Gaussian processes: prediction at a noisy input and application to iterative multiple-step ahead forecasting of time-series

With the Gaussian Process model, the predictive distribution of the output corresponding to a new given input is Gaussian. But if this input is uncertain or noisy, the predictive distribution becomes non-Gaussian. We present an analytical approach that consists of computing only the mean and variance of this new distribution (<i>Gaussian</i> <i>approximation</i>). We show how, depending on the form of the covariance function of the process, we can evaluate these moments exactly or approximately (within a Taylor approximation of the covariance function). We apply our results to the iterative multiple-step ahead prediction of non-linear dynamic systems with propagation of the uncertainty as we predict ahead in time. Finally, using numerical examples, we compare the <i>Gaussian</i> <i>approximation</i> to the numerical approximation of the true predictive distribution by simple Monte-Carlo