Pseudo-Orthogonal Arrays: Definition, Construction, Comparison, and Application in Designs of Experiments

Inspired by orthogonal arrays OA(n,k,s,t), arrays whose inner product between any two column vectors being 0 are discussed. Such arrays can be divided into two categories: One can be derived from orthogonal arrays OA(n,k,s,t) with strength t≥2; the other have random experiment points spreading in the experimental space, thus are more flexible but may not be balanced regarding the level combinations, and we denote them by pseudo-orthogonal arrays (pOA). Orthogonal Latin hypercubes can be treated as a class of pOAs. Introduced by Ye (1998), it take advantage of both Latin hypercube designs and property of orthogonality. Similar to the idea of OA, we can also generalize the Orthogonal Latin hypercube to fixed-level and mixed-level pOAs. With a more flexible structure, pseudo-orthogonal arrays have relatively higher factor-to-run ratios. The existence conditions and construction methods for fixed-level and mixed-level pseudo-orthogonal arrays are discussed and proposed. The relationship between the proposed criteria for construction algorithm and other optimality criteria is discussed. Further, when a pseudo-orthogonal array cannot be found, an array with weak correlations between different factors are constructed to approximate the pseudo-orthogonal array. How and why the correlations should be controlled are discussed. We also give some examples of the application of pseudo-orthogonal arrays and their approximation in designs of experiments.