A novel online adaptive time series prediction model with input and output uncertainties

Only the output uncertainty is usually concerned in many regression models for noisy time series prediction, and a near optimal smooth model will be a close approximation to input–output process function if the input data is noise free and only the output data is corrupted by noise in a smooth input–output process. However, if the input data is also corrupted by noise then the best predictive smooth model based on noisy data need not be an approximation to the actual underlying process; rather, the best predictive model depends on both the underlying process and the noise. In this study, a novel strategy with noise addition is proposed by combining iterated nonlinear filters and the single multiplicative neuron (SMN) model, in which the additive noises describe the internal state uncertainty which can be used to explain the input noise and the output uncertainty. The state vector and the observation equation of nonlinear filters are presented by using the weights and the biases of SMN model and the output of SMN model, and the input vector data of the SMN model are composed of known sequential noisy time series data. To verify the effectiveness of the proposed methods, the noisy Mackey–Glass time series, the noisy Box–Jenkins dataset and the noisy electroencephalogram data are employed. The experimental results have demonstrated that the proposed methods are effective for noisy time series prediction.