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.