Uniform projection nested Latin hypercube designs

Abstract

Computer experiments usually involve many factors, but only a few of them are active. In such a case, it is desirable to construct designs with good projection properties. Maximum projection designs and uniform projection designs have been developed for common experimental situations, however, there has been little study on constructing projection designs for high-accuracy computer experiments (HEs) and low-accuracy computer experiments (LEs) so far. In this paper, we propose a weighted uniform projection criterion, and construct uniform projection nested Latin hypercube designs to suit such computer experiment situations. We show that the obtained designs have good projection properties in all sub-dimensions, and we also discuss how to choose a proper value for the weight. Simulated examples are available to illustrate the effectiveness of the proposed designs.

Appendices

Appendix

Proof of Theorem 1

An LHD $D=L(n,q)$ is a balanced design, thus it satisfies the condition of Theorem 3 in Sun et al. (2019). Note that the LB of $\phi (D)$ in Sun et al. (2019) is obtained based on squared centered $L_2$-discrepancy, denoted by CD(D), then $CD_2(D)=\sqrt{CD(D)}$. For the criterion (1), we have

$$\begin{aligned} \phi (D)= & {} \frac{2}{q(q-1)}\sum _{|u|=2}\sqrt{CD(D_u)}\\\ge & {} \sqrt{\frac{2}{q(q-1)}\sum _{|u|=2}CD(D_u)}\\\ge & {} \sqrt{LB}. \end{aligned}$$

This completes the proof of Theorem 1.

To prove Theorem 2, we need the conclusion of the Theorem 1 in Sun et al. (2019). For more details, please refer to the paper.

Proof of Theorem 2

From the proof of Theorem 1 in Sun et al. (2019), we have

$$\begin{aligned} \frac{1}{\left( {\begin{array}{c}q\\ r\end{array}}\right) }\sum _{|u|=r}\phi (D_u)=\phi (D) \end{aligned}$$

(12)

for any $2 \le r \le q$, even if there is no balanced condition for the design D. Thus for both $D_h$ and $D_l$ in an UPNLHD, Eq. (12) holds as well. According to the definition of criterion in (4), we have

$$\begin{aligned} \frac{1}{\left( {\begin{array}{c}q\\ r\end{array}}\right) }\sum _{|u|=r}\Phi (D_u)= & {} \frac{1}{\left( {\begin{array}{c}q\\ r\end{array}}\right) }\sum _{|u|=r} \left( \alpha \phi (D_{h,u})+(1-\alpha )\phi (D_{l,u})\right) \\= & {} \alpha \cdot \frac{1}{\left( {\begin{array}{c}q\\ r\end{array}}\right) } \sum _{|u|=k}\phi (D_{h,u})+(1-\alpha ) \cdot \frac{1}{\left( {\begin{array}{c}q\\ r\end{array}}\right) } \sum _{|u|=r}\phi (D_{l,u}) \\= & {} \alpha \phi (D_h) +(1-\alpha ) \phi (D_l) \\= & {} \Phi (D), \end{aligned}$$

where $D_{h,u}$ is the projected deign of $D_h$ onto r-dimensional space indexed by u for any $2\le r \le q$, and $D_{l,u}$ is similarly defined. This completes the proof of Theorem 2.