Gro¨bner bases in experimental design
In this chapter we use the methods of algebraic geometry to study the identiﬁability problem in experimental design: given a design which model(s) can we identify? As mentioned the starting point is to represent a design D as a variety, namely the solution of a set of polynomial equations, or equivalently the design ideal, that is, the set of all polynomials interpolating the design points at zero. The principal result is that starting with a class of modelsM (usuallyM will be the set of all polynomials in d indeterminates) the quotient vector space M/ Ideal(D) yields a class of identiﬁable terms. The theory of Gro¨bner bases is used to characterise the design ideal and the quotient space. The following problems will be addressed in particular. (i) Which classes of polynomial models does a given design identify? (ii) Is a given model identiﬁable by a given design? (iii) What is confounding/aliasing in this context? (iv) What conditions must M satisfy so that the theory applies? This algebraic approach to identiﬁability in experimental design was introduced by Pistone and Wynn (1996).