On the construction of nested orthogonal Latin hypercube designs

Abstract

Latin hypercube designs are widely used in designing computer experiments. In recent years, nested orthogonal Latin hypercube designs have been proposed in the literature. In this article, two general methods of constructing nested orthogonal Latin hypercube designs have been developed. The methods give many new nested orthogonal Latin hypercube designs with fewer number of runs as compared to existing nested orthogonal Latin hypercube designs.

Appendix: Proof of Theorem 2

First we prove one dimensional uniformity of the constructed design for each layer. One dimensional uniformity of pth layer is obvious. Consider $(p-1)$th layer $L_{p-1}=(2^{p-1}L_1^{*T},2^{p-1}L_2^{*T})$. Since $n = 0 \text { mod } 4$, let $n=2^2t$ where t is a positive integer. Then, each column of $L_{p-1}$ contains elements belonging to set $\{-(2^{p}t-2^{p-2}), -(2^{p}t-3 \times 2^{p-2}),\ldots ,(2^{p}t-2^{p-2}), (2^{p}t-3 \times 2^{p-2}), -2^{p}t, -(2^{p}t-2 \times 2^{p-2}),\ldots ,2^{p}t, (2^{p}t-2 \times 2^{p-2})\}=\{-2^{p}t, -(2^{p}t-2^{p-2}), -(2^{p}t-2 \times 2^{p-2}),\ldots ,(2^{p}t-2 \times 2^{p-2}), (2^{p}t-2^{p-2}),2^{p}t\}$. Clearly, the elements are equally spaced with spacing $2^{p-2}$. Consider $(p-2)$th layer $L_{p-2}$. It may be verified easily that elements of each column of $L_{p-2}$ are in the set $\{-2^{p}t, -(2^{p}t-2^{p-3}), -(2^{p}t-2 \times 2^{p-3}),\ldots ,(2^{p}t-2 \times 2^{p-3}), (2^{p}t-2^{p-3}),2^{p}t\}$, the spacing between the elements being $2^{p-3}$. On similar lines, elements of each column of ith layer $L_i, i=1,2,\ldots ,p-1$ are in the set $\{-2^{p}t, -(2^{p}t-2^{i-1}), -(2^{p}t-2 \times 2^{i-1}),\ldots ,(2^{p}t-2 \times 2^{i-1}), (2^{p}t-2^{i-1}),2^{p}t\}$ with spacing $2^{i-1}$ between the elements. Proof of orthogonality is straightforward extension of the proof of orthogonality given in Theorem 1. Also, $L_p \subset L_{p-1} \subset \ldots \subset L_2\subset L_1$. This completes the proof.