Application of Nonlinear Filtering Trained RBF Networks to Multi-step Prediction of Time-series with Delayed Observations

The multi-step prediction problem of chaotic time series with one sampling delay is investigated in this paper. The delay is considered to be random and is modelled by a binary white noise with values of zero or one, and these values indicate that the observation arrives on time or that it is delayed by one sampling time. Based on the original extended Kalman filtering (EKF) and the Unscented Kalman filtering (UKF), we can obtain corresponding additional two kinds of nonlinear filtering methods with one observation sampling delay which are shortened as DEKF and DUKF in this paper. Using the radial basis function (RBF) neural network prototypes and the network weights as state equation, and the output of RBF neural network to present the observation equation, the input vector to the network is composed of predicted chaotic signal with given length, and the multi-step prediction results are represented by the predicted observation value of nonlinear filtering methods. To show the advantage of DEKF and DUKF, we applied them to the five-step prediction of Mackey-Glass time-series with one sampling delay and compared them with the original EKF and UKF. Experimental results have demonstrated that the DEKF and DUKF are proportionally superior to the original EKF and UKF. Moreover, DUKF is a better choice for Mackey-Glass time-series five-step prediction in comparison with DEKF.