EEG TIME SERIES ANALYSIS WITH EXPONENTIAL AUTOREGRESSIVE MODELLING Tugce Balli and Ramaswamy Palaniappan Department of Computing and Electronic Systems, University of Essex, Wivenhoe Park, CO4 3SQ, United Kingdom ABSTRACT This paper proposes the use of exponential autoregressive (EAR) model for modelling of time series that are known to exhibit non-linear dynamics such as random fluctuations of amplitude and frequency. Biological signal (bio-signal) such as electroencephalogram (EEG) is known to exhibit non- linear dynamics. Such signals cannot be modelled with traditional linear modelling techniques like autoregressive (AR) models as these models are known to provide only an approximation to the underlying properties of the non-linear signals. In this study, the suitability of EAR models as compared to AR models is shown using EEG signals in addition to several non-linear benchmark time series data where improved signal to noise ratio (SNR) values are indicated by the EAR models. Overall, the results indicate that use of EAR modelling which has yet to be exploited for bio-signal time series analysis has the huge potential in the characterisation and classification of EEG signals. Index Terms— Exponential autoregressive, Electroencephalogram, Non-linear time series analysis, Genetic algorithm 1. INTRODUCTION Linear modelling techniques have been widely used for analysis of time series as they are simpler in principle as compared to their non-linear counterpart. Among these modelling techniques, autoregressive (AR) modelling is one of the most commonly used method [1,2]. Basically this approach assumes that the time series is a regressive process and estimates the model coefficients according to that assumption. However the time series with non-linear fluctuations cannot be modelled precisely with linear modelling techniques like AR. In these situations, a technique that can model the non-linear fluctuations, which will provide a better representation of time series, is required. Haggan and Ozaki [3,4] introduced exponential autoregressive (EAR) model for modelling non-linear fluctuations in time series. They stated that the analysis of stochastic processes have been mostly done using some form of linear time series modelling and this can only provide an approximation to the underlying properties of the signals. Besides it is found that many signals exhibiting random vibrations display non-linear behaviour, hence a non-linear model that gives a good approximation to the underlying properties of a signal is required. Accordingly, Haggan and Ozaki proposed EAR model which exhibits certain features of random vibrations that does not occur in linear models namely the amplitude-dependent frequency, jump phenomena and limit cycle. In this study, we set out to show the improved signal to noise ratio (SNR) of a common biological signal (bio- signal), namely electroencephalogram (EEG) signal plus several other standard time series data reconstructed using coefficients from EAR as compared to AR modelling techniques. Binary genetic algorithm (BGA) hybridised with recursive least squares (RLS) method has been used to obtain the non-linear and linear coefficients of the EAR model. The rest of this paper is organised into three sections. In section 2, the EAR model and estimation of EAR model coefficients are discussed in detail. Section 3 presents the experimental results of fitting EAR and AR models to some of the benchmark non-linear time series and EEG signals. And in the final section, the conclusions and possible future improvements to this study are presented. 2. METHODOLOGY In this section, conventional AR model, EAR model and method of using GA to obtain the non-linear EAR model coefficients will be discussed. 2.1 Autoregressive model An autoregressive model of order p is defined by ∑ = - + ⋅ - = p k t k t k t e x a x 1 , (1) where x t is the data at sampled point n, a k are coefficients of the AR model and e t is Gaussian white noise with mean zero. The model order, p can be selected as order with minimum Akaike Information Criterion (AIC) [5], defined by ), 1 2 ( 2 ln ) ( 2 + + - = p p N AIC e σ (2) 978-1-4244-1643-1/08/$25 © 2008 IEEE