On using standard residuals as a metric of kriging model quality

An interpolating kriging model, though it will always return the observations exactly, may not provide a good representation of the computer simulation at other values within the input domain. Without access to additional and potentially costly validation observations, it is difficult to determine if a kriging model is a good representation of the original computer model. One method to determine the predictive quality of a kriging model is to use leave-one-out cross-validation. A second difficulty with creating kriging models is a lack of diagnostic tests to determine how to improve the kriging model to result in a better estimation of the original computer model. This paper presents developments of diagnostic tools for creating kriging models. A computationally efficient form for the leave-one-out cross-validation residual and the variance at the left out location is presented. The standardized residuals can then be used to test if all of the observations appear to come from the Gaussian spatial process specified by the kriging model. This lack of fit may be the result of: 1) erroneous data, 2) the form of the kriging model is not sufficient to estimate the observations as a Gaussian process, 3) or the range of the model is not well represented by a single spatial random process. Two practical examples are provided to demonstrate how to interpret the results and make decisions on how to improve the predictive capability of the kriging model. The first example is a one-dimensional adiabatic flame temperature calculation. The second problem is a two-dimensional Branin test function.