Model based method for estimating an attractor dimension from uni/multivariate chaotic time series with application to Bremen climatic dynamics M. Ataei a, * , B. Lohmann a,1 , A. Khaki-Sedigh b , C. Lucas c a Institute of Automation, University of Bremen, Otto-Hahn-Allee./NW1, D-28359 Bremen, Germany b Department of Electrical Engineering, K. N. Toosi University of Technology, Sayyed Khandan Bridge, P.O. Box 16315-1355, Tehran, Iran c Department of Electrical Engineering, Faculty of Engineering, University of Tehran, North Kargar Avenue, P.O. Box 14395-515, Tehran, Iran Accepted 11 June 2003 Abstract In this paper, a method for estimating an attractor embedding dimension based on polynomial models and its application in investigating the dimension of Bremen climatic dynamics are presented. The attractor embedding di- mension provides the primary knowledge for analyzing the invariant characteristics of the attractor and determines the number of necessary variables to model the dynamics. Therefore, the optimality of this dimension has an important role in computational efforts, analysis of the Lyapunov exponents, and efficiency of modeling and prediction. The smoothness property of the reconstructed map implies that, there is no self-intersection in the reconstructed attractor. The method of this paper relies on testing this property by locally fitting a general polynomial autoregressive model to the given data and evaluating the normalized one step ahead prediction error. The corresponding algorithms are de- veloped in uni/multivariate form and some probable advantages of using information from other time series are dis- cussed. The effectiveness of the proposed method is shown by simulation results of its application to some well-known chaotic benchmark systems. Finally, the proposed methodology is applied to two major dynamic components of the climate data of the Bremen city to estimate the related minimum attractor embedding dimension. Ó 2003 Elsevier Ltd. All rights reserved. 1. Introduction The basic idea of chaotic time series analysis is that, a complex system can be described by a strange attractor in its phase space. Therefore, the reconstruction of the equivalent attractorÕs state space by embedding the time series in a vector space is the first step of the analysis. This is accomplished from the observations of a single coordinate by some techniques outlined in [1] and method of delays as proposed by Takens [2] which is extended in [3]. However, TakensÕ theorem is only valid for infinite noise free data and does not directly answer how to select embedding dimension. On the other hand, computational efforts, Lyapunov exponents estimation, and efficiency of modelling and prediction is influenced significantly by the optimality of embedding dimension. There are several methods proposed in the literature for the estimation of dimension from a chaotic time series. The three basic approaches are as follow. * Corresponding author. Tel.: +49-421-2182494; fax: +49-421-2184707. E-mail addresses: ataei@iat.uni-bremen.de (M. Ataei), bl@iat.uni-bremen.de (B. Lohmann), sedigh@eetd.kntu.ac.ir (A. Khaki- Sedigh), lucas@karun.ipm.ac.ir (C. Lucas). 1 Tel.: +49-421-2184688; fax: +49-421-2184596. 0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00300-X Chaos, Solitons and Fractals 19 (2004) 1131–1139 www.elsevier.com/locate/chaos