An Interdisciplinary Journal

2003, Vol.6, No.4, pp.842-851

Model Based Method for Determining the Minimum Embedding Dimension from Chaotic Time Series - Univariate and Multivariate Cases
M. Ataei, B. Lohmann, A. Khaki-Sedigh, C. Lucas

The problem of embedding dimension estimation from chaotic time series based on polynomial models is considered. The optimality of embedding dimension has an important role in computational efforts, Lyapunov exponents analysis, and efficiency of prediction. The method of this paper is based on the fact that the reconstructed dynamics of an attractor should be a smooth map, i.e. with no self intersection in the reconstructed attractor. To check this property, a local general polynomial autoregressive model is fitted to the given data and a canonical state space realization is considered. Then, the normalized one step ahead prediction error for different orders and various degrees of nonlinearity in polynomials is evaluated. This procedure is also extended to a multivariate form to include information from other time series and resolve the shortcomings of the univariate case. Besides the estimation of the embedding dimension, a predictive model is obtained which can be used for prediction and estimation of the Lyapunov exponents. To show the effectiveness of the proposed method, simulation results are provided which present its application to some well-known chaotic benchmark systems.
Key words: chaotic uni/multivariate time series, state space reconstruction, embedding dimension, polynomial models

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