New Research Directions in Computer Experiments: -Clustered Designs
Selden B. Crary · 2012
We explore the existence, properties, and applications of exact optimal designs for computer experiments, under Gaussian-process (GP), fixed-Gaussian-covarianceparameter, zero-nugget assumptions, that prescribe a cluster of two or more design points as closely spaced as practical, without being identically located. We define such designs as -clustered and define subcases, e.g., twin-point, triplet-point, etc. designs and review the history of these designs. We also define the phase of a design, based on its symmetry properties, and we introduce the concept of phase transitions between phases. We prove that the 0 th - and 1 st -degree terms in the expansion of the determinant of the covariance matrix, in powers of the separation distance from the center of a twin-point cluster to one of the twins, are zero. Using this fact, we outline a proof that, in two or more factors, the IMSE function is a truncated rational function, with leading powers of at least two in the series expansion in powers of the separation of the points, for numerator or denominator. We outline applications of the theory to extrapolation and to inversion of covariance matrices. We demonstrate the use of a nonuplet-point (9-point) design as the first stage of a sequential GP fit. We conjecture the form of the power-series expansion of the determinant of the covariance matrix for triplet-point, quadruplet-point, etc. designs. Finally, we conjecture that standard GP fitting does not support triplet-point designs, but a renormalization fixes this problem. We use the renormalization conjecture as a basis for performing GP fits to functions with regions of closely-spaced design points and show that the proposed method avoids the numerical errors observed via standard approaches.