Fractional Brownian Fields for Response Surface Metamodeling

Abstract

Kriging, a widely used metamodeling method for computer-simulation data, models the response surface as a realization of a random field. Stationary covariance functions such as the Gaussian, power exponential, or Matérn class are the most common choice for the underlying random field model. Nonstationary versions of these same covariance functions with scale parameters that vary spatially are also sometimes used. Fractional Brownian fields (FBFs) are a different form of nonstationary random field model having stationary increments, an example of more general intrinsic stationary processes. Although FBFs have been considered for intrinsic kriging in the spatial statistics literature, they have received little attention for computer-simulation response surface metamodeling. For use in the latter context, we argue that they have some attractive (as well as some unattractive) properties that mitigate certain problems inherent to many stationary covariance models, such as reversion to the mean, numerical issues due to near-singularity of covariance matrices, and difficulties in handling abrupt response surface features.