Optimal Latin hypercube designs for the Kullback–Leibler criterion

Space-filling designs are commonly used for selecting the input values of time-consuming computer codes. Computer experiment context implies two constraints on the design. First, the design points should be evenly spread throughout the experimental region. A space-filling criterion (for instance, the maximin distance) is used to build optimal designs. Second, the design should avoid replication when projecting the points onto a subset of input variables (non-collapsing). The Latin hypercube structure is often enforced to ensure good projective properties. In this paper, a space-filling criterion based on the Kullback–Leibler information is used to build a new class of Latin hypercube designs. The new designs are compared with several traditional optimal Latin hypercube designs and appear to perform well.     

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.     

Experimental designs and their efficiencies for spatially correlated observations in two dimensions

Optimality of experimental designs for spatially correlated observations is investigated.come two dimensional correlation structures are discussed and an attempt has been made to find optimal or nearly optimal design for each sitution.The solution lend to designs similar to that used for repeated measurements.The relative efficiency of the proposed designs in comparison to randomized latin square designs is tabulated for some cases.     

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.     

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.     

Nested Latin Hypercube Designs with Sliced Structures

In computer experiments, space-filling designs with a sliced structure or nested structure have received much recent interest and been studied separately. However, it is likely that designs with both structures are needed in some situations, but there are no suitable designs so far. In this paper, we construct a special class of nested Latin hypercube designs with sliced structures, in such a design, a small sliced Latin hypercube design is nested within a large one. The construction method is easy to implement and the number of factors is flexible. Numerical simulations show the usefulness of the newly proposed designs.     

Optimal minimax designs for prediction in heteroscedastic models

We construct optimal designs for heteroscedastic models when the goal is to make efficient prediction over a compact interval. It is assumed that the point or points which are interesting to predict are not known before the experiment is run. Two minimax strategies for minimizing the maximum fitted variance and maximum predictive variance across the interval of interest are proposed and, optimal designs are found and compared. An algorithm for generating these designs is included.     

An easy method of constructing latin square designs balanced for the immediate residual and other order effects

Bradley (1958) proposed a very simple procedure for constructing latin square designs to counterbalance the immediate sequential effect for an even number of treatments. When the number of treatments is odd, balance in a single latin square is not possible. In the present note we have developed an analogous method for the construction of such designs which may be used for an even or odd number of treatments. A proof has also been offered to assure the general validity of the procedure.     

Computer experiments: a review

In this paper we provide a broad introduction to the topic of computer experiments. We begin by briefly presenting a number of applications with different types of output or different goals. We then review modelling strategies, including the popular Gaussian process approach, as well as variations and modifications. Other strategies that are reviewed are based on polynomial regression, non-parametric regression and smoothing spline ANOVA. The issue of multi-level models, which combine simulators of different resolution in the same experiment, is also addressed. Special attention is given to modelling techniques that are suitable for functional data. To conclude the modelling section, we discuss calibration, validation and verification. We then review design strategies including Latin hypercube designs and space-filling designs and their adaptation to computer experiments. We comment on a number of special issues, such as designs for multi-level simulators, nested factors and determination of experiment size.     

Projection array based designs for computer experiments

In this paper we define a new class of designs for computer experiments. A projection array based design defines sets of simulation runs with properties that extend the conceptual properties of orthogonal array based Latin hypercube sampling, particularly to underlying design structures other than orthogonal arrays. Additionally, we illustrate how these designs can be sequentially augmented to improve the overall projection properties of the initial design or focus on interesting regions of the design space that need further exploration to improve the overall fit of the underlying response surface. We also illustrate how an initial Latin hypercube sample can be expressed as a projection array based design and show how one can augment these designs to improve higher dimensional space filling properties.     

Comparison of designs for computer experiments

For many complex processes laboratory experimentation is too expensive or too time-consuming to be carried out. A practical alternative is to simulate these phenomena by a computer code. This article considers the choice of an experimental design for computer experiments. We illustrate some drawbacks to criteria that have been proposed and suggest an alternative, based on the Bayesian interpretation of the alias matrix in Draper and Guttman (Ann. Inst. Statist. Math. 44 (1992) 659). Then we compare different design criteria by studying how they rate a variety of candidate designs for computer experiments such as Latin hypercube plans, U-designs, lattice designs and rotation designs.     

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.     

Design of experiments for locally weighted regression

We consider the design of experiments when estimation is to be performed using locally weighted regression methods. We adopt criteria that consider both estimation error (variance) and error resulting from model misspecification (bias). Working with continuous designs, we use the ideas developed in convex design theory to analyze properties of the corresponding optimal designs. Numerical procedures for constructing optimal designs are developed and applied to a variety of design scenarios in one and two dimensions. Among the interesting properties of the constructed designs are the following: (1) Design points tend to be more spread throughout the design space than in the classical case. (2) The optimal designs appear to be less model and criterion dependent than their classical counterparts.(3) While the optimal designs are relatively insensitive to the specification of the design space boundaries, the allocation of supporting points is strongly governed by the points of interest and the selected weight function, if the latter is concentrated in areas significantly smaller than the design region. Some singular and unstable situations occur in the case of saturated designs. The corresponding phenomenon is discussed using a univariate linear regression example.     

