Chaotic Time Series Prediction Using a Neuro-Fuzzy System with Time-Delay Coordinates Jun Zhang, Member, IEEE, Henry Shu-Hung Chung, Senior Member, IEEE, and Wai-Lun Lo, Member, IEEE Abstract—This paper presents an investigation into the use of the delay coordinate embedding technique in the multi-input- multioutput-adaptive-network-based fuzzy inference system (MANFIS) for chaotic time series prediction. The inputs to the MANFIS are embedded-phase-space (EPS) vectors preprocessed from the time series under test, while the output time series is extracted from the output EPS vectors from the MANFIS. A moving root-mean-square error is used to monitor the error over the prediction horizon and to tune the membership functions in the MANFIS. With the inclusion of the EPS preprocessing step, the prediction performance of the MANFIS is improved significantly. The proposed method has been tested with one periodic function and two chaotic functions including Mackey-Glass chaotic time series and Duffing forced-oscillation system. The prediction performances with and without EPS preprocessing are statistically compared by using the t-test method. The results show that EPS preprocessing can help improve the prediction performance of a MANFIS significantly. Index Terms—Chaotic time series prediction, neuro-fuzzy systems, time-delay coordinate embedding. Ç 1 INTRODUCTION A time series is a sequence of regularly sampled quantities out of an observed system. It is a useful source of extracts for discovering and studying the behaviors of the system such as periodicity and stochastic distribution. In addition, a reliable time series prediction method can help researchers model the system and forecast its behaviors. Since 1970s, many prediction methods in time or frequency domain have been proposed [1]. Among the different types of time series, chaotic time series can be commonly found in natural phenomena [2], [3], [4]. Thus, starting from 1980s, chaotic time series prediction has been a popular subject [5] for understanding and controlling the chaotic behaviors to advantage, such as the stock market forecasting [6]. The prediction of a chaotic time series requires a representative model. In recent years, many new prediction approaches, such as the wavelet networks [7], neural networks [8], [9], [10], hierarchical Bayesian approach [11], fuzzy [12], [13], [14], [15], [16], [17] and neuro-fuzzy predictor [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], and time-delay embedding technique [31], [32], [33], [34], have emerged. Also, past research has also been done based on a master- slave synchronization scheme for prediction of chaotic behavior [35], multiple model predictor and genetic algo- rithm to reconstruct piecewise chaotic dynamics [36], and an effective digital tracker for continuous-time chaotic orbit tracking [37]. They provide new insight into this type of systems not available in the traditional methods such as linear regression technique and autoregressive integrated moving average models. As discussed in [18], the adaptive-network-based fuzzy inference system (ANFIS) with multi-input and single- output in [19], which uses Gaussian membership functions and employs hybrid backpropagation learning of a chaotic time series, gives the smallest root-mean-square errors among various fuzzy predictors. However, similar to the other single-scale chaotic time series prediction methods, the prediction horizon is usually limited by the fast-varying components, because the methods are based on the finite neighborhood relationships in the prediction. Nevertheless, it forms the best basis for further enhancement. Time-delay coordinate embedding methods [31], [32], [33], [34] use the relationships between the delay coordi- nates of a point and the points that appear at some time later in the phase space. Its trajectories behave with quasiperiodicity. The nearest trajectories can contribute to the neighboring set with more than one point, resulting in an increased weighting of the contribution coming from the nearest trajectories. This paper investigates the use of delay coordinate embedding technique with multi-input-multioutput-ANFIS (denoted by MANFIS) to learn and predict the continuation of chaotic signals ahead of time. The methodology hybridizes the advantages of ANFIS and the time-delay 956 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 20, NO. 7, JULY 2008 . J. Zhang is with the Department of Computer Science, Sun Yat-Sen University, China. E-mail: junzhang@ieee.org. . H.S.H. Chung is with the Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong. E-mail: eeshc@cityu.edu.hk. . W.-L. Lo is with the Department of Computer Science, Chu Hai College of Higher Eduction, Yi Lok Street, Riviera Garden, Tsuen Wan, Hong Kong. E-mail: wllo@chuhai.edu.hk. Manuscript received 15 Mar. 2006; revised 23 Dec. 2006; accepted 17 Jan. 2008; published online 28 Jan. 2008. For information on obtaining reprints of this article, please send e-mail to: tkde@computer.org, and reference IEEECS Log Number TKDE-0137-0306. Digital Object Identifier no. 10.1109/TKDE.2008.35. 1041-4347/08/$25.00 ß 2008 IEEE Published by the IEEE Computer Society