A new non-iterative deterministic algorithm for constructing asymptotically orthogonal maximin distance Latin hypercube designs

Latin hypercube designs (LHDs), maximin distance designs (MDDs) and orthogonal designs (ODs) are becoming popular and preferred choices in many areas of scientific investigation. A LHD has good projective properties on any single dimension for its uniform coverage of each individual factor, but it does not guarantee good space-filling properties in higher dimensions. A MDD maximizes the distances between its points and thus achieves the space-filling property in the full-dimensional space, but it does not guarantee the orthogonality of its factors. ODs are useful because they ensure the estimates of linear effects are uncorrelated. Since each of these three designs has pros and cons from different perspectives, a design that combines their benefits will be far superior to each design on its own. This paper gives a new non-iterative deterministic algorithm for constructing asymptotically orthogonal maximin distance LHDs. Compared with the existing results, the newly constructed LHDs have a much better performance in any dimension. Theoretical and numerical justifications for the optimality behavior of the newly constructed LHDs are given. Moreover, an iterative algorithm utilizing a mixture of criteria is provided for further improvement of the performance of the newly constructed LHDs from multiple perspectives.