Sampling schemes for model visualization

This article illustrates a technique for visualizing nonlinear mappings f : R-k --> R-m that arise frequently in engineering applications. The idea is based on viewing sections f(-1)(B) of the domain R-k, and f (A) of the ran ge R-m, respectively. After suitable discretization, such sections are easi ly approximated with familiar brushing operations in scatterplot matrices. An obvious approach to discretization is to evaluate f on a factorial grid in Rk and view the sections by restriction to the grid and its image. The p roblem is that factorial grids grow large quickly for desirable numbers of grid points (knots) and even moderate dimensions k. The problem can be solv ed by thinning factorial grids using techniques familiar from experimental design: orthogonal arrays constructed from sets of orthogonal Latin squares . As a result, one obtains manageable sets of domain points with desirable visual properties, high density in each variable pair, and the ability to c apture pairwise variable interactions in f. The benefits of this scheme are illustrated using a model of stresses in the manufacture of lamp filaments . The use of experimental designs in the analysis of computer-generated qua ntitative models is well established in the literature. The main applicatio n is for estimating high-dimensional integrals, but Owen (1992) mentioned a lso the use for visualizing such models. The purpose of this article is to execute in detail one possible approach to model visualization, namely, one based on familiar brushing operations on scatterplot matrices.