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arXiv1994

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.

comp-gas↗

arXiv1995

Nonlinear stability analysis of plane Poiseuille flow by normal forms

In the subcritical interval of the Reynolds number 4320\leq R\leq R_c\equiv 5772, the Navier--Stokes equations of the two--dimensional plane Poiseuille flow are approximated by a 22--dimensional Galerkin representation formed from eigenfunctions of the Orr--Sommerfeld equation. The resulting dynamical system is brought into a generalized normal form which is characterized by a disposable parameter controlling the magnitude of denominators of the normal form transformation. As rigorously proved, the generalized normal form decouples into a low--dimensional dominant and a slaved subsystem. {}From the dominant system the critical amplitude is calculated as a function of the Reynolds number. As compared with the Landau method, which works down to R=5300, the phase velocity of the critical mode agrees within 1 per cent; the critical amplitude is reproduced similarly well except close to the critical point, where the maximal error is about 16 per cent. We also examine boundary conditions which partly differ from the usual ones.

comp-gas↗

arXiv1995

Coarse-grained entropy rates for characterization of complex time series

A method for classification of complex time series using coarse-grained entropy rates (CER's) is presented. The CER's, which are computed from information-theoretic functionals -- redundancies, are relative measures of regularity and predictability, and for data generated by dynamical systems they are related to Kolmogorov-Sinai entropy. A deterministic dynamical origin of the data under study, however, is not a necessary condition for the use of the CER's, since the entropy rates can be defined for stochastic processes as well. Sensitivity of the CER's to changes in data dynamics and their robustness with respect to noise are tested by using numerically generated time series resulted from both deterministic -- chaotic and stochastic processes. Potential application of the CER's in analysis of physiological signals or other complex time series is demonstrated by using examples from pharmaco-EEG and tremor classification.

comp-gas↗

arXiv1994

Testing For Nonlinearity Using Redundancies: Quantitative and Qualitative Aspects

A method for testing nonlinearity in time series is described based on information-theoretic functionals -- redundancies, linear and nonlinear forms of which allow either qualitative, or, after incorporating the surrogate data technique, quantitative evaluation of dynamical properties of scrutinized data. An interplay of quantitative and qualitative testing on both the linear and nonlinear levels is analyzed and robustness of this combined approach against spurious nonlinearity detection is demonstrated. Evaluation of redundancies and redundancy-based statistics as functions of time lag and embedding dimension can further enhance insight into dynamics of a system under study.

comp-gas↗

arXiv1996

Constrained-Realization Monte-Carlo Method for Hypothesis Testing

We compare two theoretically distinct approaches to generating artificial (or ``surrogate'') data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single ``best'' model to the original data, and then to generate surrogate data sets that are ``typical realizations'' of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics. Examples of these two approaches are provided for two simple cases: a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) fits to the data. The comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal. A statistic is ``pivotal'' if its distribution is the same for all processes consistent with the null hypothesis. The typical-realization method requires that the discriminating statistic satisfy this property. The constrained-realization approach, on the other hand, does not share this requirement, and can provide an accurate and powerful test without having to sacrifice flexibility in the choice of discriminating statistic.

comp-gas↗

arXiv1996

State Space Reconstruction Parameters in the Analysis of Chaotic Time Series - the Role of the Time Window Length

The most common state space reconstruction method in the analysis of chaotic time series is the Method of Delays (MOD). Many techniques have been suggested to estimate the parameters of MOD, i.e. the time delay $τ$ and the embedding dimension $m$. We discuss the applicability of these techniques with a critical view as to their validity, and point out the necessity of determining the overall time window length, $τ_w$, for successful embedding. Emphasis is put on the relation between $τ_w$ and the dynamics of the underlying chaotic system, and we suggest to set $τ_w \geq τ_p$, the mean orbital period; $τ_p$ is approximated from the oscillations of the time series. The procedure is assessed using the correlation dimension for both synthetic and real data. For clean synthetic data, values of $τ_w$ larger than $τ_p$ always give good results given enough data and thus $τ_p$ can be considered as a lower limit ($τ_w \geq τ_p$). For noisy synthetic data and real data, an upper limit is reached for $τ_w$ which approaches $τ_p$ for increasing noise amplitude.

