs, and a vanishing fraction of nodes having more than three of them. It follows that all relevant components apart from a finite number are simple loops, and that the mean number and length of attractors increases faster than any power law with network size.

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PubMed2000

Bayesian approach to inverse quantum statistics

A nonparametric Bayesian approach is developed to determine quantum potentials from empirical data for quantum systems at finite temperature. The approach combines the likelihood model of quantum mechanics with a priori information on potentials implemented in the form of stochastic processes. Its specific advantages are the possibilities to deal with heterogeneous data and to express a priori information explicitly in terms of the potential of interest. A numerical solution in maximum a posteriori approximation is obtained for one-dimensional problems. As nonparametric estimates, the results depend strongly on the implemented a priori information.

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