Identifiability of Polynomial Models from First Principles and via a Gr\"obner Basis Approach

TL;DR

This paper investigates the conditions under which hierarchical polynomial models are identifiable from a given design, comparing a practical approach with Gr"obner basis methods, and providing illustrative examples.

Contribution

It introduces necessary and sufficient conditions for model identifiability and develops a practitioner-led approach, contrasting it with Gr"obner basis techniques.

Findings

Derived conditions for model terms to be identifiable

Developed a practical approach for building identifiable models

Compared the approach with Gr"obner basis methods

Abstract

The relationship between a set of design points and the class of hierarchical polynomial models identifiable from the design is investigated. Saturated models are of particular interest. Necessary and sufficient conditions are derived on the set of design points for specific terms to be included in leaves of the statistical fan. A practitioner led approach to building hierarchical saturated models that are identifiable is developed. This approach is compared to the method of model building based on Gr\"{o}bner bases. The main results are illustrated by examples.