Latin hypercube designs for computer experiments.

Abstract

A class of Orthogonal Latin Hypercubes (OLH) which preserves the orthogonality among columns is constructed. Applying an Orthogonal Latin Hypercube design in a computer experiment benefits the data analysis in two ways. First, it retains the orthogonality of traditional experimental designs. The estimates of linear effects of all factors are not only uncorrelated with each other, but also uncorrelated with the estimates of all quadratic effects and bi-linear interactions. Second, it can facilitate non-parametric fitting procedures, since a good space filling design within the class of Orthogonal Latin Hypercubes can be selected. Furthermore, the structure of Orthogonal Latin Hypercubes is established in the vector space $\{0,1\}\sp{m}.$ An upper bound on the number of columns of an OLH is given. OLHs whose number of columns reaches the upper bound, are found for $m=3,4$ while the cases for $m>5$ are still unknown. A computer experiment using an OLH design conducted at Ford Motor Co. which actually motivated this research is also presented. Optimal Latin hypercube designs are widely used in computer experiments. However, existing search methods are slow, especially for Latin hypercubes with large dimensions. A class of Symmetric Latin Hypercube which has good geometric properties is proposed. The CP algorithm due to Li and Wu (Technometric, 1997) is used to search for optimal Symmetric Latin Hypercubes. Compared with the search in the family of Latin hypercubes using the CP algorithm, its search time is drastically reduced and the Latin hypercubes that can be realistically obtained are likely to be better. In addition, the structure of Symmetric Latin Hypercube provides good properties for fitting a polynomial model.