Abstract

A basic application of algebraic statistics to design and analysis of experiments considers a design as a zero-dimensional variety and identifies it with the ideal of the variety. Then, a subset of a standard basis of the design ideal is used as support for identifiable regression models. Estimation of the model parameters is performed by standard least square techniques. We consider this identifiability problem in the case where more than one measurement is taken at a design point.

Introduction

The application of algebraic geometry to design and analysis of experiments started with (Pistone and Wynn 1996). There a design D, giving settings for experiments, is seen as a finite set of distinct points in ℝk. This is interpreted as the zero set of a system of polynomial equations, which in turn are seen as the generator set of a polynomial ideal (see Chapter 1). The design D is uniquely identified with this ideal called the design ideal and indicated with Ideal (D). Operations over designs find a correspondence in operations over ideals, e.g. union of designs corresponds to intersection of ideals; problems of confounding are formulated in algebraic terms and computer algebra software is an aid in finding their solutions; and a large class of linear regression models identifiable by D is given by vector space bases of a ring, called the quotient ring modulo Ideal (D) and indicated as R/ Ideal(D). This was the beginning of a successful stream of research which, together with the application of algebraic geometry to contingency table analysis covered in the first part of this volume, went under the heading of Algebraic Statistics (Pistone et al. 2001).