Space-Filling Fractional Factorial Designs

Fractional factorial designs are widely used in various scientific investigations and industrial applications. Level permutation of factors could alter their geometrical structures and statistical properties. This article studies space-filling properties of fractional factorial designs under two commonly used space-filling measures, discrepancy and maximin distance. When all possible level permutations are considered, the average discrepancy is expressed as a linear combination of generalized word length pattern for fractional factorial designs with any number of levels and any discrepancy defined by a reproducing kernel. Generalized minimum aberration designs are shown to have good space-filling properties on average in terms of both discrepancy and distance. Several novel relationships between distance distribution and generalized word length pattern are derived. It is also shown that level permutations can improve space-filling properties for many existing saturated designs. A two-step construction procedure is proposed and three-, four-, and five-level space-filling fractional factorial designs are obtained. These new designs have better space-filling properties, such as larger distance and lower discrepancy, than existing ones, and are recommended for use in practice. Supplementary materials for this article are available online.