Construction of orthogonal and nearly orthogonal latin hypercube designs from orthogonal designs

Abstract

Latin hypercube designs (LHDs), widely used for computer experiments, are a very large class of designs with desirable properties. Recently, a number of methods have been proposed to construct orthogonal LHDs. In this paper, we introduce an approach to constructing 2 r-order orthogonal designs. The methods are simple and easy to implement. Using orthogonal designs, we propose some methods for constructing orthogonal and nearly orthogonal LHDs so that the elementwise square of each column and the elementwise product of any two distinct columns are orthogonal to all columns. Further, the resulting nearly orthogonal LHDs with 2 r+1 +2 runs and 2 r factors have the minimum correlation between any two distinct columns.

Nearly orthogonal latin hypercube designs for many design columns

“…The condition that S X is column-orthogonal is rather common for an orthogonal or nearly orthogonal LHD X. Most of the fold-over orthogonal LHDs with even run sizes constructed by, for example, Ye (1998), Lin (2009, 2010), Georgiou (2009), Georgiou and Stylianou (2011), Yang and Liu (2012), and Georgiou and Efthimiou (2014) satisfy this condition. Even if S X is not column-orthogonal, the value ρ ij (S X ) in (2.3) is usually quite small.…”

“…For first-order polynomial models, orthogonal LHDs are useful because they ensure the estimates of linear effects are uncorrelated. Construction of orthogonal LHDs has been widely studied, see e.g., Ye (1998), Steinberg and Lin (2006), Cioppa and Lucas (2007), Bingham, Sitter, and Tang (2009), Pang, Liu, and Lin (2009), Georgiou (2009), Lin, Mukerjee, and Tang (2009), Lin (2009, 2010), Lin et al (2010), Sun, Pang, and Liu (2011), Georgiou and Stylianou (2011) (and its corrigendum Georgiou and Stylianou (2012)), Yang and Liu (2012), and Georgiou and Efthimiou (2014), among others. Note that some of them considered LHDs with fold-over structures, which makes sure that the sum of elementwise product of any three columns is zero.…”

“…However, the resulting LHDs of these methods usually have severe restrictions on the design dimensions. Georgiou and Stylianou (2011), Yang and Liu (2012), and Efthimiou, Georgiou, and Liu (2014) constructed nearly orthogonal LHDs by adding runs to existing LHDs. Inevitably, these methods increase experimental costs.…”

Construction of orthogonal symmetric Latin hypercube designs

“…In addition, with T 2 and T 4 in (3.1), and the OSLHD(5, 2) and OSLHD(17, 8) constructed by Yang and Liu (2012), the proposed construction yields saturated OSLHD(25, 12), OSLHD(625, 312) and OSLHD(289, 144). As another example, we could have the OSLHD(11, 3) and OSLHD(13, 3) shown in Table 3, which are obtained by computer search and are apparently new.…”

A general construction for space-filling Latin hypercubes

“…To achieve multi-dimensional space-filling property, various optimality criteria have been proposed. These include maximin distance criterion (Morris and Mitchell (1995)), multi-dimensional projection (Tang (1993); Moon, Dean, and Santner (2011)), orthogonality and near orthogonality (see, for example, Sun, Liu, and Lin (2009) ;Yang and Liu (2012); Georgiou and Efthimiou (2014)), and the discrepancy criterion (Fang et al (2000)). Attempts have also been made to seek designs based on multiple optimality criteria.…”