The analysis of uni- or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical eld of study. Considerable progress has also been made in interpreting the information so obtained in terms of dynamical systems theory.

Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.

: This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed. For signals with weakly nonlinear structure, the presence of nonlinearity in a general sense has to be inferred statistically. The paper reviews the relevant methods and discusses the implications for deterministic modeling. Most field measurements yield nonstationary time series, which poses a severe problem for their analysis. Recent progress in the detection and understanding of nonstationarity is reported. If a clear signature of approximate determinism is found, the notions of phase space, attractors, invariant manifolds etc. provide a convenient framework for time series analysis. Although the results have to be interpreted with great care, superior performance can be achieved for typical signal processing tasks. In particular, prediction and filtering of signals are discussed, as well as the classification of system states by means of time series recordings.

We review the thermodynamics of estimating the statistical fluctuations of an observed process. Since any statistical analysis involves a choice of model class -- either explicitly or implicitly -- we demonstrate the benefits of a careful choice. For each of three classes a particular model is reconstructed from data streams generated by four sample processes. Then each estimated model's thermodynamic structure is used to estimate the typical behavior and the magnitude of deviations for the observed system. These are then compared to the known fluctuation properties. The type of analysis advocated here, which uses estimated model class information, recovers the correct statistical structure of these processes from simulated data. The current alternative -- direct estimation of the Renyi entropy from time series histograms -- uses neither prior nor reconstructed knowledge of the model class. And, in most cases, it fails to recover the process's statistical structure from finite data -- unpredictability is overestimated.

Certain deterministic non-linear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows constructing more realistic and better models and thus improved predictive capabilities. This paper provides a review of two main key features of chaotic systems, the dimensions of their strange attractors and the Lyapunov exponents. The emphasis is on state space reconstruction techniques that are used to estimate these properties, given scalar observations. Data generated from equations known to display chaotic behaviour are used for illustration. A compilation of applications to real data from widely different fields is given. If chaos is found to be present, one may proceed to build non-linear models, which is the topic of the second paper in this series.

Abstract—This paper and its companion (Part II) are devoted to the analysis of the application of a chaotic piecewise-linear onedimensional (PL1D) map as random number generator (RNG). Piecewise linearity of the map enables us to mathematically find parameter values for which a generating partition is Markov and the RNG behaves as a Markov information source, and then to mathematically analyze the information generation process and the RNG. In the companion paper we discuss practical aspects of our chaos-based RNGs. Index Terms—Chaos, random number generator, symbolic dynamics. I.

High frequency forecasts have always been considered difficult to generate and generation of forecast distributions instead of point forecasts has always been desired by the practitioners in asset allocation and investment management fields. This thesis looks at the problem of high frequency forecast generation by extending the traditional nonlinear dynamics technique by Farmer and Sidorowich and implementing it in a new connectionist architecture based on Correlation Matrix Memories. The architecture employs distribution generation and produces point forecasts as maximum probability points. Distributions are generated from nearest neighbour interpolations by a Bayesian technique. Testing of the architecture's utility has been performed by looking at a selection of representative time series including currency, index and commodity series and their forecast implied efficiency which was compared to standard efficiency estimates. Efficiency was measured with respect to Henriksson--Merton ...

After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its set-theoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness ” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measure-theoretical codification of stochastic processes genuine chance processes can be defined rigorously as so-called regular processes which do not allow a long-term prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes.

A known cycle length has hitherto been considered an indispensable requirement for pseudorandom number generators. This requirement has restricted random number generators to suboptimal designs with known deficiencies. The present article shows that the requirement for a known cycle length can be avoided by including a self-test facility. The distribution of cycle lengths is analyzed theoretically and experimentally. As an example, a class of chaotic random number generators based on bit-rotation and addition are analyzed theoretically and experimentally. These generators, suitable for Monte Carlo applications, have good randomness, long cycle lengths, and higher speed than other generators of similar quality.

A model of an orthogonal cutting system is described as an elastic structure deformable in two directions. In the system, a cutting force is generated by material flow against the tool. Nonlinear dependency of the cutting force on the cutting velocity can cause chaotic vibrations of the cutting tool which influence the quality of a manufactured surface. The intensity and the characteristics of vibrations are determined by the values of cutting parameters. The influence of cutting depth on system dynamics is described by bifurcation diagrams. The properties of oscillations are illustrated by time dependence of tool displacement, corresponding frequency spectra and phase portraits. The corresponding strange attractors are characterized by correlation dimension. The vibrations are characterized by maximum Lyapunov exponent. The manufactured surface at the first cut is taken as the incoming surface in the second cut, thus incorporating the influence of rough surface in the model...