By Michael Small

ISBN-10: 981256117X

ISBN-13: 9789812561176

Nonlinear time sequence tools have constructed swiftly over 1 / 4 of a century and feature reached a complicated nation of adulthood over the past decade. Implementations of those tools for experimental facts at the moment are generally permitted and reasonably regimen; even if, surely important purposes stay infrequent. This booklet specializes in the perform of employing those ways to remedy actual difficulties. to demonstrate the usefulness of those equipment, a wide selection of actual and physiological platforms are thought of. The technical instruments used in this ebook fall into 3 special, yet interconnected parts: quantitative measures of nonlinear dynamics, Monte-Carlo statistical speculation trying out, and nonlinear modeling. Ten hugely distinct purposes function case experiences of fruitful purposes and illustrate the mathematical thoughts defined within the textual content.

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6 Irregular embeddings The methods to estimate de and r described in the previous sections assume that a single embedding lag is sufficient and having chosen de and r, the embedding defined by Xn > {Xn,Xn-T,Xn-2T, • • • ,Xn-(dt-l)T) (1-8) 20 Applied Nonlinear Time Series Analysis Fig. 8 Typical time series and attractor for the Ikeda map system. e. xn against xn—\, right panel). is adequate. e. nonlinear modelling), there is no reason to suppose that this would be the case. In the first part of this book, we are concerned primarily with the estimation of dynamic invariants, and therefore irregular embeddings are not necessary.

For each of these systems, we estimated the optimal embedding strategy using a genetic algorithm and (where necessary) the sub-sample selection scheme 30 times. All the data sets except the longest (the ECG recording and the laser system) produced identical results on repeated execution. For the two longest data sets, the most often observed embedding strategy was also the best (indicating that the sub-sample selection scheme is expedient but perhaps not always accurate). e. Ik < der) and is often of lower dimension (k < de).

Often, the purpose of time delay embedding is to estimate correlation dimension [167] or other dynamic invariants [2] (to be described in Chapter Time series embedding and reconstruction 25 2). 8) are usually adequate. But, what if one is interested in the more complex problem of estimating the underlying evolution operator of the dynamical system. Hence, we are interested in obtaining the most accurate prediction of the observed data values. By doing so, we hope to capture the long term dynamics of the underlying system.

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