Centered L2-discrepancy Research Articles

A Lower Bound for the Centered L 2-Discrepancy on Asymmetric Factorials and its Application

The role of uniformity measured by the centered L2-discrepancy (Hickernell 1998a) has been studied in fractional factorial designs. The issue of a lower bound for the centered L2-discrepancy is crucial in the construction of uniform designs. Fang and Mukerjee (2000) and Fang et al. (2002, 2003b) derived lower bounds for fractions of two- and three-level factorials. In this paper we report some new lower bounds for the centered L2-discrepancy for a set of asymmetric fraction factorials. Using these lower bounds helps to measure uniformity of a given design. In addition, as an application of these lower bounds, we propose a method to construct uniform designs or nearly uniform designs with asymmetric factorials.

Evaluation of inequivalent projections of Hadamard matrices of order 24

Screening designs are useful for situations where a large number of factors (q) is examined but only few (k) of these are expected to be important. It is of practical interest for a given k to know all the inequivalent projections of the design into the k dimensions. In this paper we give all the (combinatorially) inequivalent projections of inequivalent Hadamard matrices of order 24 into k=3,4 and 5 dimensions, as well as their frequencies. Then, we sort these projections according to their generalized resolution, generalized aberration and centered L2-discrepancy measure of uniformity. Then, we study the hidden projection properties of these designs as they are introduced by Wang and Wu (1995). The hidden projection property suggests that complex aliasing allows some interactions to be estimated without making additional runs.

A note on construction of nearly uniform designs with large number of runs

Uniform designs have been used in computer experiments (Fang et al., Technometrics 42 (2000) 237). A uniform design seeks its design points to be uniformly scattered on the experimental domain. When the number of runs is large, to search a related uniform design is a NP hard problem. Therefore, the number of runs of most existing uniform designs is small (⩽50). In this article, we propose a way to construct nearly uniform designs with large number of runs by collapsing two uniform designs in the sense of low-discrepancy. The number of runs of the novel design is the product of the two numbers of runs of both original designs. Two measures of uniformity, the centered L 2-discrepancy (CD) and wrap-around L 2-discrepancy (WD) are employed. Analytic formulas of CD- and WD-values between the novel design and both original designs are obtained.

On the Isomorphism of Fractional Factorial Designs

Two fractional factorial designs are called isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and switching the levels of factors. To identify the isomorphism of two s-factor n-run designs is known to be an NP hard problem, when n and s increase. There is no tractable algorithm for the identification of isomorphic designs. In this paper, we propose a new algorithm based on the centered L2-discrepancy, a measure of uniformity, for detecting the isomorphism of fractional factorial designs. It is shown that the new algorithm is highly reliable and can significantly reduce the complexity of the computation. Theoretical justification for such an algorithm is also provided. The efficiency of the new algorithm is demonstrated by using several examples that have previously been discussed by many others.