1,367 research outputs found
On Prediction Properties of Kriging: Uniform Error Bounds and Robustness
Kriging based on Gaussian random fields is widely used in reconstructing
unknown functions. The kriging method has pointwise predictive distributions
which are computationally simple. However, in many applications one would like
to predict for a range of untried points simultaneously. In this work we obtain
some error bounds for the (simple) kriging predictor under the uniform metric.
It works for a scattered set of input points in an arbitrary dimension, and
also covers the case where the covariance function of the Gaussian process is
misspecified. These results lead to a better understanding of the rate of
convergence of kriging under the Gaussian or the Mat\'ern correlation
functions, the relationship between space-filling designs and kriging models,
and the robustness of the Mat\'ern correlation functions
Projection Pursuit Gaussian Process Regression
A primary goal of computer experiments is to reconstruct the function given
by the computer code via scattered evaluations. Traditional isotropic Gaussian
process models suffer from the curse of dimensionality, when the input
dimension is high. Gaussian process models with additive correlation functions
are scalable to dimensionality, but they are very restrictive as they only work
for additive functions. In this work, we consider a projection pursuit model,
in which the nonparametric part is driven by an additive Gaussian process
regression. The dimension of the additive function is chosen to be higher than
the original input dimension. We show that this dimension expansion can help
approximate more complex functions. A gradient descent algorithm is proposed to
maximize the likelihood function. Simulation studies show that the proposed
method outperforms the traditional Gaussian process models
Uncertainty Quantification for Bayesian Optimization
Bayesian optimization is a class of global optimization techniques. It
regards the underlying objective function as a realization of a Gaussian
process. Although the outputs of Bayesian optimization are random according to
the Gaussian process assumption, quantification of this uncertainty is rarely
studied in the literature. In this work, we propose a novel approach to assess
the output uncertainty of Bayesian optimization algorithms, in terms of
constructing confidence regions of the maximum point or value of the objective
function. These regions can be computed efficiently, and their confidence
levels are guaranteed by newly developed uniform error bounds for sequential
Gaussian process regression. Our theory provides a unified uncertainty
quantification framework for all existing sequential sampling policies and
stopping criteria
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