Characterizing Uncertain Site-Specific Trend Function by Sparse Bayesian Learning

AbstractThis paper addresses the statistical uncertainties associated with the estimation of a depth-dependent trend function and spatial variation about the trend function using limited site-specific geotechnical data. Specifically, the statistical uncertainties associated with the following elements are considered: (1) the functional form (shape) of the trend function; (2) the parameters of the trend function (e.g., intercept and gradient); and (3) the random field parameters describing spatial variation about the trend function, namely standard deviation (σ) and scale of fluctuation (δ). The problem is resolved with a two-step Bayesian framework. In Step 1, a set of suitable basis functions that parameterize the trend function is selected using sparse Bayesian learning. In Step 2, an advanced Markov chain Monte Carlo method is adopted for the Bayesian analysis. The two-step approach is shown to be consistent in the well-defined sense that the resulting 95% Bayesian confidence interval (or region) contains the actual trend (or actual σ and δ) with a chance that is close to 0.95. Inconsistency can occur when the spatial variability has a large σ or a large δ relative to data record length.