William Li

Algorithmic construction of optimal symmetric Latin hypercube designs

Abstract We propose symmetric Latin hypercubes for designs of computer experiment. The goal is to offer a compromise between computing effort and design optimality. The proposed class of designs has some advantages over the regular Latin hypercube design with respect to criteria such as entropy and the minimum intersite distance. An exchange algorithm is proposed for constructing optimal symmetric Latin hypercube designs. This algorithm is compared with two existing algorithms by Park (1994. J. Statist. Plann. Inference 39, 95–111) and Morris and Mitchell (1995. J. Statist. Plann. Inference 43, 381–402). Some examples, including a real case study in the automotive industry, are used to illustrate the performance of the new designs and the algorithms.

Columnwise-pairwise algorithms with applications to the construction of supersaturated designs

Motivated by the construction of supersaturated designs, we develop a class of algorithms called columnwise-pairwise exchange algorithms. They differ from the k-exchange algorithms in two respects: (1) They exchange columns instead of rows of the design matrix, and (2) they employ a pairwise adjustment in the search for a “better” column. The proposed algorithms perform very well in the construction of supersaturated designs both for a single criterion and for multiple criteria. They are also applicable to the construction of designs that are not supersaturated.

Optimal Foldover Plans for Two-Level Fractional Factorial Designs

A commonly used follow-up experiment strategy involves the use of a foldover design by reversing the signs of one or more columns of the initial design. Defining a foldover plan as the collection of columns whose signs are to be reversed in the foldover design, this article answers the following question: Given a 2k−p design with k factors and p generators, what is its optimal foldover plan? We obtain optimal foldover plans for 16 and 32 runs and tabulate the results for practical use. Most of these plans differ from traditional foldover plans that involve reversing the signs of one or all columns. There are several equivalent ways to generate a particular foldover design. We demonstrate that any foldover plan of a 2k−p fractional factorial design is equivalent to a core foldover plan consisting only of the p out of k factors. Furthermore, we prove that there are exactly 2k−p foldover plans that are equivalent to any core foldover plan of a 2k−p design and demonstrate how these foldover plans can be constructed. A new class of designs called combined-optimal designs is introduced. An n-run combined-optimal 2k−p design is the one such that the combined 2k−p+1 design consisting of the initial design and its optimal foldover has the minimum aberration among all 2k−p designs.

Variable Selection With the Strong Heredity Constraint and Its Oracle Property

In this paper, we extend the LASSO method (Tibshirani 1996) for simultaneously fitting a regression model and identifying important interaction terms. Unlike most of the existing variable selection methods, our method automatically enforces the heredity constraint, that is, an interaction term can be included in the model only if the corresponding main terms are also included in the model. Furthermore, we extend our method to generalized linear models, and show that it performs as well as if the true model were given in advance, that is, the oracle property as in Fan and Li (2001) and Fan and Peng (2004). The proof of the oracle property is given in online supplemental materials. Numerical results on both simulation data and real data indicate that our method tends to remove irrelevant variables more effectively and provide better prediction performance than previous work (Yuan, Joseph, and Lin 2007 and Zhao, Rocha, and Yu 2009 as well as the classical LASSO method).

Optimal Foldover Plans for Two-Level Nonregular Orthogonal Designs

This article considers optimal foldover plans for nonregular designs. By using the indicator function, we define words with fractional lengths. The extended word-length pattern is then used to select among nonregular foldover designs. Some general properties of foldover designs are obtained using the indicator function. We prove that the full-foldover plan that reverses the signs of all factors is optimal for all 12-run and 20-run orthogonal designs. The optimal foldover plans for all 16-run (regular and nonregular) orthogonal designs are constructed and tabulated for practical use. Optimal foldover plans for higher-order orthogonal designs can be constructed in a similar manner.

Model-robust factorial designs

In industrial experimentation, experimental designs are frequently constructed to estimate all main effects and a few prespecified interactions. The robust-product-design literature is replete with such examples. A major limitation of this approach is the requirement that the experimenter know which interactions are likely to be active in advance. In this article, we develop a class of balanced designs that can be used for estimation of main effects and any combination of up to CJin teractions, where g is specified by the user. We view this as an issue of model-robust design: We construct designs that are highly efficient for all models involving main effects and g (or fewer) interactions. We compare the performances of these designs with the standard alternatives from the class of maximum-resolution fractional factorial designs for several criteria. The comparison reveals that the new designs are surprisingly robust to model misspecification, something that is generally not true for maximum-resolution fractional factorial designs. This robustness comes at a price: The new designs are frequently not orthogonal. We demonstrate, however, that the loss of orthogonality is, in general, quite small.

Blocked Nonregular Two-Level Factorial Designs

This article discusses the optimal blocking criteria for nonregular two-level designs. We extend the optimal blocking criteria of Cheng and Wu to nonregular designs by adapting the G- and G2-minimum aberration criteria discussed by Tang and Deng. To define word-length pattern for nonregular designs, we extend the notion of “word” to nonregular designs through a polynomial representation of factorial designs. We define treatment resolution and block resolution for evaluating the degrees of aliasing and confounding. We propose four new criteria, which we use to search for optimal blocking schemes of 12-run, 16-run, and 20-run two-level orthogonal arrays.

Measuring the VC-Dimension Using Optimized Experimental Design

VC-dimension is the measure of model complexity (capacity) used in VC-theory. The knowledge of the VC-dimension of an estimator is necessary for rigorous complexity control using analytic VC generalization bounds. Unfortunately, it is not possible to obtain the analytic estimates of the VC-dimension in most cases. Hence, a recent proposal is to measure the VC-dimension of an estimator experimentally by fitting the theoretical formula to a set of experimental measurements of the frequency of errors on artificially generated data sets of varying sizes (Vapnik, Levin, & Le Cun, 1994). However, it may be difficult to obtain an accurate estimate of the VC-dimension due to the variability of random samples in the experimental procedure proposed by Vapnik et al. (1994). We address this problem by proposing an improved design procedure for specifying the measurement points (i.e., the sample size and the number of repeated experiments at a given sample size). Our approach leads to a nonuniform design structure as opposed to the uniform design structure used in the original article (Vapnik et al., 1994). Our simulation results show that the proposed optimized design structure leads to a more accurate estimation of the VC-dimension using the experimental procedure. The results also show that a more accurate estimation of VC-dimension leads to improved complexity control using analytic VC-generalization bounds and, hence, better prediction accuracy.

AN INTEGRATED METHOD OF PARAMETER DESIGN AND TOLERANCE DESIGN

Taguchi proposed a two-stage approach to quality improvement: parameter design followed by tolerance design. A novel approach is proposed for integrating parameter design and tolerance design into a single stage of design optimization. It enjoys a de..

A Class of Optimal Robust Parameter Designs

In robust designs, the control-by-noise interactions are usually considered to be more important than other two-factor interactions. We extend the work on model-robust factorial designs of Li and Nachtsheim (2000) to robust designs, where not all factors are treated equally. A new criterion is proposed to maximize a designs ability to estimate models with at least one control-by-noise interaction. Optimal designs are chosen from the following three groups: regular orthogonal arrays (OAs), non-regular OAs, and balanced designs. Designs with economic run-sizes are constructed algorithmically and are tabulated for practical use.