Blind Kriging: A New Method for Developing Metamodels

Abstract

Kriging is a useful method for developing metamodels for product design optimization. The most popular kriging method, known as ordinary kriging, uses a constant mean in the model. In this article, a modified kriging method is proposed, which has an unknown mean model. Therefore, it is called blind kriging. The unknown mean model is identified from experimental data using a Bayesian variable selection technique. Many examples are presented, which show remarkable improvement in prediction using blind kriging over ordinary kriging. Moreover, a blind kriging predictor is easier to interpret and seems to be more robust against mis-specification in the correlation parameters.

Recent advances in surrogate-based optimization

“…Hopefully, if the underlying trends can be identified, the ensuing model will be more accurate than ordinary Kriging. Joseph et al [42] have certainly shown this to be the case for their engineering design examples. We will outline the process of building a blind Kriging prediction and leave the reader to consult Joseph et al [42] for more details (our description is drawn from this reference).…”

“…Joseph et al [42] have certainly shown this to be the case for their engineering design examples. We will outline the process of building a blind Kriging prediction and leave the reader to consult Joseph et al [42] for more details (our description is drawn from this reference). The above reference suggests identifying the ν's through a Bayesian forward selection technique [41] and uses candidate variables of linear effects, quadratic effects, and two-factor interactions.…”

“…We use the cautious term 'promising' because here we are looking at an analytical test function in just two dimensions, with a large quantity of observed data. In this case, and in Joseph et al [42] (who use a small amount of data in high dimensional, non analytical problems), blind Kriging performs very well, but we would like to see it applied to more problems before we commit ourselves entirely. One should also bear in mind that the blind Kriging process is more computationally expensive and this may outweigh increased accuracy.…”

“…Predictions can now be used to calculate a cross-validation error and the above process iterated to reduce this error up to m = 2k 2 times. Joseph et al [42] stop iterating when the cross-validation error begins to rise consistently and choose the m variables corresponding to the smallest error. They also note that it is not necessary to estimate θ and p at every step, just the first and the last.…”

Interpolation of spatial data – A stochastic or a deterministic problem?

“…The eight input variables and their respective ranges of interest are summarized in Table 4. We rescale these variables to the range [1,3] and use the same orthogonal sampling design as Joseph et al [39] with 27 locations. We now assume f to be a realization of a stationary Gaussian random field with covariance function Φ(h) = σ 2 e − 8 j=1 θ j h 2 j of the Gaussian type.…”