Abstract—In this paper, we study the application of Extreme Learning Machine (ELM) algorithm for single layered feedforward neural networks to non-linear chaotic time series problems. In this algorithm the input weights and the hidden layer bias are randomly chosen. The ELM formulation leads to solving a system of linear equations in terms of the unknown weights connecting the hidden layer to the output layer. The solution of this general system of linear equations will be obtained using Moore-Penrose generalized pseudo inverse. For the study of the application of the method we consider the time series generated by the Mackey Glass delay differential equation with different time delays, Santa Fe A and UCR heart beat rate ECG time series. For the choice of sigmoid, sin and hardlim activation functions the optimal values for the memory order and the number of hidden neurons which give the best prediction performance in terms of root mean square error are determined. It is observed that the results obtained are in close agreement with the exact solution of the problems considered which clearly shows that ELM is a very promising alternative method for time series prediction. Keywords—Chaotic time series, Extreme learning machine, Generalization performance. I. INTRODUCTION RTIFICIAL Neural Networks (ANNs) have been extensively applied for pattern classification and regression problems. The major reason for the success of ANNs is their ability in obtaining a non-linear approximation model function describing the association between the dependent and independent variables using the given input samples. Since ANNs adaptively select the model from the features presented in the input data, they are applied to a large number of classes of problems of importance like optical character recognition [7], face detection [11], gene prediction [14], credit scoring [6] and time series forecasting [12], [17].Though ANNs have many advantages such as better approximation capabilities and simple network structures, however, it suffers from several problems such as presence of local minima's, imprecise learning rate, selection of the number of hidden neurons and over fitting. Moreover, the gradient descent based learning algorithms such as Back Propagation (BP) will generally lead to slow convergence during the training of the networks. Rampal Singh is with Department of Computer Science, Deen Dayal Upadhyaya College, University of Delhi, New Delhi-110015, India (phone:+91-11-27570620, 09350647546, e-mail: rpsrana@ddu.du.ac.in). S. Balasundaram is with School of Computer & Systems Sciences, Jawaharlal Nehru University, New Delhi-110067, India (phone: +91-11- 26704724, e-mail: balajnu@hotmail.com). Time series forecasting is an important and challenging problem of regression. In the regression problem by analyzing the given input samples the best fit functional model describing the relationship between the dependent and independent variables is obtained. There are many prediction models exist in the literature [4], [9], [12] for time series. The important and widely used among them are Auto Regressive Integrated Moving Average (ARIMA) [1], ANNs [12], [17] and Support Vector Regression (SVR) [8], [9], [13], [15] methods. Among the above methods, ARIMA assumes the existence of a linear relationship in the time series values, i.e. its prediction value will be a linear function of the past observations and therefore it is not always suitable for complex real world problems [18]. Also it is proposed to combine several methods in order to obtain improved forecasting accuracy. For the study of a hybrid approach of combining ARIMA and ANN for time series forecasting we refer the reader to [18]. For time series involving seasonality, combining Seasonal time series ARIMA (SARIMA) and ANN is discussed in [16] and for the study of a combined SARIMA and SVR approach see [3]. Huang et al [5] have proposed a new learning algorithm for Single hidden Layer Feedforward Neural Network (SLFN) architecture called Extreme Learning Machine (ELM) which overcomes the problems caused by gradient descent based algorithms such as BP applied in ANNs. In this algorithm the input weights and the hidden layer bias are randomly chosen. The ELM formulation leads to solving a system of linear equations in terms of the unknown weights connecting the hidden layer to the output layer. The solution of this general system of linear equations is obtained using Moore-Penrose generalized pseudo inverse [10]. In this work we discuss briefly the ELM algorithm and study its feasibility of application for chaotic time series prediction problems. Throughout this paper, we assume all vectors to be column vectors. For any two vectors x, y in the m- dimensional real space m ℜ , we denote the inner product of the two vectors by y x . ′ where x ′ is the transpose of the vector x and the norm of a vector by . || || ⋅ The paper is organized as follows. In Section 2, we define Moore- Penrose generalized inverse, the minimum norm least squares solution of a general linear system of equations and state the relation between them. In Section 3, we revive the ELM algorithm for SLFN. For the application of this algorithm we considered Mackey Glass delay differential equation with different time delays, Santa Fe-A and UCR heart beat rate (ECG) time series in Section 4 and the results Application of Extreme Learning Machine Method for Time Series Analysis Rampal Singh, and S. Balasundaram A International Journal of Intelligent Systems and Technologies 2;4 © www.waset.org Fall 2007 256