By Cees Diks
Tools of nonlinear time sequence research are mentioned from a dynamical platforms standpoint at the one hand, and from a statistical viewpoint at the different. After giving a casual assessment of the idea of dynamical structures correct to the research of deterministic time sequence, time sequence generated via nonlinear stochastic platforms and spatio-temporal dynamical platforms are thought of. numerous statistical tools for the research of nonlinear time sequence are offered and illustrated with functions to actual and physiological time sequence.
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Additional info for Nonlinear Time Series Analysis: Methods And Applications
6: Estimated correlation integrals (lower panel) and local slopes (upper panel) versus r for the x-variable of Rossler's model, with the embedding dimension m ranging from 2 to 16. 7). g. Grassberger, 1990). For time series with relatively long decoherence times, t h e estimated cor relation integrals may be biased as a result of the dependence among recon struction vectors t h a t have their time indices close. This bias can be easily corrected for in practice by excluding contributions from pairs of points t h a t are close in time.
To facilitate the discussion, we start by giving a definition of stationarity that is closely related to the existence of reconstruction measures (Takens, 1996). 20) n—►oo fl *—' *:=1 exist for each m and each continuous function g : R m —> R. g. Feller, 1966; Lasota and Mackay, 1994) states that there is an associated probability measure /im on R m , which we will refer to as the m-dimensional reconstructionlmeasure. Note that a two-sided definition of stationarity can be obtained similarly upon replacing the average over positive times by an average over time indices run ning from — n to +n.
It can be driven out of this state, or excited, by external forces or by interaction with neighboring excited parts of the media. The latter mech anism enables waves of excitation to propagate into un-excited regions. In many models of excitable media, the excitation inhibits itself after some time, so that previously excited parts of the media return to the rest state. 18) — at = G(u,v), where u(x,y) is a fast activator variable, v(x,y) a slow inhibitor variable, D a diffusion constant and A = #rj + #-?