Minimal average degree aberration and the state polytope for experimental designs

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.

💡 Research Summary

The paper addresses the fundamental problem of determining which polynomial regression models are estimable from a given experimental design. Traditional design assessment relies heavily on the concept of “aberration,” which orders designs by the presence of low‑order interaction terms. While useful, aberration does not directly quantify the overall complexity of the models that can be identified, especially when variable importance varies across factors.

The authors propose an algebraic framework based on Gröbner bases of the design ideal. For a design D ⊂ ℝ^k, the set of design points defines an ideal I(D) in the polynomial ring K