Nonlinear Time Series Analysis: Methods And Applications by Cees Diks

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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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 + #-?

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