Orthogonal Array-based Latin Hypercube Research Articles

Orthogonal Array-Based Latin Hypercube Designs for Computer Experiments

Orthogonal Array-based Latin Hypercube Designs (OALHDs) have not only become popular in practice among strategies used in the development of computer experiments but also useful whenever interest is focused on performing some physical experiments. Design construction for computer experiments is a new issue in this part of the world since it is more about experimental planning rather than modelling aspect in which some progress has been made. The Bush Construction Type II method was presented in this paper to construct a strong Orthogonal Array (OA) of strength three, using Galois Fields (GF) of order s which gave rise to the constructed Orthogonal Array-Based Latin Hypercube Designs (OALHD) for computer experiments. Orthogonal Array-based Latin Hypercube Design was used in this paper as a Latin hypercube design constructed based on orthogonal array in order to achieve better space-filling properties that would otherwise not be possessed by a random Latin hypercube design (LHD). Orthogonal Array (N, k) LHD were constructed at parameter values of OA (N, k)=(64, 6) and (125, 7). This study is an improvement on the early paper which adopted the Bush Construction Type I technique and it therefore aimed at proposing a novel approach that employed the maximin criterion in the k-Nearest Neighbour with Euclidean distance for constructing strong orthogonal arrays along with the Orthogonal Array-Based Latin Hypercube Designs (OALHDs). The OA (64, 6) LHD and OA (125, 7) LHD constructed have better space-filling properties and they achieve uniformity in each dimension. This study concludes that the constructed OALHDs can be used whenever interest is focused on performing either a conventional or computer experiment on real life situations. A program implementation for the construction of OALHDs was done using MATLAB 2016 computer package.

Estimates of the coverage of parameter space by Latin Hypercube and Orthogonal Array-based sampling

• A rigorous analysis of high dimensional parameter sampling techniques. • New theoretical bounds for percentage coverage of parameter space by sampling. • Numerical simulations confirming bounds on percentage coverage of parameter space and applications of the coverage formula. • Results verifying t-way interactions coverage estimates in an experimental design setting depend on t not the total dimension. In this paper we use counting arguments to prove that the expected percentage coverage of a d dimensional parameter space of size n when performing k trials with either Latin Hypercube sampling or Orthogonal Array-based Latin Hypercube sampling is the same. We then extend these results to an experimental design setting by projecting onto a t < d dimensional subspace. These results are confirmed by simulations. The theory presented has both theoretical and practical significance in modelling and simulation science when sampling over high dimensional spaces.

Sliced Full Factorial-Based Latin Hypercube Designs as a Framework for a Batch Sequential Design Algorithm

When fitting complex models, such as finite element or discrete event simulations, the experiment design should exhibit desirable properties of both projectivity and orthogonality. To reduce experimental effort, sequential design strategies allow experimenters to collect data only until some measure of prediction precision is reached. In this article, we present a batch sequential experiment design method that uses sliced full factorial-based Latin hypercube designs (sFFLHDs), which are an extension to the concept of sliced orthogonal array-based Latin hypercube designs (OALHDs). At all stages of the sequential design, good univariate stratification is achieved. The structure of the FFLHDs also tends to produce uniformity in higher dimensions, especially at certain stages of the design. We show that our batch sequential design approach has good sampling and fitting qualities through both empirical studies and theoretical arguments. Supplementary materials are available online.

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 general theory for orthogonal array based Latin hypercube sampling

A general theory for orthogonal array based Latin hypercube sampling

Approximation of the exponential integral (well function) using sampling methods

• This study proposes a new method for exponential integral approximation. • The methods are based on probabilistic sampling. • Three different sampling methods have been used; LHS, OA and MCS. • The results of sampling methods are good and accurate compared to other methods. Exponential integral (also known as well function) is often used in hydrogeology to solve Theis and Hantush equations. Many methods have been developed to approximate the exponential integral. Most of these methods are based on numerical approximations and are valid for a certain range of the argument value. This paper presents a new approach to approximate the exponential integral. The new approach is based on sampling methods. Three different sampling methods; Latin Hypercube Sampling (LHS), Orthogonal Array (OA), and Orthogonal Array-based Latin Hypercube (OA-LH) have been used to approximate the function. Different argument values, covering a wide range, have been used. The results of sampling methods were compared with results obtained by Mathematica software, which was used as a benchmark. All three sampling methods converge to the result obtained by Mathematica, at different rates. It was found that the orthogonal array (OA) method has the fastest convergence rate compared with LHS and OA-LH. The root mean square error RMSE of OA was in the order of 1E−08. This method can be used with any argument value, and can be used to solve other integrals in hydrogeology such as the leaky aquifer integral.

Samurai Sudoku-Based Space-Filling Designs for Multi-Source Inference

Pooling data from multiple sources plays an increasingly vital role in today's world. 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: (1) the complete design achieves attractive uniformity in both univariate and bivariate margins; (2) it can be divided into groups of subdesigns with overlaps such that each subdesign achieves maximum uniformity in both univariate and bivariate margins; (3) each of the overlaps achieves maximum 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.

