Advances and Applications in Statistics
Volume 3, Issue 1, Pages 1 - 13
(April 2003)
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APPLICATION OF GRÖBNER BASES TO THE ANALYSIS OF CERTAIN TWO OR THREE LEVEL FACTORIAL DESIGNS
H. Evangelaras (Greece), I. Kotsireas (Canada) and C. Koukouvinos (Canada)
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Abstract: 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. Hadamard
matrices have traditionally been used for this
purpose. Since these designs are only main
effects plans and since the number of runs are
greater than the number of active factors (main
effects), there are plenty degrees of freedom
unused for identifying and estimating
interactions of factors.Computational
Algebraic Geometry can be used to solve
identifiability problems in design of experiments in
Statistics. The key idea is to view the design as a
solution set of a system of non-linear polynomial
equations. Then, the theory of Gröbner
bases allows one to identify the whole set of
estimable effects (main or interactions) of the
factors of the design. Modern Computer Algebra
Systems such as Maple, Magma, Mathematica and
Computational Algebra packages such as CoCoA and
Singular contain efficient implementations of the
algorithms needed to carry out the computations to
solve these problems. In this paper, we discuss the
application of Gröbner
bases theory in certain two and three level
factorial designs. |
Keywords and phrases: Gröbner bases, Hadamard matrices, inequivalent projections, ideals, leading terms, divisibility condition, estimable effects. |
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