Multi-step prediction of chaotic time-series with intermittent failures based on the generalized nonlinear filtering methods

There are many practical situations that the chaotic signal appears in a random manner so that there are intermittent failures in the observation mechanism at certain times. These random interruptions, which are called as multiplicative noises, can be modeled by a sequence of independent Bernoulli random variables. Considering the observed chaotic signal perturbed by additive and multiplicative noises at the same time, this paper generalizes the original extended Kalman filtering (EKF), the Unscented Kalman filtering (UKF) and the Gaussian particle filtering (GPF) to the case in which there is a positive probability that the observation with intermittent failures in each time consists of additive noises alone. The shortened forms of these generalized new filtering algorithms are written as GEKF, GUKF and GGPF correspondingly. Using weights and network output of perceptron neural network to constitute state transition equation and observation equation, the input vector to the network is composed of predicted chaotic signal with given length (see Section 2 for details), and the multi-step prediction results are represented by the predicted observation value of nonlinear filtering methods. To show the advantage of these generalized new filtering algorithms, we applied them to the five-step prediction of Mackey–Glass time-series and equipment’s temperature (The corresponding time series can be found at http://robjhyndman.com/TSDL) with additive and multiplicative noises, respectively and compared them with the original EKF, UKF and GPF. Experimental results have demonstrated that the GEKF, GUKF and GGPF are proportionally superior to the original EKF, UKF and GPF. Moreover, GGPF is a better choice for multi-step prediction in comparison with GEKF and GUKF.