Featured Researches

Cellular Automata And Lattice Gases

Estimating Functions of Probability Distributions from a Finite Set of Samples, Part 1: Bayes Estimators and the Shannon Entropy

We present estimators for entropy and other functions of a discrete probability distribution when the data is a finite sample drawn from that probability distribution. In particular, for the case when the probability distribution is a joint distribution, we present finite sample estimators for the mutual information, covariance, and chi-squared functions of that probability distribution.

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Cellular Automata And Lattice Gases

Estimating Predictability: Redundancy and Surrogate Data Method

A method for estimating theoretical predictability of time series is presented, based on information-theoretic functionals---redundancies and surrogate data technique. The redundancy, designed for a chosen model and a prediction horizon, evaluates amount of information between a model input (e.g., lagged versions of the series) and a model output (i.e., a series lagged by the prediction horizon from the model input) in number of bits. This value, however, is influenced by a method and precision of redundancy estimation and therefore it is a) normalized by maximum possible redundancy (given by the precision used), and b) compared to the redundancies obtained from two types of the surrogate data in order to obtain reliable classification of a series as either unpredictable or predictable. The type of predictability (linear or nonlinear) and its level can be further evaluated. The method is demonstrated using a numerically generated time series as well as high-frequency foreign exchange data and the theoretical predictability is compared to performance of a nonlinear predictor.

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Cellular Automata And Lattice Gases

Estimating the Fractal Dimension, K_2-entropy, and the Predictability of the Atmosphere

The series of mean daily temperature of air recorded over a period of 215 years is used for analysing the dimensionality and the predictability of the atmospheric system. The total number of data points of the series is 78527. Other 37 versions of the original series are generated, including ``seasonally adjusted'' data, a smoothed series, series without annual course, etc. Modified methods of Grassberger and Procaccia are applied. A procedure for selection of the ``meaningful'' scaling region is proposed. Several scaling regions are revealed in the ln C(r) versus ln r diagram. The first one in the range of larger ln r has a gradual slope and the second one in the range of intermediate ln r has a fast slope. Other two regions are settled in the range of small ln r. The results lead us to claim that the series arises from the activity of at least two subsystems. The first subsystem is low-dimensional (d_f=1.6) and it possesses the potential predictability of several weeks. We suggest that this subsystem is connected with seasonal variability of weather. The second subsystem is high-dimensional (d_f>17) and its error-doubling time is about 4-7 days. It is found that the predictability differs in dependence on season. The predictability time for summer, winter and the entire year (T_2 approx. 4.7 days) is longer than for transition-seasons (T_2 approx. 4.0 days for spring, T_2 approx. 3.6 days for autumn). The role of random noise and the number of data points are discussed. It is shown that a 15-year-long daily temperature series is not sufficient for reliable estimations based on Grassberger and Procaccia algorithms.

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Cellular Automata And Lattice Gases

Exact Results for the Asymmetric Simple Exclusion Process with a Blockage

We present new results for the current as a function of transmission rate in the one dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one to r < 1. Exact finite volume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a gap in allowed density corresponding to a nonequilibrium ``phase transition'' in the infinite system. A series expansion in r, derived from the finite systems, is proven to be asymptotic for all sufficiently large systems. Pade approximants based on this series, which make specific assumptions about the nature of the singularity at r = 1, match numerical data for the ``infinite'' system to a part in 10^4.

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Cellular Automata And Lattice Gases

Exact results for deterministic cellular automata traffic models

We present a rigorous derivation of the flow at arbitrary time in a deterministic cellular automaton model of traffic flow. The derivation employs regularities in preimages of blocks of zeros, reducing the problem of preimage enumeration to a well known lattice path counting problem. Assuming infinite lattice size and random initial configuration, the flow can be expressed in terms of generalized hypergeometric function. We show that the steady state limit agrees with previously published results.

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Cellular Automata And Lattice Gases

Flow in the Driven Cavity Calculated by the Lattice Boltzmann Method

The lattice Boltzmann method with enhanced collisions and rest particles is used to calculate the flow in a two-dimensional lid-driven cavity. The abilitity of this method to compute the velocity and the pressure of an incompressible fluid in a geometry with Dirichlet and Neumann boundary conditions is verified by calculating a test-problem where the analytical solution is known. Different parameter configurations have been tested for Reynolds numbers from Re=10 to Re=2000 . The vortex structure for a more generalized lid-driven cavity problem with a non-uniform top speed has been studied for various acpect ratios.

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Cellular Automata And Lattice Gases

Generalized Boltzmann Equation for Lattice Gas Automata

In this paper, for the first time a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for the n -particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlinear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard sphere systems, describing the time evolution of pair correlations. As a quantitative test we calculate equal time correlation functions in equilibrium for two models that violate semi-detailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid type model on a triangular lattice. The numerical predictions agree very well with computer simulations.

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Cellular Automata And Lattice Gases

Generalized Thermal Lattice Gases

We show how to employ thermal lattice gas models to describe non-equilibrium phenomena. This is achieved by relaxing the restrictions of the usual micro-canonical ensemble for these models via the introduction of thermal ``demons'' in the style of Creutz. Within the Lattice Boltzmann approximation, we then derive general expressions for the usual transport coefficients of such models, in terms of the derivatives of their equilibrium distribution functions. To illustrate potential applications, we choose a model obeying Maxwell-Boltzmann statistics, and simulate Rayleigh-Bénard convection with a forcing term and a temperature gradient, both of which are continuously variable.

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Cellular Automata And Lattice Gases

Generalized redundancies for time series analysis

Extensions to various information theoretic quantities used for nonlinear time series analysis are discussed, as well as their relationship to the generalized correlation integral. It is shown that calculating redundancies from the correlation integral can be more accurate and more efficient than direct box counting methods. It is also demonstrated that many commonly used nonlinear statistics have information theory based analogues. Furthermore, the relationship between the correlation integral and information theoretic statistics allows us to define ``local'' versions of many information theory based statistics; including a local version of the Kolmogorov-Sinai entropy, which gives an estimate of the local predictability.

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Cellular Automata And Lattice Gases

Generating high-dimensional chaotic signals by the sum

The method of generating of high-dimensional oscillations on the basis of summing of low-dimensional chaotic signals of noncoupled dynamical systems is investigated. It is shown that the correlation dimension of attractor reconstructed from such oscillations is equal to the sum of dimensions of the original signals. The possibilities for obtaining of homogeneous high-dimensional chaotic attractor are numerically demonstrated with the original signals of R{ö}ssler system.

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