Orthogonal Latin hypercube designs from generalized orthogonal designs

Latin hypercube designs is a class of experimental designs that is important when computer simulations are needed to study a physical process. In this paper, we proposed some general criteria for evaluating Latin hypercube designs through their alias matrices. Moreover, a general method is proposed for constructing orthogonal Latin hypercube designs. In particular, links between orthogonal designs (ODs), generalized orthogonal designs (GODs) and orthogonal Latin hypercube designs are established. The generated Latin hypercube designs have some favorable properties such as uniformity, orthogonality of the first and some second order terms, and optimality under the defined criteria.     

Some new classes of orthogonal Latin hypercube designs

Orthogonal Latin hypercube (OLH) is a good design choice in a polynomial function model for computer experiments, because it ensures uncorrelated estimation of linear effects when a first-order model is fitted. However, when a second-order model is adopted, an OLH also needs to satisfy the additional property that each column is orthogonal to the elementwise square of all columns and orthogonal to the elementwise product of every pair of columns. Such class of OLHs is called OLHs of order two while the former class just possessing two-dimensional orthogonality is called OLHs of order one. In this paper we present a general method for constructing OLHs of orders one and two for n=s^m$n=s^m$ runs, where s and m may be any positive integers greater than one, by rotating a grouped orthogonal array with a column-orthogonal rotation matrix. The Kronecker product and the stacking methods are revisited and combined to construct some new classes of OLHs of orders one and two with other flexible numbers of runs. Some useful OLHs of order one or two with larger factor-to-run ratio and moderate runs are tabulated and discussed.     

Orthogonal Latin hypercube designs for Fourier-polynomial models

Latin hypercube designs (LHDs) are widely used in computer experiments because of their one-dimensional uniformity and other properties. Recently, a number of methods have been proposed to construct LHDs with properties that all linear effects are mutually orthogonal and orthogonal to all second-order effects, i.e., quadratic effects and bilinear interactions. This paper focuses on the construction of LHDs with the above desirable properties under the Fourier-polynomial model. A convenient and flexible algorithm for constructing such orthogonal LHDs is provided. Most of the resulting designs have different run sizes from that of Butler (2001), and thus are new and very suitable for factor screening and building Fourier-polynomial models in computer experiments as discussed in Butler (2001).     

Maximum expected entropy transformed Latin hypercube designs

Existing projection designs (e.g. maximum projection designs) attempt to achieve good space-filling properties in all projections. However, when using a Gaussian process (GP), model-based design criteria such as the entropy criterion is more appropriate. We employ the entropy criterion averaged over a set of projections, called expected entropy criterion (EEC), to generate projection designs. We show that maximum EEC designs are invariant to monotonic transformations of the response, i.e. they are optimal for a wide class of stochastic process models. We also demonstrate that transformation of each column of a Latin hypercube design (LHD) based on a monotonic function can substantially improve the EEC. Two types of input transformations are considered: a quantile function of a symmetric Beta distribution chosen to optimize the EEC, and a nonparametric transformation corresponding to the quantile function of a symmetric density chosen to optimize the EEC. Numerical studies show that the proposed transformations of the LHD are efficient and effective for building robust maximum EEC designs. These designs give projections with markedly higher entropies and lower maximum prediction variances (MPV''s) at the cost of small increases in average prediction variances (APV''s) compared to state-of-the-art space-filling designs over wide ranges of covariance parameter values.     

Optimizing Latin hypercube designs by particle swarm

Latin hypercube designs (LHDs) are widely used in many applications. As the number of design points or factors becomes large, the total number of LHDs grows exponentially. The large number of feasible designs makes the search for optimal LHDs a difficult discrete optimization problem. To tackle this problem, we propose a new population-based algorithm named LaPSO that is adapted from the standard particle swarm optimization (PSO) and customized for LHD. Moreover, we accelerate LaPSO via a graphic processing unit (GPU). According to extensive comparisons, the proposed LaPSO is more stable than existing approaches and is capable of improving known results.     

Block-circulant matrices for constructing optimal Latin hypercube designs

Computer simulations are usually needed to study a complex physical process. In this paper, we propose new procedures for constructing orthogonal or low-correlation block-circulant Latin hypercube designs. The basic concept of these methods is to use vectors with a constant periodic autocorrelation function to obtain suitable block-circulant Latin hypercube designs. A general procedure for constructing orthogonal Latin hypercube designs with favorable properties and allowing run sizes being different from a power of 2 (or a power of 2 plus 1), is presented here for the first time. In addition, an expansion of the method is given for constructing Latin hypercube designs with low correlation. This expansion is useful when orthogonal Latin hypercube designs do not exist. The properties of the generated designs are further investigated. Some examples of the new designs, as generated by the proposed procedures, are tabulated. In addition, a brief comparison with the designs that appear in the literature is given.     

Latin hypercube sampling with multidimensional uniformity

Complex models can only be realized a limited number of times due to large computational requirements. Methods exist for generating input parameters for model realizations including Monte Carlo simulation (MCS) and Latin hypercube sampling (LHS). Recent algorithms such as maximinLHS seek to maximize the minimum distance between model inputs in the multivariate space. A novel extension of Latin hypercube sampling (LHSMDU) for multivariate models is developed here that increases the multidimensional uniformity of the input parameters through sequential realization elimination. Correlations are considered in the LHSMDU sampling matrix using a Cholesky decomposition of the correlation matrix. Computer code implementing the proposed algorithm supplements this article. A simulation study comparing MCS, LHS, maximinLHS and LHSMDU demonstrates that increased multidimensional uniformity can significantly improve realization efficiency and that LHSMDU is effective for large multivariate problems.     

