Kriging (also known as Gaussian process regression) is a Bayesian technique that aim at approximating functions (most often in order to surrogate it because it is expensive to evaluate). In the following it is assumed we aim at creating a surrogate model of a scalar-valued model . Note the implementation of Kriging deals with vector-valued functions (), without simply looping over each output. It is also assumed the model is obtained over a design of experiments in order to produce a set of observations gathered in the following dataset: . Ultimately Kriging aims at producing a predictor (also known as a response surface or metamodel) denoted as .
We put the following Gaussian process prior on the model :
is a generalized linear model based upon a functional basis and a vector of coefficients ,
is a zero-mean stationary Gaussian process whose covariance function reads:
where is the variance and is the correlation function that solely depends on the Manhattan distance between input points and a vector of parameters .
Under the Gaussian process prior assumption, the observations and a prediction at some unobserved input are jointly normally distributed:
is the regression matrix,
is the observations’ correlation matrix, and:
is the vector of cross-correlations between the prediction and the observations.
As such, the Kriging predictor is defined as the following conditional distribution:
where and are the maximum likelihood estimates of the correlation parameters and variance (see references).
It can be shown (see references) that the predictor is also Gaussian:
where is the generalized least squares solution of the underlying linear regression problem:
Kriging may also be referred to as Gaussian process regression.
Lophaven, H. Nielsen and J. Sondergaard, 2002, “DACE, A Matlab kriging toolbox”, Technichal University of Denmark. http://www2.imm.dtu.dk/projects/dace/
Santner, B. Williams and W. Notz, 2003. “The design and analysis of computer experiments”, Springer, New York.
Rasmussen and C. Williams, 2006, T. Dietterich (Ed.), “Gaussian processes for machine learning”, MIT Press.