E(s2)-optimal and minimax-optimal cyclic supersaturated designs via multi-objective simulated annealing

doi:10.1016/j.jspi.2007.05.044

Journal of Statistical Planning and Inference

Volume 138, Issue 6, 1 July 2008, Pages 1639-1646

https://doi.org/10.1016/j.jspi.2007.05.044 Get rights and content

Abstract

In this paper, we are interested in finding $E (s^{2})$ -optimal and minimax-optimal, two level cyclic structured supersaturated designs through a metaheuristic approach guided via multi-objective simulated annealing (SA). Our construction method is based on cyclic generators. This class of metaheuristics enabled us to build supersaturated designs for $q = 2, 4, \dots, 14$ generators. Comparisons are made with previous works and it is shown that SA gives promising results for supersaturated designs that satisfy more than one optimality property. Furthermore, we provide some lower bounds and explicit formulas for the frequency of the elements with maximum absolute values that appear in the information matrix, when these values are 2, 4 or 6.

Introduction

A two level design is called saturated when the number of factors (columns) m equals to $n - 1$ , where n is the number of experimental runs (rows). A two level supersaturated design is a two level factorial design in which the number of experimental runs n is smaller or equal to the number of factors m. For each factor of a two level design there are two possible settings known as levels, which can be coded as $\pm 1$ . Any combination of the levels of all factors under consideration is called a treatment combination. Let $X =$ [ $c_{1}$ , $c_{2}, \dots, c_{m}]$ be the design matrix of the experiment in which, each row represents an m treatment combination and each column gives the sequence of factor levels. For each factor, both level values are of equal interest. Thus we consider designs with the equal occurrence property, where all columns consist of $n / 2$ elements equal to 1 and $n / 2$ elements equal to $- 1$ , when n is even. The designs with the equal occurrence property are called balanced designs. The last row of the designs presented in this paper is a row of $+ 1$ 's. If we omit this row, we obtain designs for odd number of rows, with the number of $+ 1$ 's to be one less than the number of $- 1$ 's in each column. These designs are nearly-balanced designs.

A supersaturated design can save considerable cost when the number of factors is large and a small number of runs is desired or available. The usefulness of these designs relies upon the realism of effect sparsity, namely, that the number of dominant active factors is small. Therefore, for situations where there is no prior knowledge of the effects of factors, but a strong belief in factor sparsity, and the aim is to screen out active factors, experimenters should use supersaturated designs. For more details regarding the usage of supersaturated designs see Holcomb et al. (2006) and Gilmour (2006).

Section snippets

Optimality criteria for supersaturated designs

Orthogonality between all pairs of columns of the model matrix, which is formed from the design matrix by appending a column of 1's as the first column, is desirable for estimating main effects or factors. This condition cannot be satisfied for all pairs of columns in a supersaturated design where $m ⩾ n$ . Therefore, we try to construct designs as near orthogonal as possible. We present here three optimality criteria for constructing and evaluating supersaturated designs.

$E (s^{2})$ criterion: Let $s_{ij}$ be

Cyclic structured supersaturated designs

To accommodate a large number of factors in a supersaturated design, Nguyen and Cheng (2006) proposed the use of cyclic balanced incomplete block designs (BIBDs). Their approach involved the creation of columns as cyclically generated blocks of a BIBD.

In our approach, we maintained the cyclic structure of generators although our construction is not based on BIBDs. Our goal is to construct supersaturated designs with n rows and $m = q \cdot (n - 1)$ columns, where n and q are even. In particular, each

Simulated annealing search for cyclic supersaturated designs

Single objective simulated annealing or simply simulated annealing (SA), is commonly said to be the oldest among metaheuristics and has its origins in statistical mechanics (Metropolis et al., 1958), and it was Kirkpatrick et al. (1983) and Cerny (1985) that translated it as a search algorithm for combinatorial optimization problems. We assume some basic familiarity with SA concepts. The concepts necessary for a description of the SA can be found in Laarhoven and Aarts (1987). In this section

Results

In this section, the $E (s^{2})$ -optimal and minimax-optimal cyclic structured supersaturated designs that we constructed with the aid of the multi-objective SA algorithm, given in the previous section, are presented. The balanced supersaturated designs given below have n runs and $m = q \cdot (n - 1)$ factors, where q is even. Furthermore, we present the corresponding near balanced supersaturated designs for $n - 1$ runs and $m = q \cdot (n - 1)$ factors.

Table~1, Table~2 list the values of the optimality criteria of

Finding bounds for $f_{s_{\max}}$ via bounds on $E (s^{2})$ and $s_{\max}$

In the following theorem we derive a lower bound on $f_{s_{\max}}$ for certain values of $s_{\max}$ . Furthermore, assuming $E (s^{2})$ -optimality we establish an explicit formula for $f_{s_{\max}}$ .

Theorem 4

Let $X$ be a balanced supersaturated design with n runs and $m = q \cdot (n - 1)$ factors.

(1)
Let $n \equiv 0 (\mod 4)$ and $s_{\max} = 4$ . Then $f_{s_{\max}} ⩾ {mn}^{2} (m - n + 1) / (32 (n - 1))$ .
(2)
Let $n \equiv 2 (\mod 4)$ and $s_{\max} = 2$ . Then $f_{s_{\max}} = m (m - 1) / 2$ .
(3)
Let $n \equiv 2 (\mod 4)$ and $s_{\max} = 6$ . Then $f_{s_{\max}} ⩾ m (n - 2) ((n - 2) (m - 1) - n^{2}) / (64 (n - 1))$ .
In addition, if the design is $E (s^{2})$ -optimal (1) becomes equality for any positive

Acknowledgments

We would like to thank two anonymous referees for their detailed comments and suggestions. The research of the second author was financially supported by Greek State Scholarships Foundation (IKY). The research of the first and third author was financially supported from the General Secretariat of Research and Technology by a grant PENED 03ED740.

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View full text

E(s2)-optimal and minimax-optimal cyclic supersaturated designs via multi-objective simulated annealing

Abstract

Introduction

Section snippets

Optimality criteria for supersaturated designs

Cyclic structured supersaturated designs

Simulated annealing search for cyclic supersaturated designs

Results

Finding bounds for fsmax via bounds on E(s2) and smax

Acknowledgments

Chemometrics Intell. Lab. Systems

European J. Oper. Res.

Comput. Statist. Data Anal.

Comput. Oper. Res.

J. Statist. Plann. Inference

J. Statist. Plann. Inference

Some systematic supersaturated designs

Technometrics

Construction of E(s2)-optimal supersaturated designs

Ann. Statist.

A general method of constructing E(s2)-optimal supersaturated designs

J. Roy Statist. Soc. Ser. B

A thermodynamical approach to the travelling salesman problem: an efficient simulation algorithm

J. Optim. Theory Appl.

E(s2)-optimality of supersaturated designs

Statist. Sinica

$E (s^{2})$ -optimal and minimax-optimal cyclic supersaturated designs via multi-objective simulated annealing

Finding bounds for $f_{s_{\max}}$ via bounds on $E (s^{2})$ and $s_{\max}$

Construction of $E (s^{2})$ -optimal supersaturated designs

A general method of constructing $E (s^{2})$ -optimal supersaturated designs

$E (s^{2})$ -optimality of supersaturated designs