Benchmarking Low-Dimensional Projection Uniformity in Mixed Two- and Three-Level Designs
Keywords:
Uniformity, Discrepancy, Lower bound, ProjectionAbstract
The uniformity of low-dimensional projections is a fundamental consideration in the construction of experimental designs, particularly in screening experiments where the effect hierarchy principle dictates that lower-order effects dominate. This paper provides comprehensive benchmarks for assessing this uniformity in mixed two- and three-level balanced designs—a class of designs ubiquitous in industrial, pharmaceutical, and computer experiments. We investigate the projection-weighted symmetric discrepancy (PWSDisc) as a uniformity criterion that explicitly prioritizes low-dimensional projections. A closed-form analytical expression for PWSDisc is derived in terms of elementary coincidence counts, revealing its intrinsic combinatorial structure. Building on this representation, we establish a sharp, easily computable lower bound that serves as a gold standard for design optimality. A catalog of lower bounds for practical parameter combinations is provided, offering immediate benchmarks for practitioners. Furthermore, an updated threshold-accepting algorithm incorporating these bounds as a stopping rule is presented, enabling efficient construction of uniform mixed two- and three-level designs. The results fill a critical gap in the theory of projection-based uniformity and have direct applications in computer experiments, robust parameter design, and beyond.
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