Efficient Spatial Prediction Using Gaussian Markov Random Fields under Uncertain Localization

In this paper, we develop efficient spatial prediction algorithms using Gaussian Markov random fields (GMRFs) under uncertain localization and sequential observations. We first review a GMRF as a discretized Gaussian process (GP) on a lattice, and justify the usage of maximum a posteriori (MAP) estimates of noisy sampling positions in making inferences. We show that the proposed approximation can be viewed as a discrete version of Laplace’s approximation for GP regression under localization uncertainty. We then formulate our problem of computing prediction and propose an approximate Bayesian solution, taking into account observations, measurement noise, uncertain hyperparameters, and uncertain localization in a fully Bayesian point of view. In particular, we present an efficient scalable approximation using MAP estimates of noisy sampling positions with a controllable tradeoff between approximation error and complexity. The effectiveness of the proposed algorithms is illustrated using simulated and real-world data.