Construction of Uniform Designs on Arbitrary Domains by Inverse Rosenblatt Transformation

Abstract

The uniform design proposed by Fang [6] and Wang and Fang [17] has become an important class of designs for both traditional industrial experiments and modern computer experiments. There exist established theory and methods for constructing uniform designs on hypercube domains, while the uniform design construction on arbitrary domains remains a challenging problem. In this paper, we propose a deterministic construction method through inverse Rosenblatt transformation, as a general approach to convert the uniformly designed points from the unit hypercubes to arbitrary domains. To evaluate the constructed designs, we employ the central composite discrepancy as a uniformity measure suitable for irregular domains. The proposed method is demonstrated with a class of flexible regions, constrained and manifold domains, and the geographical domain with very irregular boundary. The new construction results are shown competitive to traditional stochastic representation and acceptance-rejection methods.

Appendix: Good Lattice Point Method

The uniform designs constructed on the unit hypercubes by the GLP method are also known as the NT-nets [9], which uses the classical star-discrepancy for evaluating the uniformity of the candidate designs. In Algorithm 4 we write the GLP method using the centered-$\ell _2$ discrepancy (CD2), a more popular criterion proposed by Hickernell [11]. Meanwhile, it is easy to check that the GLP designs (7.15) always include a point ${\varvec{x}}_n=(1-1/2n, \ldots , 1-1/2n )\in C^s$. The leave-one-out (LOO) GLP method is to remove such a dummy point, then scale the remaining points by $n/(n-1)$ in all coordinates. Thus, in order to construct an n-run uniform design, we can use a mixed GLP method by selecting the lower-CD2 design between the GLP (with input n) and LOO-GLP (with input $n+1$) outputs.

Fig. 7.8

Scatter plots of LOO-Fibonacci designs for $n=20, 88$ and 143 runs

Table 7.3 LOO-Fibonacci designs with $h_1=1$ and $h_2=F_{k-1}$ minimize the CD2 criterion for the mixed GLP method, where $h_2^*$ represents the best found generating vectors per each method

It is well-known that for $s=2$ and $n=F_k$ (Fibonacci numbers 5, 8, 13, 21, ...), the lattice designs generated by $h_1 = 1$ and $h_2 =F_{k-1}$ enjoy the remarkable low star-discrepancy property [18]. It is of our interest to investigate whether such Fibonacci designs may also attain low discrepancy with respect to the CD2 criterion. As a key difference, the star-discrepancy is anchored at the origin of the unit hypercube, while the CD2 is anchored at the center. It turns out the Fibonacci designs are sub-optimal under CD2. Nevertheless, we find that the LOO-Fibonacci designs with $n=F_k-1$ ($F_k\le 1597$) runs remarkably minimize the CD2 criterion among all the generating vectors for the mixed GLP method. See Fig. 7.8 about the LOO-Fibonacci designs with 20, 88 and 143 runs. See Table 7.3 for the numerical results based on exhaustive search up to $F_k=1597$. From Table 7.3, it can be found that the LOO-Fibonacci designs with $n=F_k-1$ also include $h_1=1$ and $h_2=F_{k-2}$ as the optimal generating vector. This can be actually justified by the reflection-invariant property of the CD2 criterion.