Online multivariate time series prediction using SCKF-γESN model

Bayesian Cramér-Rao Lower Bounds for Prediction and Smoothing of Nonlinear TASD Systems

The performance evaluation of state estimators for nonlinear regular systems, in which the current measurement only depends on the current state directly, has been widely studied using the Bayesian Cramér-Rao lower bound (BCRLB). However, in practice, the measurements of many nonlinear systems are two-adjacent-states dependent (TASD) directly, i.e., the current measurement depends on the current state as well as the most recent previous state directly. In this paper, we first develop the recursive BCRLBs for the prediction and smoothing of nonlinear systems with TASD measurements. A comparison between the recursive BCRLBs for TASD systems and nonlinear regular systems is provided. Then, the recursive BCRLBs for the prediction and smoothing of two special types of TASD systems, in which the original measurement noises are autocorrelated or cross-correlated with the process noises at one time step apart, are presented, respectively. Illustrative examples in radar target tracking show the effectiveness of the proposed recursive BCRLBs for the prediction and smoothing of TASD systems.

Relevance Vector Machines-Based Time Series Prediction for Incomplete Training Dataset: Two Comparative Approaches

Considering that real-life time series mixed with missing points cannot be directly modeled by using most of the supervised machine learning methods, this paper proposes a novel time series prediction method based on relevance vector machines for incomplete training dataset. Given the regularity between the missing inputs and outputs constructed by the phase space reconstruction, this paper imputes the missing inputs during the learning process by the values of their corresponding missing outputs such that the elements in kernel matrix related to the missing inputs are capable of being updated. This paper designs two strategies to estimate the missing outputs. The first one is based on the expectation maximization formulation in which a joint posterior distribution over the missing outputs and the weights vector is derived as a multivariate Gaussian form, and the another maximizes the marginal likelihood function with respect to the missing outputs and other hyperparameters. To verify the performance of the two proposed computing strategies, two synthetic time series and a real-life dataset are employed. The results indicate that the proposed methods have robust and better performance over the other methods when dealing with incomplete time series training dataset.