Invariant Image Reparameterisation: Bridging Symbolic and Numerical Methods for Identifiability Analysis and Model Reduction

TL;DR

This paper introduces Invariant Image Reparameterisation (IIR), a numerical method for model reparameterisation and reduction that simplifies identifiability analysis by using derivative calculations at a single point.

Contribution

It develops a new approach to replace symbolic reparameterisation conditions with numerical Jacobian calculations, enabling efficient model reduction and identifiability analysis.

Findings

Invariant image provides a basis-independent parameter representation.

Single Jacobian determines reparameterisation space for certain models.

Method distinguishes minimal and non-minimal reductions, and strong and weak parameter identifiability.

Abstract

Structural and practical parameter non-identifiability issues are common when mathematical models are used to interpret data. Such issues motivate model reparameterisation and reduction methods. Here, we consider Invariant Image Reparameterisation (IIR), which asks when symbolic reparameterisation conditions can be replaced by numerical derivative calculations at a single reference point. The central object is the invariant image: a reduced, basis-independent representation of the parameter combinations controlling observable model behaviour. We show that when a one-to-one componentwise transformation makes observable behaviour depend only on fixed linear combinations of the transformed parameters, a single numerical Jacobian determines the associated lower-dimensional reparameterisation space. This includes models depending on monomial combinations of the original parameters. We also give a first-order invariance condition that distinguishes minimal from non-minimal but exact reductions via the invariant part of the local null space. In structurally identifiable but practically weakly informed settings, the same calculations separate strongly and weakly informed parameter combinations. The invariant image admits multiple coordinate representations: the SVD gives a default orthonormal basis ordered by local identifiability, while sparse monomial bases are often more interpretable. Treating these coordinates as interest parameters in Profile-Wise Analysis gives likelihood-based uncertainty quantification. We demonstrate the method on parameterised normal models with Poisson-limit, extended Poisson-limit, and non-limit cases, and on the repressilator, a nonlinear differential equation model of gene regulation. A Julia implementation of IIR, with these and further examples, is available at https://github.com/omaclaren/reparam.