Atlas: Combining D-Optimal and Covering Array Design Properties by Dean S. Hoskins

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FIMXII-SCMA2005@AUBURN, Twelfth Annual International Conference on Statistics, Combinatorics, Mathematics and Applications
December 2-4, 2005
Auburn University
Auburn, Alabama, USA

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Forum for Interdisciplinary Mathematics

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Combining D-Optimal and Covering Array Design Properties
by
Dean S. Hoskins
Department of Computer Science and Engineering, Arizona State University, Tempe Arizona
Coauthors: Charles J. Colbourn

Research into combining properties of D-Optimal designs and covering arrays achieves synergy in that the desirable properties of both types of designs can be realized together in one design matrix. Historically, D-Optimal and covering array designs offer a common mechanism for analyzing multi-level categorical factors, one from the quantitative standpoint and one from the qualitative standpoint. Our research has indicated that covering arrays can also be applied to quantitative analysis of factors and factor interactions and in several analyzed cases can estimate full factorial results better than D-Optimal designs. Since D-Optimal designs are based on the minimization of the regression coefficient variation from the standard regression equation in matrix form,

b = (X¢X)^-1X¢y

the minimization of the error squared difference between Type III sum of squares to the full factorial data may not always be achieved. The equation relating the error of a full factorial as compared to a smaller design can be represented by the equation e² = å^ME+2FI( r_ff - r_rf )²
where r_ff and r_rf are the percentage contributions to variance for the full and reduced designs (D-Optimal or covering array). This equation is used as a comparison mechanism to determine how well full factorial designs are approxiamated.

A covering array, CA_l(N;t, k, v), is an N ×k array for which every N ×t subarray has the property that every t-tuple appears at least l times. In this application, t is the strength, k is the number of factors (degree), and v is the number of symbols for each factor (order).

Earlier research [ASU: Hoskins, Colbourn, Montgomery] has shown that a hybrid covering array / D-Optimal design generator can be constructed that enhances D-Optimal designs that estimate main effects plus two-factor interactions to include full covering to strength t=3. The benefit of this hybrid design is that it (1) maintains reasonable D-efficiencies with a full strength 3 covering, allowing quantitative estimation of main effects and two-factor interactions, (2) offers the ability to qualitatively see three factor interactions, (3) has the added benefit of providing a closer estimation of full factorial results than a D-optimal design does alone, and (4) maintains full rank of the final design matrix.

Current research goes one step further and modifies the hybrid generator to generate D-Optimal / covering array designs that maintain a higher D-efficiency than earlier designs, effectively the same as designs generated by the SAS Optex procedure with very little loss in D-efficiency. The construction of this enhanced generator is discussed along with algorithms used and actual designs are constructed as examples comparing results to the earlier generator. The key to the success of the generator is that as the covering array is being built row by row |X¢X| (X is the design matrix) is constantly being evaluated and eliminating non-optimal rows from the design.

The presentation will cover past, present, and future aspects of our research at Arizona State University which is a collaboration between the Computer Science and Industrial Engineering Departments.

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Date received: October 14, 2005