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Algorithm for globally identifiable reparametrizations of ODEs

Sebastian Falkensteiner, Alexey Ovchinnikov, J. Rafael Sendra

TL;DR

This paper presents a procedure for replacing, if possible, the ODE model with an equivalent one that has globally identifiable parameters, and derives this as an algorithm for one-dimensional ODE models and then reuse this approach for higher-dimensional models.

Abstract

Structural global parameter identifiability indicates whether one can determine a parameter's value in an ODE model from given inputs and outputs. If a given model has parameters for which there is exactly one value, such parameters are called globally identifiable. Given an ODE model involving not globally identifiable parameters, first we transform the system into one with locally identifiable parameters. As a main contribution of this paper, then we present a procedure for replacing, if possible, the ODE model with an equivalent one that has globally identifiable parameters. We first derive this as an algorithm for one-dimensional ODE models and then reuse this approach for higher-dimensional models.

Algorithm for globally identifiable reparametrizations of ODEs

TL;DR

This paper presents a procedure for replacing, if possible, the ODE model with an equivalent one that has globally identifiable parameters, and derives this as an algorithm for one-dimensional ODE models and then reuse this approach for higher-dimensional models.

Abstract

Structural global parameter identifiability indicates whether one can determine a parameter's value in an ODE model from given inputs and outputs. If a given model has parameters for which there is exactly one value, such parameters are called globally identifiable. Given an ODE model involving not globally identifiable parameters, first we transform the system into one with locally identifiable parameters. As a main contribution of this paper, then we present a procedure for replacing, if possible, the ODE model with an equivalent one that has globally identifiable parameters. We first derive this as an algorithm for one-dimensional ODE models and then reuse this approach for higher-dimensional models.
Paper Structure (20 sections, 11 theorems, 111 equations, 3 algorithms)

This paper contains 20 sections, 11 theorems, 111 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential algebra and identifiability
  4. State-space realizations of IO-equations and parametrizations
  5. Simplifying coefficients of parametrizations
  6. Polynomial realizations
  7. First-order IO-equations
  8. Technical preparation: polynomial parametrizations of curves
  9. Technical results: polynomial realizations
  10. Main theorem and algorithm for first-order equations
  11. Higher order realizations
  12. Additional subtlety
  13. General approach, originating from first-order equations
  14. Realizations with identifiable parameters
  15. Preliminaries
  16. ...and 5 more sections

Key Result

Theorem 1

Let be a rational parametrization of $\mathbb{V}(\mathcal{F})$ with the invertible $d \times d$ submatrix $\mathcal{M}$ as in eq-invertiblesubmatrix. Then $\mathcal{P}$ defines a realization of $\mathcal{F}$ if and only if In the affirmative case, the realization is

Theorems & Definitions (44)

  • Definition 1
  • Definition 2: IO-identifiability
  • Definition 3: IO-equations
  • Example 1
  • Definition 4: Realizability
  • Example 2: Lotka-Volterra
  • Definition 5: Parametrizations
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 34 more