Orthogonal Latin Hypercube Designs Research Articles

Construction of orthogonal-MaxPro Latin hypercube designs

Orthogonal Latin hypercube designs (LHDs) and maximum projection (MaxPro) LHDs are widely used in computer experiments. They are efficient for estimating the trend part and the Gaussian process part of the universal Kriging (i.e., the Gaussian process) model, respectively, especially when only some of the factors are active. Yet, the orthogonality and the MaxPro criteria often do not agree with each other. In this work, we propose a new class of optimal designs, called orthogonal-MaxPro LHDs, optimizing a well-defined multi-objective criterion combining the correlation and the MaxPro metrics. An efficient parallel algorithm via level permutations and expansions is developed, whose efficiency is guaranteed by theories. Numerical results are presented to show that the construction is fast and the obtained designs are attractive, especially for large computer experiments.

On Construction of Nearly Orthogonal Latin Hypercube Designs

Orthogonal Latin hypercube designs are becoming popular in designing computer experiments. Available literature on construction of orthogonal or nearly orthogonal LHDs has one or more restriction in terms of either runs or factors. In this article, we have proposed a method of construction for obtaining nearly orthogonal Latin hypercube designs capable of accommodating flexible number of runs or factors.

Orthogonal Latin Hypercube designs for Eight Factors

Orthogonal Latin Hypercube designs for Eight Factors

Construction of Column-Orthogonal Designs with Two-Dimensional Stratifications

For the design of computer experiments, column orthogonality and space-filling are two desirable properties. In this paper, we develop methods for constructing a new class of column-orthogonal designs (ODs) with two-dimensional stratifications on finer grids, including orthogonal Latin hypercube designs (OLHDs) as special cases. In addition to being column-orthogonal, these designs have good space-filling properties in two dimensions. The resulting designs achieve stratifications on s2×s or s×s2 grids, and most column pairs satisfy stratifications on s2×s2 grids. Moreover, many column pairs can achieve stratifications on s4×s2 and s2×s4 grids. Furthermore, the obtained space-filling ODs can have s6 levels, s4 levels, and mixed levels, as required for different needs.

A new rotation method for constructing orthogonal Latin hypercube designs

A new rotation method for constructing orthogonal Latin hypercube designs

Orthogonal designs with branching and nested factors

Computer experiments with branching and nested factors are a common class of computer experiments, but it is challenging to construct designs for this type of experiments. In this paper, we define a special type of design called branching orthogonal Latin hypercube design (BOLHD). Such a design has an appealing structure, that is, no matter at each level of a branching factor or the level‐combination of branching factors, the corresponding design points of nested factors form an orthogonal Latin hypercube design (OLHD). This structure makes it a good choice for designing computer experiments with branching and nested factors. We propose several deterministic construction methods when branching factors have the same number of levels. Based on sliced Latin hypercube designs (SLHDs), the proposed methods are easy to operate. Some construction results are tabulated for practical use.

Construction of space-filling orthogonal Latin hypercube designs

Latin hypercube designs (LHDs) play an important role in computer experiments, because they can achieve the maximum stratification when projected onto any one dimension. The orthogonal LHD (OLHD), as a special kind of LHDs, has been widely studied and used. OLHDs ensure that the estimates of main effects in linear models are uncorrelated. Also, it is crucial to use a design with good stratifications in order to explore the experimental region efficiently and build a high-quality metamodel. This paper first proposes a method to construct OLHDs with sd runs and d⌊(sd−1)/(d(s−1))⌋/2 factors, which achieve a stratification on an s2×s or s×s2 grid when projected onto any two dimensions. Moreover, most column pairs achieve stratifications on s2×s2 grids. Another method is further provided to construct OLHDs with more factors, which achieve the aforementioned stratifications in each sub-design. The resulting OLHDs are more space-filling than existing OLHDs.

