Benchmarking Low-Dimensional Projection Uniformity in Mixed Two- and Three-Level Designs

Abstract

The uniformity of low-dimensional projections is a fundamental considerationin the construction of experimental designs, particularly in screeningexperiments where the effect hierarchy principle dictates that lower-ordereffects dominate. This paper provides comprehensive benchmarks for assessingthis uniformity in mixed two- and three-level balanced designs—a class ofdesigns ubiquitous in industrial, pharmaceutical, and computer experiments.We investigate the projection weighted symmetric discrepancy (PWSDisc) asa uniformity criterion that explicitly prioritizes low-dimensional projections.A closed-form analytical expression for PWSDisc is derived in terms ofelementary coincidence counts, revealing its intrinsic combinatorial structure.Building on this representation, we establish a sharp, easily computable lowerbound that serves as a gold standard for design optimality. A catalog of lowerbounds for practical parameter combinations is provided, offering immediatebenchmarks for practitioners. Furthermore, an updated threshold-acceptingalgorithm incorporating these bounds as a stopping rule is presented, enablingefficient construction of uniform mixed two- and three-level designs. Theresults fill a critical gap in the theory of projection-based uniformity and havedirect applications in computer experiments, robust parameter design, and beyond.