A clustering-based coordinate exchange algorithm for generating G-optimal experimental designs

In the optimal experimental design literature, the G-optimality is defined as minimizing the maximum prediction variance over the entire experimental design space. Although the G-optimality is a highly desirable property in many applications, there are few computer algorithms developed for constructing G-optimal designs. Some existing methods employ an exhaustive search over all candidate designs, which is time-consuming and inefficient. In this paper, a new algorithm for constructing G-optimal experimental designs is developed for both linear and generalized linear models. The new algorithm is made based on the clustering of candidate or evaluation points over the design space and it is a combination of point exchange algorithm and coordinate exchange algorithm. In addition, a robust design algorithm is proposed for generalized linear models with modification of an existing method. The proposed algorithm are compared with the methods proposed by Rodriguez et al. [Generating and assessing exact G-optimal designs. J. Qual. Technol. 2010;42(1):3–20] and Borkowski [Using a genetic algorithm to generate small exact response surface designs. J. Prob. Stat. Sci. 2003;1(1):65–88] for linear models and with the simulated annealing method and the genetic algorithm for generalized linear models through several examples in terms of the G-efficiency and computation time. The result shows that the proposed algorithm can obtain a design with higher G-efficiency in a much shorter time. Moreover, the computation time of the proposed algorithm only increases polynomially when the size of model increases.     

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).     

Computer experiments with functional inputs and scalar outputs by a norm-based approach

A framework for designing and analyzing computer experiments is presented, which is constructed for dealing with functional and scalar inputs and scalar outputs. For designing experiments with both functional and scalar inputs, a two-stage approach is suggested. The first stage consists of constructing a candidate set for each functional input. During the second stage, an optimal combination of the found candidate sets and a Latin hypercube for the scalar inputs is sought. The resulting designs can be considered to be generalizations of Latin hypercubes. Gaussian process models are explored as metamodel. The functional inputs are incorporated into the Kriging model by applying norms in order to define distances between two functional inputs. We propose the use of B-splines to make the calculation of these norms computationally feasible.     

Construction of orthogonal Latin hypercube designs with flexible run sizes

Latin hypercube designs (LHDs) have recently found wide applications in computer experiments. A number of methods have been proposed to construct LHDs with orthogonality among main-effects. When second-order effects are present, it is desirable that an orthogonal LHD satisfies the property that the sum of elementwise products of any three columns (whether distinct or not) is 0. The orthogonal LHDs constructed by Ye (1998), Cioppa and Lucas (2007), Sun et al. (2009) and Georgiou (2009) all have this property. However, the run size n of any design in the former three references must be a power of two (n=2^c) or a power of two plus one (n=2^c+1), which is a rather severe restriction. In this paper, we construct orthogonal LHDs with more flexible run sizes which also have the property that the sum of elementwise product of any three columns is 0. Further, we compare the proposed designs with some existing orthogonal LHDs, and prove that any orthogonal LHD with this property, including the proposed orthogonal LHD, is optimal in the sense of having the minimum values of ave(|t|), t_max, ave(|q|) and q_max.     

Adaptive grid semidefinite programming for finding optimal designs

We find optimal designs for linear models using a novel algorithm that iteratively combines a semidefinite programming (SDP) approach with adaptive grid techniques. The proposed algorithm is also adapted to find locally optimal designs for nonlinear models. The search space is first discretized, and SDP is applied to find the optimal design based on the initial grid. The points in the next grid set are points that maximize the dispersion function of the SDP-generated optimal design using nonlinear programming. The procedure is repeated until a user-specified stopping rule is reached. The proposed algorithm is broadly applicable, and we demonstrate its flexibility using (i) models with one or more variables and (ii) differentiable design criteria, such as A-, D-optimality, and non-differentiable criterion like E-optimality, including the mathematically more challenging case when the minimum eigenvalue of the information matrix of the optimal design has geometric multiplicity larger than 1. Our algorithm is computationally efficient because it is based on mathematical programming tools and so optimality is assured at each stage; it also exploits the convexity of the problems whenever possible. Using several linear and nonlinear models with one or more factors, we show the proposed algorithm can efficiently find optimal designs.     

Nearly optimal truncated group sequential test on binomial proportions

In this article, we systematically study the optimal truncated group sequential test on binomial proportions. Through analysis of the cost structure, average test cost is introduced as a new optimality criterion. According to the new criterion, the optimal tests on different design parameters including the boundaries, success discriminant value, stage sample vector, stage size, and the maximum sample size are defined. Since the computation time in finding optimal designs by exhaustive search is intolerably long, group sequential sample space sorting method and procedures are developed to find the near-optimal ones. In comparison with the international standard ISO2859-1, the truncated group sequential designs proposed in this article can reduce the average test costs around 20%.     

D-optimal weighted paired comparison designs

As a generalization of the paired comparison designs, the experimental design which considers the weighted difference between responses is studied. D-optimal designs are constructed for a model with main effects and first-order interactions. The experimental region is a hypercube or a hypersphere.