Robust Kriging Models

Kriging models have proven useful in estimating complex and computationally expensive analyses. They are capable of interpolating a set of observations by quantifying both longer range variations with parametric trends and shorter range variations with spatial correlations. Kriging models have had some difficulty with robustness in situations when there are a larger number of input dimensions and few observations as well as a larger number of observations with few dimensions. This paper will detail how to add a parameter to the kriging model that will account for random or measurement errors. The result is a model that will no longer interpolate all of the observations and may be a function of fewer of input dimensions. The resulting model may be better able to approximate the orignal function as demonstrated in four two-dimensional examples. Kriging models provide a very flexible and computationally efficient metamodel form that can be adapted to reproduce many complex response surfaces. It is useful for the approximation of computer analyses since it is capable of interpolating all of the observations used to create it. The kriging model is a statistics-based model that can incorporate the trend model properties from more typical linear regression with the spatial correlation properties of kernel-based approximation methods such as radial basis functions. The kriging model parameters can be objectively estimated given the set of observations from the process it is intended to approximate. With the potential strengths of the kriging model form at approximating complex responses also come some weaknesses. These possible weaknesses include computational difficulty and expense at estimating the optimal model parameters. These weaknesses are the result of a lack of robustness with the kriging model. The robustness of a kriging model can be compromised in three practical situations. In the first situation, a kriging model is desired to fit a set of observations in which there are few observations (n) relative to the number of input dimensions (d)) (e.g. n < 10d). In this situation, there is not enough information to adequately estimate the spatial correlation that may exist between the observations. A second situation may arrise when there are a larger number of observations to interpolate relative to the number of input dimensions. In this situation there may be additional information present in the observations that can’t be adequately quantified by the kriging model form. The last situation may arrise when a number of observations are located very close to each other, as may occur when using a kriging model as part of an iterative optimization process. In this situation, the kriging model can become numerically unstable. This work presents a method to adresses these three situations of robustness by demonstrating the impact on including a nugget parameter in the model that relaxes the interpolation constraint and gives the universal kriging model the ability to smoothly transition from the purely parametric trend model of linear regression to the strict interpolation model of universal kriging. This article presents a possible option for dealing with the lack of robustness that has been experienced by many users of the interpolating kriging model form in the area of design and design automation. The next section of this article details the background of the interpolating kriging model formulation and provides a small addition of that form to loosen the requirement to interpolate all of the observations, the nugget parameter. One difficulty with this non-interpolating form may be the requirement to estimate at least one additional parameter to define the (spatial) covariance of the observed data used in the resulting model form. The Discussion of Formulation section provides a more general explanation of the formulation to give users a more intuitive feel for the mechanisms being used in the altered model form. Four demonstration problems are described and used to demonstrate the effectiveness of the new model form, highlighting the impact of the new nugget parameter on the resulting kriging model. Finally, the article closes with some conclusions about this robust kriging model form and provides a list of future developments required for the further use of this kriging model form.