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This study implements a chaos-based model to predict the foreign exchange rates. In the first stage, the delay coordinate embedding is used to reconstruct the unobserved phase space (or state space) of the exchange rate dynamics. The phase space exhibits the inherent essential characteristic of the exchange rate and is suitable for financial modeling and forecasting. In the second stage, kernel predictors such as support vector machines (SVMs) are constructed for forecasting. Compared with traditional neural networks, pure SVMs or chaos-based neural network models, the proposed model performs best. The root-mean-squared forecasting errors are significantly reduced.

International transactions are usually settled in the near future. Exchange rate forecasting is very important to evaluate the benefit and risk attached to the international business environment. Owing to the high risk associated with the international transactions, exchange rate forecasting is one of the challenging and important fields in modern time series analysis. The difficulty of forecasting arises from the inherent non-linearity and non-stationarity in exchange rate or financial time series (Cao, 2003 and Huang and Wu, 2010). To solve the problem, this study develops a new forecasting strategy that employs the phase space reconstruction from chaos theory and support vector regression from kernel methods to extract the above financial characteristics for making a good prediction. Namely, this study will develop a chaos-based nonparametric model to predict the exchange rate’s future behavior. For forecasting strategies, Box and Jenkins’ Auto-Regressive Integrated Moving Average (ARIMA) technique has been widely used for time series forecasting. However, ARIMA is a general univariate model and it is developed based on the assumption that the time series being forecasted are linear and stationary, usually not satisfied for financial data. In recent years, neural networks (NN) has found useful applications in financial time series forecasting, including Hill et al., 1996, Kamruzzaman and Sarker, 2003, Wang and Leu, 1996, Yao and Tan, 2000, Zhang and Hu, 1998 and Zimmermann et al., 2001. Recently, the support vector machine (SVM) method (e.g., Cristianini and Shawe-Taylor, 2000, Schoelkopf et al., 1999, Vapnik, 1995 and Wang, 2005), another form of neural networks, has been gaining popularity and has been regarded as the state-of-the-art technique for regression and classification applications. It is believed that the formulation of SVM embodies the structural risk minimization principle, thus combining excellent generalization properties with a sparse model representation. Recent applications of SVM in financial forecasting include: Cao, 2003, Chang and Tsai, 2008, Huang, 2008, Huang and Wu, 2008, Kim, 2003 and Ince and Trafalis, 2005. Chaos theory is relatively new in science. It is only very recently, in the late 1980s, that interest in chaos theory as a financial analysis tool has emerged. This is so since the theory offers a new way by which the behavior of financial markets can be predicted (at least, over the short-term) and that this non-linear, financial model goes beyond statistics for it can reveal hidden patterns and trends in financial data which could not be captured by conventional statistical techniques. Scheindman and Lebaron, 1989 and Frank and Stengos, 1988 have found the chaotic behavior in financial market such as stock market, foreign exchange markets and futures market. Chaotic time series prediction becomes an extremely important research area and obtains widespread application (Liu et al., 2007, Shangfei and Peiling, 1998 and Xu and Xiong, 2003). Recent applications of chaos theory in financial markets include: Federici and Gandolfo, 2002, Kumar et al., 1999, Ma and Xu, 2007, Pavlidis et al., 2005 and Torkamani et al., 2007. For time series feature extraction, using chaos theory a chaotic attractor may be obtained by measuring the chaotic exchange rate series. The properties of the chaotic attractor can be retained through a reconstruction procedure. This procedure is known as the delay coordinate embedding (Takens, 1981) resulting in a reconstructed phase space (or state space) which contains a reconstructed chaotic attractor preserving both geometrical and dynamical properties of the original chaotic attractor. In forecasting, one can only observe the exchange rate series, which are unable to exhibit the inherent essential character of the exchange rate dynamic system. The phase space reconstruction provides us a mean to study the unobserved variables of the system, and thus is suitable for financial forecasting. The major innovation of this paper lies in combing the phase space reconstruction with kernel regressors for exchange rate forecasting. In the first stage, the delay coordinate embedding transforms the input space (raw data) to a feature space (or state space) suitable for financial modeling and forecasting. In the second stage of the new method, kernel regressors that best fit the transformed series are constructed for final forecasting. Compared with neural networks, pure SVMs or chaos-based neural network models, the proposed model performs best. The root-mean-squared forecasting errors are significantly reduced. The remainder of the paper is organized as follows. Section 2 introduces the new prediction model, including the chaos theory, delay coordinate embedding, and the support vector regression. Section 3 describes the data used in the study, and discusses the experimental findings. Conclusions are given in Section 4.

This study developed a novel hybrid model for exchange rate forecasting, which operates on unobserved phase space and utilizes support vector regressions for nonparametric modeling and prediction. Extracting the time series features from phase space, the new method could predict the future evolution of an exchange rate more effectively and thus performs better than pure models like SVR and BPNN. By using MI and FNN this study obtained the optimal delay time and embedding dimension. Then, the original time series are transformed into multi-dimension phase space, which contains the reconstructed chaotic attractor preserving both geometrical and dynamical properties of the original chaotic attractor. Experimental results on six exchange rates and three performance measures confirmed that Chaos-SVR is the best predictors, which substantially reduces the prediction errors and performs better then other pure or hybrid models. In summary, Chaos-SVR model extracts key features in exchange rate dynamics, and is more suitable in describing data relationship between input and output so that it performs better than other conventional methods. The highly effective forecasting framework can also be applied to other problems involving financial forecasting. Results of this study can also be used to perform good hedges on global investments.