Comparison of trend models for geotechnical spatial variability: Sparse Bayesian Learning vs. Gaussian Process Regression

This paper compares two probabilistic models for the trend function of geotechnical spatial variability: sparse Bayesian learning (SBL) vs. Gaussian process regression (GPR). For the SBL model (MSBL), the spatial trend is represented as the weighted sum of a sparse set of basis functions (BFs). For the GPR model (MGPR), the spatial trend is represented as a stationary normal random field. The comparison between these two models is based on their Bayesian evidence. The comparison results show that MSBL usually outperforms MGPR with a larger Bayesian evidence when the underlying trend function can be represented by sparse BFs. This usually happens for one-dimensional (1D) simulated examples and 1D real cone penetration test (CPT) examples. However, MGPR usually outperforms MSBL when the spatial trend can not be well represented by sparse BFs. This usually happens for 2D and 3D real CPT examples. Another important contribution of the current paper is the derivations of Kronecker-product formulae for the GPR method. These formulae resolve the issue of the high computational cost for 3D GPR analyses.