Invariant Image Reparameterisation

This repository contains a Julia implementation of methods described in "Invariant Image Reparameterisation: A Unified Approach to Structural and Practical Identifiability and Model Reduction". A preprint is available from arxiv.org/abs/2502.04867.

Overview

The code implements methods for

• Analysing structural and practical parameter identifiability in mathematical models in a unified way
• Finding identifiable and nonidentifiable nonlinear (monomial) parameter combinations
• Model reparameterisation techniques based on identifiable/nonidentifiable parameter combinations
• Profile likelihood, in the form of Profile-Wise Analysis (PWA), for uncertainty quantification for both parameters and predictions

Installation

This package requires Julia 1.0 or higher. Install the required packages:

using Pkg
Pkg.add([
    "Distributions", 
    "ForwardDiff",
    "LaTeXStrings",
    "Measures",
    "NLopt",
    "Plots"
])

This package also uses the following standard libraries:

• LinearAlgebra
• SparseArrays

Structure

The codebase is organized as:

Main Module

• ReparamTools.jl - The primary module that users interact with, providing the interface for parameter transformations, identifiability analysis, and likelihood-based inference

Examples

• transport_model.jl - Demonstration of parameter identifiability analysis for a transport model (diffusive flow in composite medium)
• mm_model.jl - Example using the Michaelis-Menten/Monod model
• stat_model.jl - Simple statistical model example

Implementation Files

• core.jl - Internal implementation of likelihood and identifiability methods
• parameterizations.jl - Implementation of parameter transformation methods
• utils.jl - Common utility functions and helpers
• visualization.jl - Internal plotting and visualization tools

Usage

Basic model setup (using the simple stat_model.jl as an example):

include("../ReparamTools.jl") # Assuming working in examples directory
using .ReparamTools 

# Define model through auxiliary mapping (maps parameters to data distribution parameters)
ϕ_xy = xy -> [xy[1]*xy[2], xy[2]]

# Create distribution mapping (specifies how distribution depends on parameters)
distrib_xy = xy -> Normal(ϕ_xy(xy)[1], sqrt(ϕ_xy(xy)[1]*(1-ϕ_xy(xy)[2])))

# Construct likelihood using data
lnlike_xy = construct_lnlike_xy(distrib_xy, data)

See the example files for complete analyses including identifiability analysis, model reduction, parameter combinations, and inference.

Examples

The repository includes several examples demonstrating the methods:

1. Statistical Model Example (stat_model.jl)

   • Demonstrates basic structural and practical parameter identifiability analysis for model with scalar output

2. Transport Model (transport_model.jl)

   • Demonstrates basic structural and practical parameter identifiability analysis for model with vector output on a fine grid and coarser observation grid
   • Includes predictive uncertainty quantification

3. Michaelis-Menten Model (mm_model.jl)

   • Similar to diffusion example but includes the use of an ODE solver as part of model definition

Each example includes:

• Model definition
• Identifiability analysis
• Parameter transformations
• Profile likelihood calculations
• Visualisation of results

License

This project is licensed under the MIT License - see the LICENSE file for details