Spatial Function Estimation with Uncertain Sensor Locations

Abstract

In this thesis, we investigate the task of spatial function estimation from the viewpoint of Gaussian Process Regression (GPR) while accounting for uncertain training positions (uncertain sensor positions, uncertain inputs). We first present the theory behind GPR with known training positions. The theory is then applied to derive the expressions for the GPR predictive distribution at a test position under training position uncertainty. Because these expressions are intractable, they are evaluated approximately using the Monte Carlo sampling method. This method is demonstrated to improve the prediction performance over the standard usage of GPR not accounting for uncertainty and also compared to a simplified approach present in the literature. We furthermore investigate the possibilities of performing GPR under training position uncertainty while using closed form expressions for prediction reported in the literature. It turns out that significant approximations are needed to obtain these closed form expressions, which makes the resulting posterior distribution inherently approximate. In fact, the resulting GPR method uses the standard form of GPR for prediction along with a modified expression of the covariance function. A simulation shows that the prediction results of this method are similar to those of standard GPR not accounting for uncertainty. On the other hand, the posterior variance indicating the prediction uncertainty was increased, which is the desired effect of incorporating uncertainty of training positions.