Good Projective Properties Research Articles

On the identification of active factors in nonregular two‐level designs with a small number of runs

AbstractNonregular two‐level designs are attractive screening designs due to their good projection properties and flexible run sizes. In particular, the 12‐run Plackett–Burman (PB) design has become quite popular. However, existing methods struggle with the identification of active factors when the number of active factors exceeds the projectivity of the designs. This is especially the case when interactions are present, the variance is high and the number of runs is small. In this paper, we propose a method for analysing nonregular two‐level designs that particularly addresses the issues above. It exploits the projection properties of designs and is here applied on the 12‐run PB design and the 16‐run no‐confounding (NC) designs. In the construction of the method, the use of test‐ and penalty‐based procedures are avoided. Instead, the number of allowed terms in a model is restricted. The effectiveness of the method and comparison between designs are evaluated by simulations for different scenarios. Ways to evaluate the reliability of the screening procedure are pointed out. An example with real data is given to demonstrate how one might perform the analysis in practice.

Uniform projection nested Latin hypercube designs

Computer experiments usually involve many factors, but only a few of them are active. In such a case, it is desirable to construct designs with good projection properties. Maximum projection designs and uniform projection designs have been developed for common experimental situations, however, there has been little study on constructing projection designs for high-accuracy computer experiments (HEs) and low-accuracy computer experiments (LEs) so far. In this paper, we propose a weighted uniform projection criterion, and construct uniform projection nested Latin hypercube designs to suit such computer experiment situations. We show that the obtained designs have good projection properties in all sub-dimensions, and we also discuss how to choose a proper value for the weight. Simulated examples are available to illustrate the effectiveness of the proposed designs.

Formula omitted]- and [formula omitted]-optimal supersaturated designs

Supersaturated designs are useful for factor screening experiments under the factor sparsity assumption that only a small number of factors are active. The popular E(s2)-criterion for choosing two-level supersaturated designs minimizes the sum of squares of theentries of the information matrix over the designs in which the two levels of each factor appear equal number of times. Jones and Majumdar (2014) proposed the UE(s2)-criterion which is essentially the same as the E(s2)-criterion except that the requirement of factor-level-balance is dropped. Removing this constraint makes UE(s2)-optimal designs easy to construct, but it also produces many UE(s2)-optimal designs with diverse performances. It is necessary to choose better designs from them, especially those with good lower-dimensional projection properties. While E(s2)-optimal designs tend to have better lower-dimensional projections than arbitrary UE(s2)-optimal designs, they are usually very difficult to construct. We propose a secondary criterion and provide simple and systematic constructions of superior UE(s2)-optimal designs having good projection properties. We also derive conditions under which E(s2)-optimal designs are UE(s2)-optimal as well and identify several families of designs that are optimal under both criteria.

Split-plot designs for multistage experimentation

ABSTRACTMost of today’s complex systems and processes involve several stages through which input or the raw material has to go before the final product is obtained. Also in many cases factors at different stages interact. Therefore, a holistic approach for experimentation that considers all stages at the same time will be more efficient. However, there have been only a few attempts in the literature to provide an adequate and easy-to-use approach for this problem. In this paper, we present a novel methodology for constructing two-level split-plot and multistage experiments. The methodology is based on the Kronecker product representation of orthogonal designs and can be used for any number of stages, for various numbers of subplots and for different number of subplots for each stage. The procedure is demonstrated on both regular and nonregular designs and provides the maximum number of factors that can be accommodated in each stage. Furthermore, split-plot designs for multistage experiments with good projective properties are also provided.

