Brownian Integrated Covariance Functions for Gaussian Process Modeling: Sigmoidal Versus Localized Basis Functions

Ning Zhang, Daniel W. Apley*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


Gaussian process modeling, or kriging, is a popular method for modeling data from deterministic computer simulations, and the most common choices of covariance function are Gaussian, power exponential, and Matérn. A characteristic of these covariance functions is that the basis functions associated with their corresponding response predictors are localized, in the sense that they decay to zero as the input location moves away from the simulated input sites. As a result, the predictors tend to revert to the prior mean, which can result in a bumpy fitted response surface. In contrast, a fractional Brownian field model results in a predictor with basis functions that are nonlocalized and more sigmoidal in shape, although it suffers from drawbacks such as inability to represent smooth response surfaces. We propose a class of Brownian integrated covariance functions obtained by incorporating an integrator (as in the white noise integral representation of a fractional Brownian field) into any stationary covariance function. Brownian integrated covariance models result in predictor basis functions that are nonlocalized and sigmoidal, but they are capable of modeling smooth response surfaces. We discuss fundamental differences between Brownian integrated and other covariance functions, and we illustrate by comparing Brownian integrated power exponential with regular power exponential kriging models in a number of examples. Supplementary materials for this article are available online.

Original languageEnglish (US)
Pages (from-to)1182-1195
Number of pages14
JournalJournal of the American Statistical Association
Issue number515
StatePublished - Jul 2 2016


  • Fractional Brownian field
  • Gaussian random fields
  • Kriging
  • Metamodeling
  • Response surface modeling

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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