Man V. M. Nguyen

January 10, 2003

Abstract

Construction of balanced fractions of mixed factorial design using Hadamard matrices and Groebner bases.

In this talk I will present some new results in constructing resolution III mixed designs, especially some 2^{a}3^{b} series designs with a + b < 9 and a, b > 0. A necessary criterion for searching balanced fractions is stated as well [3].

This is joint work with Arjeh M. Cohen.

In the paper “Construction of 2^{a}3^{b}-design” , [1], the algebraic and statistical fundamentals for the problems of constrution of experimental designs was shown. I used the idea of the border basis and multiplication matrices [2] to solve the following inverse problem:

Suppose given an order ideal E, we construct a fraction F such that E = Est(F) . However, in practical applications, very often we want to find fractions

with some extra properties. Some useful properties are resolution III and balanced . Can we place resolution III and balanced requirements in the Groebner base setting, and how? In detail we want to answer the two following question:

Question 1: If we want to construct a resolution III fraction F , which terms should we put in an order ideal E of F to make sure that a returned fraction F has resolution III ?

Question 2: Is there any neccessary condition which is imposed in the system of equations (to give a Groebner base of the vanishing ideal I(F) [2]) to make sure F is two-balanced?

References

[1] Construction of 2^{m}3^{n}-design, Nguyen Van Minh Man, Technical report, TU/e, June 2002.

[2] M. Caboara and L. Robbiano, Families of Estimable Terms, ISSAC 2001.

[3] Finding balanced fractions in designs of experiments, preprint, Arjeh M. Cohen, Man V.M. Nguyen, TUE, Dec 2002 .