Latin hypercube sampling with inequality constraints

In some studies requiring predictive and CPU-time consuming numerical models, the sampling design of the model input variables has to be chosen with caution. For this purpose, Latin hypercube sampling has a long history and has shown its robustness capabilities. In this paper we propose and discuss a new algorithm to build a Latin hypercube sample (LHS) taking into account inequality constraints between the sampled variables. This technique, called constrained Latin hypercube sampling (cLHS), consists in doing permutations on an initial LHS to honor the desired monotonic constraints. The relevance of this approach is shown on a real example concerning the numerical welding simulation, where the inequality constraints are caused by the physical decreasing of some material properties in function of the temperature.     

Construction of effect-wise orthogonal factorial designs

Generalizing the concept of Kronecker products of designs, two distinct methods have been suggested for the construction of effect-wise orthogonal factorial designs. The methods described ensure desirable properties with respect to main effects, cover almost all cases of factorial designs and require, in most cases, a smaller number of replications than any of the existing methods.     

Construction of orthogonal factorial designs controlling interaction efficiencies

This paper employs some variants of the usual Kronecker product to construct orthogonal factorial designs controlling the interaction efficiencies. The methods suggested have a fairly wide coverage and the resulting designs involve a small number of replicates.     

Mixture designs with orthogonal blocks

Constructions of blocked mixture designs are considered in situations where BLUEs of the block effect contrasts are orthogonal to the BLUEs of the regression coefficients. Orthogonal arrays (OA), Balanced Arrays (BAs), incidence matrices of balanced incomplete block designs (BIBDs), and partially balanced incomplete block designs (PBIBDs) are used. Designs with equal and unequal block sizes are considered. Also both cases where the constants involved in the orthogonality conditions depend and do not depend on the factors have been taken into account. Some standard (already available) designs can be obtained as particular cases of the designs proposed here.     

Construction of central composite designs for balanced orthogonal blocks

Box & Hunter (1957) recommended a set of orthogonally blocked central composite designs (CCD) when the region of interest is spherical. In order to achieve rotatability along with orthogonal blocking, the block size for those designs becomes unequal and it may not be attractive or practical to use such unequally blocked designs in many practical situations. In this paper, a construction method of orthogonally blocked CCD under the assumption of equal block size is proposed and an index of block orthogonality is introduced.     

Repeated measurements designs with unequal period sizes

Summary This paper is concerned with the designs in which each experimental unit is assigned more than once to a treatment, either different or identical. An easy method of constructing balanced minimal repeated measurements designs with unequal period sizes is presented whenever the number of periods is less than the number of treatments. Strongly balanced minimal repeated measurements designs with unequal period sizes are also constructed whenever the number of periods is less than the number of treatments.     

A fast general extension algorithm of Latin hypercube sampling

A fast general extension algorithm of Latin hypercube sampling (LHS) is proposed, which reduces the time consumption of basic general extension and preserves the most original sampling points. The extension algorithm starts with an original LHS of size m and constructs a new LHS of size m?+?n that remains the original points. This algorithm is the further research of basic general extension, which cost too much time to get the new LHS. During selecting the original sampling points to preserve, time consumption is cut from three aspects. The first measure of the proposed algorithm is to select isolated vertices and divide the adjacent matrix into blocks. Secondly, the relationship of original LHS structure and new LHS structure is discussed. Thirdly, the upper and lower bounds help reduce the time consumption. The proposed algorithm is applied for two functions to demonstrate the effectiveness.     

E-optimal incomplete block designs with two distinct block sizes

Sufficient conditions are derived for the determination of E-optimal designs in the class D(v,b₁,b₂,k₁,k₂) of incomplete block designs for v treatments in b₁ blocks of size k₁ each and b₂ blocks of size k₂ each. Some constructions for E-optimal designs that satisfy the sufficient conditions obtained here are given. In particular, it is shown that E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by augmenting b₂ blocks, with k₂ − k₁ extra plots each, of a BIBD(v,b = b₁ + b₂,k₁,λ) and GDD(v,b = b₁ + b₂,k₁,λ₁,λ₂). It is also shown that equireplicate E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by combining disjoint blocks of BIBD(v,b,k₁,λ) and GDD(v,b,k₁,λ₁,λ₂) into larger blocks. As applications of the construction techniques, several infinite series of E-optimal designs with small block sizes differing by at most two are given. Lower bounds for the A-efficiency are derived and it is found that A-efficiency exceeds 99% for v ⩾ 10, and at least 97.5% for 5 ⩽v < 10.     

Orthogonal arrays of strength 3 and small run sizes

All mixed (or asymmetric) orthogonal arrays of strength 3 with run size at most 64 are determined.     

A class of asymmetrical orthogonal resolution-IV designs

The concept of d-resolvability of orthogonal arrays of strength (d+1) is introduced. This is used to construct orthogonal resolution-IV plans of the type $\text{nt·n·2}{}^{\mathrm{n(m?1)}}\text{mn}{}^2\text{t}$. These plans are minimal and a large number of these plans are new.     

A class of single replicate orthogonal designs

A simple method is given to calculate the number of degrees of freedom confounded with blocks of a specific factorial effect in a single replicate orthogonal design. Two classes of designs having partial orthogonality are also discussed     

On the orthogonal designs of order 40

Using a new method we construct all 17 remaining (unresolved for over 20 years) full orthogonal designs of order 40 in three variables. This implies that all full orthogonal designs OD(2^t5;x,y, 2^t5−x−y) exist for all t⩾3. The last two remaining orthogonal designs of order 40 in 2 variables are obtained as a special case of two of these designs.