comp-gas↗

arXiv1995

Lattice Gas Automata for Reactive Systems

Reactive lattice gas automata provide a microscopic approachto the dynamics of spatially-distributed reacting systems. After introducing the subject within the wider framework of lattice gas automata (LGA) as a microscopic approach to the phenomenology of macroscopic systems, we describe the reactive LGA in terms of a simple physical picture to show how an automaton can be constructed to capture the essentials of a reactive molecular dynamics scheme. The statistical mechanical theory of the automaton is then developed for diffusive transport and for reactive processes, and a general algorithm is presented for reactive LGA. The method is illustrated by considering applications to bistable and excitable media, oscillatory behavior in reactive systems, chemical chaos and pattern formation triggered by Turing bifurcations. The reactive lattice gas scheme is contrasted with related cellular automaton methods and the paper concludes with a discussion of future perspectives.

comp-gas↗

arXiv1994

The correlation dimension of differenced data

The effect differencing has on the estimated correlation dimension of a nongaussian time series is discussed. Two different methods for generating surrogate data sets are compared. The results suggest that any filtering should be applied to both the raw and surrogate time series, otherwise spurious results can occur.

comp-gas↗

arXiv1993

Spatial Variation of Correlation Times for 1D Phase Turbulence

For one-dimensional phase-turbulent solutions of the Kuramoto-Sivashinsky equation with rigid boundary conditions, we show that there is a substantial variation of the correlation time~$τ_c(x)$ with spatial position $x$ in moderately large systems of size $L$. These results suggest that some time-averaged properties of spatiotemporal chaos do not become homogeneous away from boundaries for large systems and for long times.

chao-dyn↗

arXiv1995

Detecting nonlinearity in multivariate time series

We propose an extension to time series with several simultaneously measured variables of the nonlinearity test, which combines the redundancy -- linear redundancy approach with the surrogate data technique. For several variables various types of the redundancies can be defined, in order to test specific dependence structures between/among (groups of) variables. The null hypothesis of a multivariate linear stochastic process is tested using the multivariate surrogate data. The linear redundancies are used in order to avoid spurious results due to imperfect surrogates. The method is demonstrated using two types of numerically generated multivariate series (linear and nonlinear) and experimental multivariate data from meteorology and physiology.

comp-gas↗

arXiv1999

A Particulate Basis for an Immiscible Lattice-Gas Model

We show that a phenomenological hydrodynamic lattice-gas model of two-phase flow, developed by Rothman and Keller in 1988 and used extensively for numerical simulations since then, can be derived from an underlying model of particle interactions. From this result, we elucidate the nature of the hydrodynamic limit of the Rothman-Keller model.

cond-mat.soft↗

arXiv1997

Artificial Neural Network Methods in Quantum Mechanics

In a previous article we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for differential and integrodifferential operators, using ANNs. We start by considering the Schrödinger equation for the Morse potential that has an analytically known solution, to test the accuracy of the method. We then proceed with the Schrödinger and the Dirac equations for a muonic atom, as well as with a non-local Schrödinger integrodifferential equation that models the $n+α$ system in the framework of the resonating group method. In two dimensions we consider the well studied Henon-Heiles Hamiltonian and in three dimensions the model problem of three coupled anharmonic oscillators. The method in all of the treated cases proved to be highly accurate, robust and efficient. Hence it is a promising tool for tackling problems of higher complexity and dimensionality.