Projection Properties of Blocked Non‐regular Two‐level Designs

Minimum resolution IV (MinResIV) and Plackett and Burman (PB) designs are popular screening designs because of their run size efficiency and good projection properties. The purpose of this investigation is to find out which projection properties that can be maintained when these designs are blocked and also the efficiency by which the effects of interest can be estimated. MinResIV designs with 10–20 runs and PB designs with 12 and 20 are investigated. For the PB designs, design factor columns were used as blocking columns, while for the MinResIV designs, the main rule consisted of finding good arrangements for allocating mirror‐image pair runs to the same blocks, a method inspired from Jacroux. As a criterion for a good blocking arrangements, we used maximum Ds ‐ efficiency, which we found to be more generally applicable and provide better projection properties than previous suggested methods. It came out that for most of these designs, it was possible to find blocking schemes satisfying that for any three factors, either all main effects and their interactions or all main effects and their two‐factor interactions could be estimated with a reasonable high efficiency when blocked. Copyright © 2016 John Wiley & Sons, Ltd.

An algorithmic foldover procedure for nearly orthogonal arrays with projection

Nearly orthogonal arrays are a class of experimental designs that are useful for accommodating multiple mixed-level factors and small run sizes for factor screening experiments. Several algorithmic approaches have been used to construct such arrays using D, A, ave(S2), and most interestingly, projection properties to evaluate and classify candidate arrays. Nearly orthogonal arrays constructed to contain good projection properties make it possible to estimate higher order interactions for subsets of design columns but can introduce significant correlation as to make some factor estimates terribly imprecise. This paper proposes a method to exploit the projection properties of these nearly orthogonal arrays while guarding against correlated factor effects. The usefulness of this foldover technique is demonstrated by a live-virtualconstructive simulation experiment.

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.

The Use of a 12‐run Plackett–Burman Design in the Injection Moulding of a Technical Plastic Component

AbstractDuring injection moulding of technical plastic components it is often the case that 15–20 variables need to be set to operational conditions when the production of a new plastic components is started. In this case study, a 12‐run Plackett–Burman design is used in the experimentation with eight factors and nine responses to obtain better operational conditions for the production of a certain product. A 12‐run Plackett–Burman design belongs to the class of non‐geometric orthogonal arrays. Most of these designs have very good projection properties compared with their run sizes, but due to partial aliasing among effects, standard methods of analysing fractional factorial two‐level designs do not apply. In this paper, part of the planning of the experiment, the method of analysis used and the results achieved are presented. Substantial improvement in the operational conditions was obtained. Copyright © 2006 John Wiley & Sons, Ltd.

Hidden projection properties of some nonregular fractional factorial designs and their applications

In factor screening, often only a few factors among a large pool of potential factors are active. Under such assumption of effect sparsity, in choosing a design for factor screening, it is important to consider projections of the design onto small subsets of factors. Cheng showed that as long as the run size of a two-level orthogonal array of strength two is not a multiple of 8, its projection onto any four factors allows the estimation of all the main effects and two-factor interactions when the higher-order interactions are negligible. This result applies, for example, to all Plackett-Burman designs whose run sizes are not multiples of 8. It is shown here that the same hidden projection property also holds for Paley designs of sizes greater than 8, even when their run sizes are multiples of 8. A key result is that such designs do not have defining words of length three or four. Applications of this result to the construction of $E(s^2)$-optimal supersaturated designs are also discussed. In particular, certain designs constructed by using Wu's method are shown to be $E(s^2)$-optimal. The article concludes with some three-level designs with good projection properties.

Exploratory designs for computational experiments

Recent work by Johnson et al. ( J. Statist. Plann. Inference 26 (1990) 131–148) establishes equivalence of the maximin distance design criterion and an entropy criterion motivated by function prediction in a Bayesian setting. The latter criterion has been used by Currin et al. ( J. Amer. Statist. Assoc. 86 (1991) 953–963) to design experiments for which the motivating application is approximation of a complex deterministic computer model. Because computer experiments often have a large number of controlled variables (inputs), maximin designs of moderate size are often concentrated in the corners of the cuboidal design region, i.e. each input is represented at only two levels. Here we will examine some maximin distance designs constructed within the class of Latin hypercube arrangements. The goal of this is to find designs which offer a compromise between the entropy/maximin criterion, and good projective properties in each dimension (as guaranteed by Latin hypercubes). A simulated annealing search algorithm is presented for constructing these designs, and patterns apparent in the optimal designs are discussed.