Improved Surrogate Data for Nonlinearity Tests.
TL;DR: It is shown that nonlinear rescalings of a Gaussian linear stochastic process cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity.
Abstract: Current tests for nonlinearity compare a time series to the null hypothesis of a Gaussian linear stochastic process. For this restricted null assumption, random surrogates can be constructed which are constrained by the linear properties of the data. We propose a more general null hypothesis allowing for nonlinear rescalings of a Gaussian linear process. We show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity. An iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution.
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TL;DR: Dynamical properties of brain electrical activity from different recording regions and from different physiological and pathological brain states are compared and strongest indications of nonlinear deterministic dynamics were found for seizure activity.
Abstract: We compare dynamical properties of brain electrical activity from different recording regions and from different physiological and pathological brain states Using the nonlinear prediction error and an estimate of an effective correlation dimension in combination with the method of iterative amplitude adjusted surrogate data, we analyze sets of electroencephalographic (EEG) time series: surface EEG recordings from healthy volunteers with eyes closed and eyes open, and intracranial EEG recordings from epilepsy patients during the seizure free interval from within and from outside the seizure generating area as well as intracranial EEG recordings of epileptic seizures As a preanalysis step an inclusion criterion of weak stationarity was applied Surface EEG recordings with eyes open were compatible with the surrogates' null hypothesis of a Gaussian linear stochastic process Strongest indications of nonlinear deterministic dynamics were found for seizure activity Results of the other sets were found to be inbetween these two extremes
2,387 citations
TL;DR: Specific as well as more general approaches to constrained randomisation, providing a full range of examples, and some implementational aspects of the realisation of these methods in the TISEAN software package are discussed.
1,556 citations
Cites background or methods from "Improved Surrogate Data for Nonline..."
...None of these techniques have been successfully implemented for the phase randomisation of surrogates since they destroy the invertibility of the transform....
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...It could also be, and this seems the most likely option after all we know about the equations governing climate phenomena, that the process is nonlinear but the single time series at this sampling covers such a poor fraction of the rich dynamics that it must appear linear stochastic to the analysis....
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...As an illustration let us perform a statistical test for nonlinearity on a monthly time series of the Southern Oscillation Index (SOI) from 1866 to 1994 (1560 samples)....
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...[31], we propose a method which iteratively corrects deviations in spectrum and distribution from the goal set by the measured data....
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TL;DR: In this paper, the authors describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos and present a variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation.
Abstract: We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature. (c) 1999 American Institute of Physics.
1,381 citations
TL;DR: A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters.
Abstract: Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from this http URL . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.
1,356 citations
Cites background from "Improved Surrogate Data for Nonline..."
...As discussed in [83], this procedure is biased towards a flatter spectrum since the inverse of s(·) is not available exactly....
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TL;DR: Interpretation of results in terms of 'functional sources' and 'functional networks' allows the identification of three basic patterns of brain dynamics: normal, ongoing dynamics during a no-task, resting state in healthy subjects, and hypersynchronous, highly nonlinear dynamics of epileptic seizures and degenerative encephalopathies.
1,226 citations
Cites methods from "Improved Surrogate Data for Nonline..."
...One of the most important tools to safeguard against this is the use of so-called surrogate data (Schreiber and Schmitz, 2000)....
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...Another classification scheme was proposed by Schreiber and Schmitz (Schreiber and Schmitz, 1997)....
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...An even more sophisticated approach has been proposed by Schreiber and Schmitz (Schreiber and Schmitz, 1996)....
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References
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Book•
18 Nov 1993
TL;DR: By reading time series prediction forecasting the future and understanding the past, you can take more advantages with limited budget.
Abstract: The book is a summary of a time series forecasting competition that was held a number of years ago. It aims to provide a snapshot of the range of new techniques that are used to study time series, both as a reference for experts and as a guide for novices.
1,450 citations
Book•
24 Oct 1991
TL;DR: In this paper, the changing structure of stock returns nonlinearity in foreign exchange summary, relation to other work, and future horizons are discussed, as well as the size and distribution of the BDS statistic quantiles.
Abstract: The BDS statistic the changing structure of stock returns nonlinearity in foreign exchange summary, relation to other work, and future horizons. Appendices: size and distribution of the BDS statistic quantiles of the BDS statistic size of the BDS statistic on GARCH residuals.
580 citations
TL;DR: The typical-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, and is found to depend on whether or not the discriminating statistic is pivotal.
252 citations