Theory of Quadrature Formulas and Design of Experiments

Abstract

It is often the case, in design of experiments for estimating the models, that we are concerned with optimal allocation of observation points under a given sample size. In particular when experimental domain is the surface or interior of a ball, there have been numerous works concerning designs with a certain rotation equivariance property called rotatability. Rotatable designs and related notions have been extensively and independently studied from the viewpoint of not only design theory but also quadrature theory in numerical analysis and Euclidean design theory in combinatorics. In this article, while exploring the theories of rotatable designs and Euclidean designs, we give an overview of the construction theory of high-dimensional quadrature formulas. We also elucidate the advantages of reconstructing various design-theoretic concepts in the framework of quadrature theory, as exemplified by the determination of maximum number t for which classical response surface designs such as Box-Behnken designs, central composite designs and Doehlert designs are of t-th-order rotatable.