Latin hypercube designs with controlled correlations and multi-dimensional stratification

Various methods have been proposed to construct Latin hypercube designs with small correlations. Orthogonal arrays have been used to construct Latin hypercube designs with multi-dimensional stratification. To integrate these two ideas, we propose a method to construct Latin hypercube designs with both controlled correlations and multi-dimensional stratification. For numerical integration, the constructed designs not only filter out lower-dimensional variance components as effectively as ordinary orthogonal array-based Latin hypercube designs, but also filter out bilinear terms more effectively. The proposed construction method entails no iterative searches. Sampling properties of the constructed designs are derived. Examples are given to illustrate the proposed construction method and the theoretical results.

A comprehensive evaluation of various sensitivity analysis methods: A case study with a hydrological model

Sensitivity analysis (SA) is a commonly used approach for identifying important parameters that dominate model behaviors. We use a newly developed software package, a Problem Solving environment for Uncertainty Analysis and Design Exploration (PSUADE), to evaluate the effectiveness and efficiency of ten widely used SA methods, including seven qualitative and three quantitative ones. All SA methods are tested using a variety of sampling techniques to screen out the most sensitive (i.e., important) parameters from the insensitive ones. The Sacramento Soil Moisture Accounting (SAC-SMA) model, which has thirteen tunable parameters, is used for illustration. The South Branch Potomac River basin near Springfield, West Virginia in the U.S. is chosen as the study area. The key findings from this study are: (1) For qualitative SA methods, Correlation Analysis (CA), Regression Analysis (RA), and Gaussian Process (GP) screening methods are shown to be not effective in this example. Morris One-At-a-Time (MOAT) screening is the most efficient, needing only 280 samples to identify the most important parameters, but it is the least robust method. Multivariate Adaptive Regression Splines (MARS), Delta Test (DT) and Sum-Of-Trees (SOT) screening methods need about 400–600 samples for the same purpose. Monte Carlo (MC), Orthogonal Array (OA) and Orthogonal Array based Latin Hypercube (OALH) are appropriate sampling techniques for them; (2) For quantitative SA methods, at least 2777 samples are needed for Fourier Amplitude Sensitivity Test (FAST) to identity parameter main effect. McKay method needs about 360 samples to evaluate the main effect, more than 1000 samples to assess the two-way interaction effect. OALH and LPτ (LPTAU) sampling techniques are more appropriate for McKay method. For the Sobol' method, the minimum samples needed are 1050 to compute the first-order and total sensitivity indices correctly. These comparisons show that qualitative SA methods are more efficient but less accurate and robust than quantitative ones.

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.

Nested orthogonal array-based Latin hypercube designs

We propose two methods for constructing a new type of design, called a nested orthogonal array-based Latin hypercube design, intended for multi-fidelity computer experiments. Such designs are two nested space-filling designs in which the large design achieves stratification in both bivariate and univariate margins and the small design achieves stratification in univariate margins. These designs have better space-filling properties than nested Latin hypercube designs in which the large design possesses uniformity in univariate margins only. The first method expands an ordinary Latin hypercube design to a larger design that achieves uniformity in any one- or two-dimensional projection. The second method uses an orthogonal array with strength two to simultaneously construct a pair of nested orthogonal array-based Latin hypercube designs. Examples are given to illustrate the proposed methods. Sampling properties of the proposed designs are derived. Copyright 2011, Oxford University Press.

Batch sequential designs for computer experiments

Computer models simulating a physical process are used in many areas of science. Due to the complex nature of these codes it is often necessary to approximate the code, which is typically done using a Gaussian process. In many situations the number of code runs available to build the Gaussian process approximation is limited. When the initial design is small or the underlying response surface is complicated this can lead to poor approximations of the code output. In order to improve the fit of the model, sequential design strategies must be employed. In this paper we introduce two simple distance based metrics that can be used to augment an initial design in a batch sequential manner. In addition we propose a sequential updating strategy to an orthogonal array based Latin hypercube sample. We show via various real and simulated examples that the distance metrics and the extension of the orthogonal array based Latin hypercubes work well in practice.

Alternative sampling methods for estimating multivariate normal probabilities

We study the performance of alternative sampling methods for estimating multivariate normal probabilities through the GHK simulator. The sampling methods are randomized versions of some quasi-Monte Carlo samples (Halton, Niederreiter, Niederreiter-Xing sequences and lattice points) and some samples based on orthogonal arrays (Latin hypercube, orthogonal array and orthogonal array based Latin hypercube samples). In general, these samples turn out to have a better performance than Monte Carlo and antithetic Monte Carlo samples. Improvements over these are large for low-dimensional (4 and 10) cases and still significant for dimensions as large as 50.