Parameter identifiability and input-output equations

TL;DR

This paper clarifies the relationship between structural parameter identifiability and input-output equations in differential models, establishing conditions under which these notions coincide and describing the algebraic structure of identifiable functions.

Contribution

It rigorously proves the connections between identifiability and input-output identifiability, and characterizes the field of identifiable functions using characteristic sets.

Findings

Identifiability implies input-output identifiability.

The notions coincide if the model lacks rational first integrals.

The field of input-output identifiable functions is generated by coefficients of a minimal characteristic set.

Abstract

Structural parameter identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. One of the standard approaches to assessing this problem is via input-output equations and, in particular, characteristic sets of differential ideals. The precise relation between identifiability and input-output identifiability is subtle. The goal of this note is to clarify this relation. The main results are: 1) identifiability implies input-output identifiability; 2) these notions coincide if the model does not have rational first integrals; 3) the field of input-output identifiable functions is generated by the coefficients of a "minimal" characteristic set of the corresponding differential ideal. We expect that some of these facts may be known to the experts in the area, but we are not aware of any articles in which these facts are stated precisely and rigorously proved.