Fuzzy clustering approach for accurate embedding dimension identification in chaotic time series

• Authors: Xiaomo Jiang;Hojjat Adeli
• Venue: Integrated Computer-Aided Engineering
• Year: 2003

The dynamics of many physically complex systems cannot be represented by a set of differential equations. Measured sensor signals in the form of time-series can be used to investigate complexities and chaotic behavior in such systems. In order to characterize the chaotic behavior of a time series, a state space is constructed from the observed time series which requires the selection of its embedding dimension. In this article, first three existing methods for determining the embedding dimension are investigated using three different examples with available analytical equations where the exact value of the optimum embedding dimension is known. They are the fill-factor method, the average integral local deformation method, and the false nearest neighbors method. Next, a fuzzy c-means clustering approach is proposed for finding the optimum embedding dimension accurately. The time lag index obtained from the average mutual information method is used as the number of clusters. The reconstructed state space vectors are clustered using the fuzzy c-means clustering approach. It is shown that the proposed approach yields the exact answer in all three examples. The proposed approach does not require the arbitrary or trial-and-error selection of parameters and is computationally efficient. It can be used effectively for chaos analysis of complex real-life time series such as electroencephalographs, cardiological arrhythmias, weather patterns, and traffic congestions where the underlying physical phenomenon cannot be described analytically.