Column-orthogonal designs with multi-dimensional stratifications

The orthogonal Latin hypercube design and its relaxation, and column-orthogonal design, are two kinds of orthogonal designs for computer experiments. However, they usually do not achieve maximum stratifications in multi-dimensional margins. In this paper, we propose some methods to construct column-orthogonal designs with multi-dimensional stratifications by rotating symmetric and asymmetric orthogonal arrays. The newly constructed column-orthogonal designs ensure that the estimates of all linear effects are uncorrelated with each other and even uncorrelated with the estimates of all second-order effects (quadratic effects and bilinear effects) when the rotated orthogonal arrays have strength larger than two. Besides orthogonality, the resulting designs also preserve better space-filling property than those constructed by using the existing methods. In addition, we provide a method to construct a new class of orthogonal Latin hypercube designs with multi-dimensional stratifications by rotating regular factorial designs. Some newly constructed orthogonal Latin hypercube designs are tabulated for practical use.

LowCon: A Design-based Subsampling Approach in a Misspecified Linear Model

We consider a measurement constrained supervised learning problem, that is, (i) full sample of the predictors are given; (ii) the response observations are unavailable and expensive to measure. Thus, it is ideal to select a subsample of predictor observations, measure the corresponding responses, and then fit the supervised learning model on the subsample of the predictors and responses. However, model fitting is a trial and error process, and a postulated model for the data could be misspecified. Our empirical studies demonstrate that most of the existing subsampling methods have unsatisfactory performances when the models are misspecified. In this paper, we develop a novel subsampling method, called “LowCon,” which outperforms the competing methods when the working linear model is misspecified. Our method uses orthogonal Latin hypercube designs to achieve a robust estimation. We show that the proposed design-based estimator approximately minimizes the so-called worst-case bias with respect to many possible misspecification terms. Both the simulated and real-data analyses demonstrate the proposed estimator is more robust than several subsample least-squares estimators obtained by state-of-the-art subsampling methods. Supplementary materials for this article are available online.

On maximin distance and nearly orthogonal Latin hypercube designs

Maximin distance Latin hypercube designs (LHDs) are frequently used in computer experiments, but their constructions are challenging. In this paper, we present some new results connecting maximin L2-distance optimality and near orthogonality for mirror-symmetric LHDs. We further propose a simple and effective method for constructing nearly orthogonal LHDs that can yield almost the largest minimum distance. The obtained designs with small and medium sizes are tabulated and their superior performances are illustrated via comparisons.

On the construction of nested orthogonal Latin hypercube designs

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.

Generalized good lattice point sets

Good lattice point (GLP) sets are frequently used in quasi-Monte Carlo method and computer experiments. However, the space-filling property of GLP sets needs to be improved especially when the number of factors is large. This paper shows that the generalized GLP (GGLP) sets, constructed simply and fast by the linear level permutation of the GLP sets, have better space-filling property than the GLP sets and the orthogonal Latin hypercube designs (OLHD) in the sense of maximin distance criterion and uniformity criterion, especially for high dimensional cases. Unlike the OLHD, the number of runs of the GGLP sets can be chosen as any integer. It is also shown that the GLP sets are better than Latin hypercube designs as the starting design for linear and nonlinear level permutation. The GGLP sets are recommended for the designs with large number of factors and/or large number of runs.

On second order orthogonal Latin hypercube designs

In polynomial regression modeling, the use of a second order orthogonal Latin hypercube design guarantees that the estimates of the first order effects are uncorrelated to each other as well as to the estimates of the second order effects. In this paper, we prove that such designs with n runs and k>2 columns do not exist if n≡4mod8. Furthermore, we prove that second order Latin hypercube designs with k≥4 columns that guarantee the orthogonality of two-factor interactions which do not share a common factor, do not exist for even n that is not a multiple of 16. Finally, we investigate the class of symmetric orthogonal Latin hypercube designs (SOLHD), which are a special subset of second order orthogonal Latin hypercube designs. We describe construction techniques for SOLHDs with n runs and (a) k≤4 columns, when n≡0mod8 or n≡1mod8, (b) k≤8 columns when n≡0mod16 or n≡1mod16 and (c) k=4 columns when n≡0mod16 or n≡1mod16, that guarantee the orthogonality of two-factor interactions which do not share a common factor. Finally, we construct and enumerate all non-isomorphic SOLHDs with n≤17 runs and k≥2 columns, as well as with 19≤n≤20 runs and k=2 columns.

