A stratified L2-discrepancy with application to space-filling designs

Space-filling designs are widely used in computer experiments. We propose a stratified L2-discrepancy to evaluate the uniformity of a design when the design domain is stratified into various subregions. Weights are used to adjust preferences for the uniformity over subregions in each stratification. The stratified L2-discrepancy is easy to compute, satisfies a Koksma–Hlawka type inequality, and overcomes the curse of dimensionality that exists for other discrepancies. It is applicable to a broad class of designs, and covers several minimum aberration-type criteria as special cases. Strong orthogonal arrays of maximum strength are shown to have low stratified L2-discrepancies, and thus are suitable for computer experiments. In addition, we develop a lower bound for the stratified L2-discrepancy and provide a construction method for designs that achieve the lower bound. We further introduce a general version of the stratified L2-discrepancy for evaluating designs with flexible stratification properties.