Minimal average degree aberration and the state polytope for experimental designs

TL;DR

This paper introduces a new algebraic approach using minimal average degree aberration and the state polytope to identify estimable polynomial models in experimental designs, providing bounds and algorithms applicable to various designs.

Contribution

It presents a novel algebraic method based on Groebner bases for selecting models with minimal average degree, extending aberration concepts to all experimental designs.

Findings

The method yields models with minimal average degree in a systematic way.

Bounds are derived for the criteria, enabling asymptotic estimability analysis.

An algorithm is provided for practical implementation.

Abstract

For a particular experimental design, there is interest in finding which polynomial models can be identified in the usual regression set up. The algebraic methods based on Groebner bases provide a systematic way of doing this. The algebraic method does not in general produce all estimable models but it can be shown that it yields models which have minimal average degree in a well-defined sense and in both a weighted and unweighted version. This provides an alternative measure to that based on "aberration" and moreover is applicable to any experimental design. A simple algorithm is given and bounds are derived for the criteria, which may be used to give asymptotic Nyquist-like estimability rates as model and sample sizes increase.