Extended and Unscented Kalman filtering based feedforward neural networks for time series prediction

Abstract

With the ability to deal with high non-linearity, artificial neural networks (ANNs) and support vector machines (SVMs) have been widely studied and successfully applied to time series prediction. However, good fitting results of ANNs and SVMs to nonlinear models do not guarantee an equally good prediction performance. One main reason is that their dynamics and properties are changing with time, and another key problem is the inherent noise of the fitting data. Nonlinear filtering methods have some advantages such as handling additive noises and following the movement of a system when the underlying model is evolving through time. The present paper investigates time series prediction algorithms by using a combination of nonlinear filtering approaches and the feedforward neural network (FNN). The nonlinear filtering model is established by using the FNN’s weights to present state equation and the FNN’s output to present the observation equation, and the input vector to the FNN is composed of the predicted signal with given length, then the extended Kalman filtering (EKF) and Unscented Kalman filtering (UKF) are used to online train the FNN. Time series prediction results are presented by the predicted observation value of nonlinear filtering approaches. To evaluate the proposed methods, the developed techniques are applied to the predictions of one simulated Mackey–Glass chaotic time series and one real monthly mean water levels time series. Generally, the prediction accuracy of the UKF-based FNN is better than the EKF-based FNN when the model is highly nonlinear. However, comparing from prediction accuracy and computational effort based on the prediction model proposed in our study, we draw the conclusion that the EKF-based FNN is superior to the UKF-based FNN for the theoretical Mackey–Glass time series prediction and the real monthly mean water levels time series prediction.