Efficient modeling of random fields by using Gaussian process inducing-point approximations

Gaussian process (GP) regression provides an elegant way to estimate the trend and associated uncertainty of soil properties from datasets by using flexible and expressive kernels. However, the exact computation is intensive due to the dense operation of the matrix that scales cubically with the size of the datapoints. This paper proposes using inducing-point approximations as an efficient method to address the problem. The approximated posterior mean and covariance are reformulated for efficient sampling. Numerical results indicate that in terms of effi- ciency, sparse approximations are a significant improvement over exact computations. The locations of arbi- trarily selected inducing points are compared in terms of approximation accuracy, and recommendations are provided based on empirical evidence. In addition, the relations between the relevance vector machine (RVM), Bayesian compressive sensing (BCS), and GP for random field modeling are clarified. The results from different models are compared. The sparse GP provides a more reasonable estimation of the uncertainty at locations far from the observations while being competitive with the RVM and BCS in terms of computational efficiency.