Construction of Uniform Designs and Complex-Structured Uniform Designs via Partitionable t-Designss

Abstract

This paper proposes several infinite classes of new uniform designs under the discrete discrepancy. The construction is based upon combinatorial configurations, namely partitionable t-designs, hence it dose not require any computer search. Moreover, certain complex structures within the proposed uniform designs are explored. It turns out that some of the uniform designs have nested or sliced subdesigns which are also uniform designs under the discrete discrepancy.The proposed uniform designs may find many applications in statistics including computer experiments, sequential experiments, drug combination studies and cross-validation.

Doubly Coupled Designs for Computer Experiments With Both Qualitative and Quantitative Factors

“…For recent work on computer experiments, refer to Chen et al (2018), , Xiao and Xu (2018), , Huang et al (2021), and reference therein. One prevailing way to select input settings for computer experiments is to use Latin hypercube designs (LHDs) proposed by McKay et al (1979), because of the desirable feature that when projected onto any factor, the resulting design points spread out uniformly and achieve the maximum stratification.…”

A method of constructing optimal sliced uniform designs under discrete discrepancy

“…In contrast, the maximum stratification criteria, the distance‐based criteria, and the orthogonality criteria have intuitive geometric or numeric interpretations. To the best of our knowledge, only the papers by Chen et al (2016) and Huang et al (2021) focused on systematic constructions for sliced uniform designs. Chen et al (2016) proposed to use a computer program called the threshold accepting algorithm (Fang et al, 2000) to search sliced uniform designs under the centered L 2 discrepancy criterion (the CL 2 discrepancy, for short).…”

“…This means that sliced uniform designs under the discrete discrepancy may also behave well under some other design criteria. Huang et al (2021) obtained a class of optimal sliced uniform designs in that not only the whole design but also each slice of such a design attain the lower bound of the discrete discrepancy (Fang et al, 2006). Their designs, however, are restricted to some specific design parameters; that is, the run sizes and the numbers of factors and levels are constraint to some specific values.…”

“…Due to this appealing property, sliced space‐filling designs are desirable for generating experimental data with batch structures. The existing literature mainly suggests four types of design criteria to measure the space‐filling property of a sliced design; they are as follows: (1) the maximum stratification criteria (Ai et al, 2014; Guo et al, 2020; Qian & Wu, 2009; Xu et al, 2011; Yang et al, 2014; Yin et al, 2014), (2) the distance‐based criteria (Ba et al, 2015; He, 2017; Li et al, 2021; Yang et al, 2016), (3) the orthogonality criteria (Cao & Liu, 2015; Huang et al, 2014; Yang et al, 2013; Wang et al, 2017), and (4) the uniformity criteria (Chen et al, 2016; Huang et al, 2021). Compared to the first three, the uniformity criteria usually yield designs with more economical run sizes and more flexible numbers of factors and levels (Fang et al., 2011,2018).…”

“…However, their sliced uniform designs are not optimal since the threshold accepting algorithm is often trapped in local pockets of the design space, which means that their designs may not have the smallest (optimal) CL 2 values. Huang et al (2021) proposed a combinatorial method for constructing sliced uniform designs under the discrete discrepancy criterion. The discrete discrepancy, first proposed by Hickernell and Liu (2002), is another commonly used uniformity criterion.…”