By Cees Diks

Tools of nonlinear time sequence research are mentioned from a dynamical structures point of view at the one hand, and from a statistical point of view at the different. After giving an off-the-cuff evaluation of the idea of dynamical platforms correct to the research of deterministic time sequence, time sequence generated by means of nonlinear stochastic structures and spatio-temporal dynamical platforms are thought of. a number of statistical tools for the research of nonlinear time sequence are offered and illustrated with purposes to actual and physiological time sequence.

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**Nonlinear Time Series Analysis: Methods And Applications**

Tools of nonlinear time sequence research are mentioned from a dynamical platforms point of view at the one hand, and from a statistical standpoint at the different. After giving a casual assessment of the idea of dynamical platforms correct to the research of deterministic time sequence, time sequence generated via nonlinear stochastic platforms and spatio-temporal dynamical structures are thought of.

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Stochastic Time Series Models 35 Note that the class of ARMA models generalizes both that of MA and AR models; an ARMA(fc, 0) model is an autoregressive model of order k, while an ARMA(0,/) model is an MA model of order /. The stability criterion and the invertibility condition require that the roots of the characteristic equations k A(z) = zk zk-Y^ajZk-J }=l (3-8) t B(z) = Y^bjz'-J, b0 = \ J=l of an ARMA(fc,Z) model are within the unit circle of the complex plane (see Tong, 1990). Linear processes for which the invariant density is jointly Gaussian are referred to as linear Gaussian random processes (LGRPs).

G. contain regions with reduced excitability) in order to produce fibrillation-like space-time behavior, Panfilov and Hogeweg (1993) introduced a model in which this behavior results from the sponta neous break-up of a spiral wave in a homogeneous model of the atrium. 4. 6, illustrating the breakup of a spiral wave. Other models of the atria showing spiral wave break-up into complex spatio-temporal behavior have been described by Bar and Eiswirth (1993) and K a r m a (1993). The most important issue concerning atrial fibrillation at this early stage in modeling atrial fibrillation appears not to be which of these models is the right one; at present, all these models have to be regarded as metaphors reproducing only some aspects of the phenomenology of atrial fibrillation.

26) 26 2. 11: Estimated values of the correlation dimension D2 and the correlation entropy A"2 as a function of the embedding dimension m for the atrial fibrillation time series. enyi, 1971) of Renyi dimensions Dq and entropies Kq. It was shown by O t t et al. (1984) t h a t the Renyi spectrum of dimensions, Dq, is preserved under smooth coordinate transformations. , 1994). A dynamical system is chaotic if K\ > 0. g. Hentschel and Procaccia, 1983). As a result, a positive value of K2 implies a positive value of R\, so t h a t positive values of K'2 imply chaos.