Estimating Shape Constrained Functions Using Gaussian Processes

Gaussian processes are a popular tool for nonparametric function estimation because of their flexibility and the fact that much of the ensuing computation is parametric Gaussian computation. Often, the function is known to be in a shape-constrained class, such as the class of monotonic or convex functions. Such shape constraints can be incorporated through the use of derivative processes, which are joint Gaussian processes with the original process, as long as the conditions of mean square differentiability in Theorem 2.2.2 of Adler [The Geometry of Random Fields, Vol. 62, SIAM, Philadelphia, 1981] hold. The possibilities and challenges of introducing shape constraints through this device are explored and illustrated through simulations and two real data examples. The first example involves emulating a computer model of vehicle crashworthiness and the second involves emulating a computer model that predicts dissolving of ingredients in a mixing tank for a product. Computation is carried out through a Gibbs sampling scheme.