Orthogonal Latin hypercube designs with special reference to four factors

Latin hypercube designs are popular now-a-days for computer experiments. We give construction methods of orthogonal Latin hypercube designs for four factors for any number of runs for which such a design exists by combining smaller orthogonal matrices. We also propose methods for obtaining orthogonal Latin hypercube designs with larger number of factors.

A Method of Constructing Orthogonal Array-Based Latin Hypercube Designs for Computer Experiments

A Method of Constructing Orthogonal Array-Based Latin Hypercube Designs for Computer Experiments

Sliced Orthogonal Array-Based Latin Hypercube Designs

We propose an approach for constructing a new type of design, called a sliced orthogonal array-based Latin hypercube design. This approach exploits a slicing structure of orthogonal arrays with strength two and makes use of sliced random permutations. Such a design achieves one- and two-dimensional uniformity and can be divided into smaller Latin hypercube designs with one-dimensional uniformity. Sampling properties of the proposed designs are derived. Examples are given for illustrating the construction method and corroborating the derived theoretical results. Potential applications of the constructed designs include uncertainty quantification of computer models, computer models with qualitative and quantitative factors, cross-validation and efficient allocation of computing resources. Supplementary materials for this article are available online.

Samurai Sudoku-Based Space-Filling Designs for Data Pooling

Pooling data from multiple sources plays an increasingly vital role in today’s world. By using a popular Sudoku game, we propose a new type of design, called a Samurai Sudoku-based space-filling design to address this issue. Such a design is an orthogonal array-based Latin hypercube design with the following attractive properties: (i) the complete design achieves uniformity in both univariate and bivariate margins; (ii) it can be divided into groups of subdesigns with overlaps such that each subdesign achieves uniformity in both univariate and bivariate margins; and (iii) each of the overlaps achieves uniformity in both univariate and bivariate margins. Examples are given to illustrate the properties of the proposed design, and to demonstrate the advantages of using the proposed design for pooling data from multiple sources.[Received August 2013. Revised July 2015.]

A new class of nested(nearly) orthogonal Latin hypercube designs

A new class of nested(nearly) orthogonal Latin hypercube designs

Enumeration and Evaluation of Small Orthogonal Latin Hypercube Designs for Polynomial Regression Models

In this paper, we construct and evaluate all nonisomorphic Latin hypercube designs with n≤16 runs, the use of which guarantee that the estimates of the first‐order effects are uncorrelated with each other and also uncorrelated with the estimates of the second‐order effects, in polynomial regression models. The produced designs are evaluated using well‐known and popular criteria, and optimal designs are presented in every case studied. An effort to construct nonisomorphic small Latin hypercubes in which only the estimates of the first‐order effects are required to be uncorrelated with each other has also been made, and new designs are presented. All the constructed designs, besides their stand‐alone properties, are useful for the construction of bigger orthogonal Latin hypercubes with desirable properties, using well‐known techniques proposed in the literature. Copyright © 2015 John Wiley & Sons, Ltd.

Nearly orthogonal latin hypercube designs for many design columns

Latin hypercube designs (LHDs) have found wide applications in com- puter experiments. Some methods have been proposed to construct orthogonal (or nearly orthogonal) LHDs. This paper proposes methods for expanding a fold-over orthogonal (or nearly orthogonal) LHD to a nearly orthogonal LHD which is able to accommodate many factors. The number of factors is flexible and can be almost as twice large as the number of factors of the original LHD, while the run size remains unchanged. It is shown that the upper bound of the maximum correlation between any two distinct columns of the resulting design is very small (smaller than 0.10 for most cases). The proposed methods can be applied to any fold-over LHDs.