Before we apply nonlinear techniques, for example those inspired by chaos theory, to dynamical phenomena occurring in nature, it is necessary to first ask if the use of such advanced techniques is justified by the data. While many processes in nature seem very unlikely a priori to be linear, the possible nonlinear nature might not be evident in specific aspects of their dynamics. The method of surrogate data has become a very popular tool to address such a question. However, while it was meant to provide a statistically rigorous, foolproof framework, some limitations and caveats have shown up in its practical use. In this paper, recent efforts to understand the caveats, avoid the pitfalls, and to overcome some of the limitations, are reviewed and augmented by new material. In particular, we will discuss specific as well as more general approaches to constrained randomisation, providing a full range of examples. New algorithms will be introduced for unevenly sampled and multivariate da...

: 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.

Extensions to various information-theoretic quantities (such as entropy, redundancy, and mutual information) are discussed in the context of their role in nonlinear time series analysis. We also discuss "linearized" versions of these quantities and their use as benchmarks in tests for nonlinearity. Many of these quantities can be expressed in terms of the generalized correlation integral, and this expression permits us to more clearly exhibit the relationships of these quantities to each other and to other commonly used nonlinear statistics (such as the BDS and Green-Savit statistics). Further, numerical estimation of these quantities is found to be more accurate and more efficient when the the correlation integral is employed in the computation. Finally, we consider several "local" versions of these quantities, including a local Kolmogorov-Sinai entropy, which gives an estimate of variability of the short-term predictability. 1 Introduction In Shaw's influential (and prize-winning)...

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. 1 Introduction A number of descriptive measures, like dimensions...

The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysis of experimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/or of the Hindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have prevailed over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions. Yet, and although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single and coupled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributed according to chaotic patterns at all levels of its various forms of hierarchy. This chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn, C. R. Acad. Sci. Paris, Ser. III 324 (2001) 773–793] that was focused on the basic concepts of nonlinear dynamics and methods of investigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioning their control using external perturbations. Here we present the data and main arguments that support the existence of chaos at all levels from the simplest to the most complex forms of organization of the nervous system.

Abstract not found

Daily records of atmospheric surface pressure, temperature and geopotential heights of 500 hPa isobaric level were tested for nonlinearity, the necessary condition for deterministic chaos, using redundancy and surrogate data techniques. While the time series of the temperature and the geopotential heights were found indiscernible from correspondent isospectral linear stochastic processes, a significant nonlinear component was detected in the dynamics of the pressure recording, however, no specific signatures of low-dimensional chaos were manifest. During the last decade many papers have been published, devoted to the problem of inferring the dynamical mechanisms of the weather and climate changes from recorded data. The measured quantities, selected for the analyses, have included, e.g., local surface pressures, relative sunshine durations, zonal wave amplitudes [4], upper-level geopotential heights [2, 14], low-level vertical velocity components [33], or, oxygen-isotope concentrations...

Linear digital signal processing based around linear, time-invariant systems theory finds substantial application in speech processing. The linear acoustic source-filter theory of speech production provides ready biomechanical justification for using linear techniques. Nonetheless, biomechanical studies surveyed in this thesis display significant nonlinearity and non-Gaussianity, casting doubt on the linear model of speech production. In order therefore to test the appropriateness of linear systems assumptions for speech production, surrogate data techniques can be used. This study uncovers systematic flaws in the de-sign and use of existing surrogate data techniques, and, by making novel improvements, develops a more reliable technique. Collating the largest set of speech signals to-date compatible with this new technique, this study next demonstrates that the linear assumptions are not appropriate for all speech signals. Detailed analysis shows that while vowel production from healthy subjects cannot be explained within the linear assumptions, consonants can. Linear assumptions also fail for most vowel production by pathological subjects with voice disorders. Combining this new empirical evidence with information from biomechanical studies concludes that the

Detection of nonlinearity in experimental time series is usually based on rejection of a linear null hypothesis by a statistical test. Typically, the null hypothesis is a Gaussian process or a Gaussian process passed through a static nonlinear transformation or a similar simple alternative. Rejection of such a null is frequently interpreted as a detection of a deterministic nonlinear relation in data under study, which is, however, only one of possible alternatives. We show how variable variance, or seasonality in variance could lead to spurious identification of deterministic nonlinearity and discuss how to distinguish actual determinism in studied time series. 1 Introduction Let fy(t)g be a time-series, i.e., a series of measurements done on a system in consecutive instants of time t = 1; 2; : : :. We will discuss approaches to processing and prediction for such a data. The time series fy(t)g can be considered as a realization of a stationary linear stochastic process fY (t)g. Witho...

this paper we demonstrate application of such entropy rates in analysis of an epileptic EEG and compare linear and nonlinear versions of this measure, as well as its results with results of a correlation dimension algorithm.