Review of Dace-Kriging Metamodel

Interdisciplinary Description of Complex Systems 21(3), 316-323, 2023 REVIEW OF DACE-KRIGING METAMODEL Muzaffer Balaban* Turkish Statistical Institute Ankara, Turkey DOI: 10.7906/indecs.21.3.8 Regular article Received: 12 June 2023. Accepted: 28 June 2023. ABSTRACT This paper presents a conceptual review of the kriging metamodel that is introduced for the design and analysis of computer experiments (DACE). Kriging is a statistical interpolation method to build an approximation model from a set of evaluations of the function at a finite set of points. The method originally developed for geostatistics, and it is now widely used in the domains of spatial data analysis and computer experiments analysis. The main difference between these domains the dimensionality of the problems. Geostatistics and spatial data are mainly deal with the coordinates. Computer experiments, simulation outputs and other engineering problems have multidimensional input variables. With this study, it is aimed to examine the limitations of the prediction performance of the DACE-kriging metamodel. The result of the study shows that the regression part of the DACE-kriging metamodel is the most important part to develop an approximation, and if there is a spatial relationship of the residuals, kriging part will also contribute to the improvement of the prediction performance. Otherwise, kriging will have no contribution to the DACE-kriging metamodel, and even worsen the prediction performance. If the regression part perfectly fit to the observations, the residual will have poor spatial relationship and the kriging part will be meaningless anymore. KEY WORDS DACE-kriging, regression, basic kriging, correlogram CLASSIFICATION JEL: C15 *Corresponding author, : ; +90 535 507 4252 *Turkish Statistical Institute, 06100, Ankara, Turkey Review of DACE-kriging metamodel INTRODUCTION Kriging term covers several spatial interpolation models. Kriging theory was originally developed as a geostatistical interpolation method [1]. The kriging model makes predictions at unobserved locations using a linearly weighted combination of observations. Each observation influences the kriging prediction is based on geographical proximity to the unobserved location, the spatial spreads and the pattern of spatial correlation of the observations. Kriging models are meaningful only if the observations are spatially correlated. The kriging weights are recalculated using the appropriate variogram or correlogram model for each prediction point. There are many kinds of kriging in the literature such as simple kriging, ordinary kriging, universal kriging, cokriging, median polish kriging etc. [2]. Sacks et al. [3] presented a modified kriging approach as a metamodel to deterministic computer experiments. The use of kriging metamodels has been remarkably effective for global metamodeling in the design and analysis of computer experiments (DACE) community when the simulation models are complex and/or very expensive to run [4]. Their approach is a hybrid method that combines a regression between the output variable and input variables with the simple kriging (SK) of the regression residuals. Firstly, a polynomial regression model is applied to the outputs and then basic kriging applied to the residuals. Their main contribution to the kriging literature is expanding of the problem dimensions from two-dimensional coordinate to the high dimensional computer experiments. Additionally, they have used high dimensional correlogram models instead of variogram mostly used in geostatistics to find the kriging weights. Prediction at each new point is performed by summing the predicted trend and residual. The parameter set used for the regression part are estimated once for the whole search space, and for the kriging part the weights are re-estimated at each new point. In the literature it is known that Regression models are local and kriging models are global. Kriging models are flexible because of the diversity of the correlogram model obtained from the experiments. Therefore, it reveals the importance of the kriging part to develop a global metamodel for the whole search space. There are several names of this method in the DACE literature. Some of them are as follows. Kriging [3, 5-16], spatial correlation metamodels [8, 17, 18], Gaussian process models [4, 19, 20], Gaussian stochastic process models [21, 22], Gaussian kriging [21] are used as the name of the method in the related references. In the geostatistical literature this method is called regression kriging [23, 24]. Some authors variously call this method as regression with residual simple kriging [25], detrended kriging [26, 27] and residual kriging [28, 29]. I prefer to use “DACE-Kriging” in the metamodeling process as the same meaning with the “regression kriging” in Geostatistics as the name for this method to prevent some misunderstanding because the kriging term refer to a general class of geostatistical interpolation methods. The aim of this study is to examine the limitations of the prediction performance of the DACE-kriging metamodel. The results of this study show that the regression part of the DACEKriging model is the most important part to develop an approximation, and if there is a spatial relationship of the residuals, kriging part will also contribute to the improvement of the prediction performance. Otherwise, kriging will have no contribution to the DACE-Kriging model, and even worsen the prediction performance. If the regression part perfectly fit to the observations, the residual will have poor spatial relationship and the kriging part will be meaningless anymore. Remaining of this article as follows. Model formulation of DACE-Kriging metamodel is presented in Section 2, numerical examples are given in Section 3, and Section 4 presents conclusions. 317 M. Balaban MODEL FORMULATION OF DACE-KRIGING METAMODEL DACE-Kriging metamodel is a mixed estimation method that is a combination of multiple regression methods and simple kriging. It can be defined as the estimation of residual values obtained from the difference between the estimation values made by methods such as regression and the observation values by kriging method [3]. Simply, prediction at each new point is performed by summing the predicted trend and residual. Predicted trend is obtained by linear or quadratic regression (or higher order) and predicted residual is obtained by simple kriging applied to regression residuals. Model assumptions of Y(𝐱) are given in the followgin equations: Y(𝐱) = M(𝐱) + Z(𝐱), (1) M(𝐱) = ∑bj=0 βj fj (𝐱), (2) Z(𝐱) = Y(𝐱) − M(𝐱), (3) Z(𝐱) = Y(𝐱) − ∑bj=0 βj fj (𝐱). (4) DACE-Kriging predictor is given in the following two equations: ŷ(𝐱 𝟎 ) = ∑bj=0 βj fj (𝐱 𝟎 ) + ∑ni λi z(𝐱 𝐢 ), (5) ŷ(𝐱 𝟎 ) = ∑bj=0 βj fj (𝐱 𝟎 ) + ∑ni λi ( y(𝐱 𝐢 ) − ∑bj=0 βj fj (𝐱 𝐢 )). (6) The first part of the model in (5) and (6) shows the regression model and the second part shows the si (...truncated)