Algebraic Tools for the Analysis of State Space Models

TL;DR

This paper surveys algebraic methods for analyzing state space models, focusing on structural identifiability, observability, and indistinguishability, and introduces new algebraic techniques for these analyses.

Contribution

It introduces novel algebraic approaches, including linear algebra for identifiability and Gr"obner bases for observability, enhancing analysis of ODE-based models.

Findings

Linear algebra method for identifiable functions

Use of Gr"obner bases for algebraic observability

Sufficient algebraic condition for indistinguishability

Abstract

We present algebraic techniques to analyze state space models in the areas of structural identifiability, observability, and indistinguishability. While the emphasis is on surveying existing algebraic tools for studying ODE systems, we also present a variety of new results. In particular: on structural identifiability, we present a method using linear algebra to find identifiable functions of the parameters of a model for unidentifiable models. On observability, we present techniques using Gr\"obner bases and algebraic matroids to test algebraic observability of state space models. On indistinguishability, we present a sufficient condition for distinguishability using computational algebra and demonstrate testing indistinguishability.