Min-Qian Liu

Construction of E(s2) optimal supersaturated designs using cyclic BIBDs

Abstract In this paper formulas for computing the E(s2) values of some kinds of E(s2) optimal supersaturated designs are given, and a general algorithm of constructing E(s2) optimal supersaturated designs from cyclic BIBDs is proposed. Within this class of designs, by further discriminating the pairwise correlations, efficient designs of runs from 6 to 24 are constructed and tabulated. Comparisons with other existing designs are made at last, demonstrating the effectiveness of our method.

Combinatorial constructions for optimal supersaturated designs

Abstract Combinatorial designs have long had substantial application in the statistical design of experiments, and in the theory of error-correcting codes. Applications in experimental and theoretical computer science, communications, cryptography and networking have also emerged in recent years. In this paper, we focus on a new application of combinatorial design theory in experimental design theory. E ( f NOD ) criterion is used as a measure of non-orthogonality of U-type designs, and a lower bound of E ( f NOD ) which can serve as a benchmark of design optimality is obtained. A U-type design is E ( f NOD )-optimal if its E ( f NOD ) value achieves the lower bound. In most cases, E ( f NOD )-optimal U-type designs are supersaturated. We show that a kind of E ( f NOD )-optimal designs are equivalent to uniformly resolvable designs. Based on this equivalence, several new infinite classes for the existence of E ( f NOD )-optimal designs are then obtained.

ON CONSTRUCTION OF OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS

Supersaturated designs (SSDs) offer a potentially useful way to investi- gate many factors with only a few experiments during the preliminary stages of experimentation. While the construction and analysis of symmetrical SSDs have been widely explored, asymmetrical (or mixed-level) SSDs deserve further inves- tigation. Mixed-level SSDs can be judged by various criteria. But, justified by existing results, the χ 2 criterion proposed by Yamada and Lin (1999) is adopted here. Optimality results for mixed-level SSDs are provided. A new construction method for χ 2 -optimal SSDs is proposed, and we discuss properties of the resulting

Using Discrepancy to Evaluate Fractional Factorial Designs

Fractional factorial design is arguably the most widely used design in experimental investigation, and uniformity has gained popularity in experimental designs in recent years. In this present paper, a suitable measure of uniformity, i.e. a discrete discrepancy, is defined by the reproducing kernel Hilbert space, and is used to evaluate the uniformity of fractional factorial designs. Some relations between orthogonality and uniformity in fractional factorial designs are obtained. The results show that orthogonality and uniformity are strongly related to each other and the discrepancy plays an important role in evaluating such experimental designs.

Construction of sliced (nearly) orthogonal Latin hypercube designs

Abstract Sliced Latin hypercube designs are very useful for running a computer model in batches, ensembles of multiple computer models, computer experiments with qualitative and quantitative factors, cross-validation and data pooling. However, the presence of highly correlated columns makes the data analysis intractable. In this paper, a construction method for sliced (nearly) orthogonal Latin hypercube designs is developed. The resulting designs have flexible sizes and most are new. With the orthogonality or near orthogonality being guaranteed, the space-filling property of the resulting designs is also improved. Examples are provided for illustrating the proposed method.

Computer Experiments With Both Qualitative and Quantitative Variables

Computer experiments have received a great deal of attention in many fields of science and technology. Most literature assumes that all the input variables are quantitative. However, researchers often encounter computer experiments involving both qualitative and quantitative variables (BQQV). In this article, a new interface on design and analysis for computer experiments with BQQV is proposed. The new designs are one kind of sliced Latin hypercube designs with points clustered in the design region and possess good uniformity for each slice. For computer experiments with BQQV, such designs help to measure the similarities among responses of different level-combinations in the qualitative variables. An adaptive analysis strategy intended for the proposed designs is developed. The proposed strategy allows us to automatically extract information from useful auxiliary responses to increase the precision of prediction for the target response. The interface between the proposed design and the analysis strategy is demonstrated to be effective via simulation and a real-life example from the food engineering literature. Supplementary materials for this article are available online.

Construction of nearly orthogonal Latin hypercube designs

Latin hypercube designs have found wide application in computer experiments. A number of methods have recently been proposed to construct orthogonal Latin hypercube designs. In this paper, we propose an approach for expanding the orthogonal Latin hypercube design in Sun et al. (Biometrika 96:971–974, 2009) to a nearly orthogonal Latin hypercube design of a larger column size. The newly added part has half number of columns of the original part. It can be shown that the upper bound of the maximum correlation between any two distinct columns of the resulting design is very small. Our method also works for expanding any symmetric Latin hypercube designs.

A note on the lower bounds on maximum number of clear two-factor interactions for 2m-pIII and 2m-pIV designs

Clear effects criterion is one of the most important rules for selecting 2 m−p designs, and it is valuable to know the maximum number of clear two-factor interactions in 2 m−p designs. This article provides lower bounds on the maximum number of clear two-factor interactions for and designs by constructing specific designs, respectively. The lower bounds improve the ones given in Tang et al. (2002) and the construction methods perform quite well.

On the construction of nested space-filling designs

Nested space-filling designs are nested designs with attractive low-dimensional stratification. Such designs are gaining popularity in statistics, applied mathematics and engineering. Their applications include multi-fidelity computer models, stochastic optimization problems, multi-level fitting of nonparametric functions, and linking parameters. We propose methods for constructing several new classes of nested space-filling designs. These methods are based on a new group projection and other algebraic techniques. The constructed designs can accommodate a nested structure with an arbitrary number of layers and are more flexible in run size than the existing families of nested space-filling designs. As a byproduct, the proposed methods can also be used to obtain sliced space-filling designs that are appealing for conducting computer experiments with both qualitative and quantitative factors.

Uniformity of incomplete block designs

Incomplete block designs are a type of experimental design layout that has had widespread use in science and engineering. A balanced incomplete block design (BIB) can be characterized by the balanced arrangement of its design points. In this article, incomplete block designs will be investigated from a new perspective: the perspective of uniformity in distribution of design points. A design is of high uniformity, or low discrepancy, if its design points distribute uniformly over the entire design space. The authors use a general discrepancy measure to prove theoretically that BIBs are the most uniform ones among all binary incomplete block designs.