quant-ph↗

arXiv1997

Quasi-Linear Cellular Automata

Simulating a cellular automaton (CA) for t time-steps into the future requires t^2 serial computation steps or t parallel ones. However, certain CAs based on an Abelian group, such as addition mod 2, are termed ``linear'' because they obey a principle of superposition. This allows them to be predicted efficiently, in serial time O(t) or O(log t) in parallel. In this paper, we generalize this by looking at CAs with a variety of algebraic structures, including quasigroups, non-Abelian groups, Steiner systems, and others. We show that in many cases, an efficient algorithm exists even though these CAs are not linear in the previous sense; we term them ``quasilinear.'' We find examples which can be predicted in serial time proportional to t, t log t, t log^2 t, and t^a for a < 2, and parallel time log t, log t log log t and log^2 t. We also discuss what algebraic properties are required or implied by the existence of scaling relations and principles of superposition, and exhibit several novel ``vector-valued'' CAs.

adap-org↗

arXiv1997

Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones

We show that a wide variety of non-linear cellular automata (CAs) can be decomposed into a quasidirect product of linear ones. These CAs can be predicted by parallel circuits of depth O(log^2 t) using gates with binary inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of inputs are allowed. Thus these CAs can be predicted by (idealized) parallel computers much faster than by explicit simulation, even though they are non-linear. This class includes any CA whose rule, when written as an algebra, is a solvable group. We also show that CAs based on nilpotent groups can be predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p'' gates respectively. We use these techniques to give an efficient algorithm for a CA rule which, like elementary CA rule 18, has diffusing defects that annihilate in pairs. This can be used to predict the motion of defects in rule 18 in O(log^2 t) parallel time.

patt-sol↗

arXiv1997

Simulating quantum mechanics on a quantum computer

Algorithms are described for efficiently simulating quantum mechanical systems on quantum computers. A class of algorithms for simulating the Schrodinger equation for interacting many-body systems are presented in some detail. These algorithms would make it possible to simulate nonrelativistic quantum systems on a quantum computer with an exponential speedup compared to simulations on classical computers. Issues involved in simulating relativistic systems of Dirac and gauge particles are discussed.

quant-ph↗

arXiv1996

Turbulent channel flow simulations using a coarse-grained extension of the Lattice Boltzmann method

A coarse-grained version of the Lattice Boltzmann (LB) method is developed with the intent of enhancing its geometrical flexibility so as to be able to tackle a wider class of flows of engineering interest. To this purpose, the original uniform LB technique is combined with standard finite-volume techniques based upon a blend of piecewise constant and piecewise linear interpolation schemes. A series of validation tests for the three dimensional channel flow with one-dimensional (cross-channel) statistical behaviour are presented. The main conclusion is that, although the method does indeed mark a significant stride forward with respect to the original uniform LB scheme, better interpolation schemes should be developed before the coarse-grain LB can become fully competitive with modern CFD schemes.

comp-gas↗

arXiv2000

The rate of entropy increase at the edge of chaos

Under certain conditions, the rate of increase of the statistical entropy of a simple, fully chaotic, conservative system is known to be given by a single number, characteristic of this system, the Kolmogorov-Sinai entropy rate. This connection is here generalized to a simple dissipative system, the logistic map, and especially to the chaos threshold of the latter, the edge of chaos. It is found that, in the edge-of-chaos case, the usual Boltzmann-Gibbs-Shannon entropy is not appropriate. Instead, the non-extensive entropy $S_q\equiv \frac{1-\sum_{i=1}^W p_i^q}{q-1}$, must be used. The latter contains a parameter q, the entropic index which must be given a special value $q^*\ne 1$ (for q=1 one recovers the usual entropy) characteristic of the edge-of-chaos under consideration. The same q^* enters also in the description of the sensitivity to initial conditions, as well as in that of the multifractal spectrum of the attractor.

cond-mat.stat-mech↗

arXiv1996

On the absence of homogeneous scalar unitary cellular automata

Failure to find homogeneous scalar unitary cellular automata (CA) in one dimension led to consideration of only ``approximately unitary'' CA---which motivated our recent proof of a No-go Lemma in one dimension. In this note we extend the one dimensional result to prove the absence of nontrivial homogeneous scalar unitary CA on Euclidean lattices in any dimension.

quant-ph↗