A supersaturated design is a design whose run size is not large enough for estimating all the main effects. The goodness of multi-level supersaturated designs can be judged by the generalized minimum aberration criterion proposed by Xu and Wu (2001). Optimal supersaturated designs are shown to have a periodic property and general methods for constructing optimal multilevel supersaturated designs are proposed. Inspired by the Addelman-Kempthorne construction of orthogonal arrays, optimal multi-level supersaturated designs are given in an explicit form: columns are labeled with linear or quadratic polynomials and rows are points over a finite field. Additive characters are used to study the properties of resulting designs. Some small optimal supersaturated designs of 3, 4 and 5 levels are listed with their properties.

A smooth response surface algorithm is developed as an elaborate data mining technique for analyzing gene expression data and constructing gene regulatory network. A three-dimensional smooth response surface is generated to capture the biological relationship between the target and activator-repressor. This new technique is applied to functionally describe triplets of activators, repressors and targets, and their regulations in gene expression data. A diagnostic strategy is built into the algorithm to evaluate the scores of the triplets so that those with low scores are kept and a regulatory network is constructed based on this information and existing biological knowledge. The predictions based on the identified triplets in two yeast gene expression data sets agree with some experimental data in the literature. It provides a novel model with attractive mathematical and statistical features that make the algorithm valuable for mining expression or concentration information, assist in determining the function of uncharacterized proteins, and can lead to a better understanding of coherent pathways.

Orthogonal arrays are widely used in industrial experiments for factor screening. Suppose only a few of the factors are important. An orthogonal array can be used not only for screening factors but also for detecting interactions among a subset of active factors. In this article, a set of optimality criteria is proposed to assess the performance of designs for factor screening, pro jection, and interaction detection, and a three-step approach is proposed to search for opti- mal designs. Combinatorial and algorithmic construction methods are proposed for generating new designs. Level permutation methods are used for improving the eligibility and estimation efficiency of the pro jected designs. The techniques are then applied to search for best three-level designs with 18 and 27 runs. Many new, efficient and practically useful nonregular designs are found and their properties discussed.