A Gaussian Approximation of Marginal Likelihood in Relevance Vector Machine for Industrial Data With Input Noise

Given that there exists input uncertainty caused by the noise embedded in industrial data, this study proposes a relevance vector machine (RVM) prediction model with input noise. Due to the fact that the marginal likelihood cannot be analytically calculated when introducing the input uncertainty, a Gaussian approximation is proposed in this study on the basis of the law of total expectation and the law of total covariance. Furthermore, to approximate the posterior distribution over the model weights, this study employs the Markov chain Monte Carlo algorithm, where a Gaussian proposal distribution is designed to draw new samples. In the prediction stage, a Gaussian approximation is also designed for a new testing input in order for the input uncertainty to be reflected in the estimation of output variance. To verify the effectiveness of the proposed method, four synthetic data sets, four benchmark data sets, and two industrial data sets are employed in the comparative experiments. The results indicate that the proposed RVM with uncertain input outperforms other approaches, and it also performs better on the time series prediction issue.