Framing local structural identifiability in terms of parameter symmetries

This paper establishes a theoretical link between standard differential algebra and Lie symmetry approaches to structural identifiability by introducing "parameter symmetries," proving that a parameter combination is locally structurally identifiable if and only if it is a differential invariant of these symmetries, and validating this framework through applications to biological models.

The Gist

Imagine you are a detective trying to solve a mystery, but you can't see the suspects (the parameters). You can only see the footprints they leave behind (the observed outputs). Your goal is to figure out exactly who the suspects are just by looking at the footprints.

In the world of science, this is called structural identifiability. It asks: "If I have perfect data, can I uniquely figure out the hidden numbers that drive my model?"

This paper introduces a new, clever way to solve this mystery using symmetries. Here is the breakdown of their discovery, explained simply.

1. The Old Way: The "Recipe" Approach

For decades, scientists used a method called Differential Algebra. Think of this like a chef trying to guess the ingredients of a soup just by tasting it.

• The chef writes down a complex recipe (the math equations).
• They simplify the recipe until it only lists the final taste (the output).
• They look at the numbers attached to the ingredients in that simplified list.
• The Problem: This tells you what you can figure out (e.g., "I know the total amount of salt and pepper combined"), but it doesn't tell you why you can't figure out the individual amounts of salt and pepper. It's like being told "the sum is 10" without knowing if the recipe allows you to swap 3 spoons of salt for 7 spoons of pepper without changing the taste.

2. The New Way: The "Shape-Shifter" Approach

The authors of this paper say, "Let's look at the soup from a different angle." They introduce a concept called Parameter Symmetries.

Imagine the hidden parameters are ingredients in a magical kitchen. A Parameter Symmetry is a magical transformation that swaps or changes the ingredients (e.g., turning 3 spoons of salt into 7 spoons of pepper) without changing the taste of the soup at all.

• The Analogy: If you can change the amount of salt and pepper in a specific way, and the soup tastes exactly the same, then you can never tell the difference between those two specific amounts. They are "indistinguishable."
• The Discovery: The authors realized that if a parameter (or a combination of parameters) cannot be changed by any of these magical transformations without altering the taste, then that parameter is identifiable. It is unique.

3. The "Universal Invariant": The Unchangeable Core

The paper introduces a term called a Universal Parameter Invariant. Let's call this the "Unchangeable Core."

• Imagine you have a set of magical rules (symmetries) that can twist and turn your ingredients.
• Some ingredients change when you apply the rules.
• Some combinations stay exactly the same.
• The Universal Invariant is the specific combination of ingredients that never changes, no matter how you twist the rules.

The Big Conclusion:
The paper proves a simple but powerful rule:

A parameter is identifiable if and only if it is part of the "Unchangeable Core."

If a number stays the same no matter how you magically reshuffle the model to keep the output the same, then you can find it. If it changes, you can't.

4. Why This Matters (The "CaLinInv" Recipe)

The authors created a three-step guide (called CaLinInv) to find these "Unchangeable Cores":

1. Canonical Coordinates (The Translation): Translate the complex model into a language that only talks about what you can see (the outputs).
2. Linearised Symmetry Conditions (The Magic Rules): Figure out exactly what magical transformations (swaps) are possible that keep the output looking the same.
3. Universal Invariants (The Core): Find the combinations that remain untouched by these magic rules.

5. Real-World Examples

The paper tested this on two real-life scenarios:

• The Glucose-Insulin Model: Imagine a model of how your body handles sugar. The old method might tell you that you can't tell the difference between two specific rates. The new method tells you exactly how those rates can swap (e.g., "If you double rate A, you must halve rate B, and the result is the same"). It gives you the map of the "magic swaps."
• The Tuberculosis Model: In a model of disease spread, they found that while you couldn't identify every single number individually, you could identify specific combinations (like the product of two rates). The new method confirmed this but also showed the exact "shape-shifting" rules that make those numbers indistinguishable.

Summary

Think of the old method as a list of answers (e.g., "You can find the sum, but not the parts").
The new method is like a map of the terrain. It shows you exactly how the hidden numbers can move around without changing the result.

• Identifiable: The number is stuck in a cage; it can't move without changing the result.
• Non-identifiable: The number is on a treadmill; it can run all day (change its value), but the result (the output) stays exactly the same.

By understanding these "treadmills" (symmetries), scientists can finally understand exactly which parts of their models are solvable and which are not, and why. This bridges the gap between two different mathematical worlds, making the process of model validation much clearer and more powerful.

Technical Summary

1. Problem Statement

Mechanistic modeling of dynamical systems (typically systems of Ordinary Differential Equations, ODEs) requires structural identifiability analysis to determine which model parameters can be uniquely estimated from observed outputs.

• The Gap: While the standard differential algebra approach is the established method for determining global structural identifiability (uniqueness over the entire parameter space), it relies on rewriting models as higher-order input-output equations. Conversely, Lie symmetry methods (specifically "full symmetries" acting on independent variables, states, outputs, and parameters) have been used for over two decades to analyze identifiability, often focusing on scaling transformations.
• The Challenge: The precise theoretical link between the differential algebra approach and full symmetry methods has remained elusive. Furthermore, existing symmetry-based approaches often rely on restrictive assumptions (e.g., only scaling symmetries) or face computational difficulties due to infinite-dimensional symmetry groups. There is a need to rigorously define how symmetries relate to local structural identifiability (uniqueness within a small neighborhood) and to unify these two methodological frameworks.

2. Methodology

The authors introduce a new theoretical framework centered on Parameter Symmetries and propose a three-step algorithmic procedure called CaLinInv.

A. Theoretical Framework: Parameter Symmetries

• Definition: A Parameter Symmetry is defined as a specific type of full Lie symmetry (a $C^\infty$ diffeomorphism) that acts solely on the model parameters ($\theta$) while leaving the independent variable (time $t$) and the observed outputs ($y$) invariant.
• Mechanism: These symmetries represent re-parametrizations of the model that preserve the observed output trajectory. If a parameter transformation $\hat{\theta}(\epsilon)$ leaves $y(t)$ unchanged, the parameters are not uniquely identifiable.
• Universal Parameter Invariants: The authors define Universal Parameter Invariants as functions of the parameters ($I(\theta)$) that remain constant under all possible parameter symmetries of a given model.
• Key Theorem (Theorem 1): A parameter $\theta_\ell$ is locally structurally identifiable if and only if it is a universal parameter invariant.
  • Proof Logic: If a parameter is identifiable, any transformation preserving the output must leave the parameter unchanged (invariant). Conversely, if a parameter is invariant under all output-preserving symmetries, it is locally identifiable.

B. The CaLinInv Algorithm

To apply this theory, the authors propose a three-step recipe:

1. Canonical Coordinates: Rewrite the original system of first-order ODEs into an equivalent system of higher-order ODEs depending only on the observed outputs ($y$) and parameters ($\theta$). This step mirrors the first step of the differential algebra approach.
2. Linearised Symmetry Conditions: Solve the linearised symmetry conditions to find the infinitesimal generators ($X^\theta = \sum \chi_\ell(\theta)\partial_{\theta_\ell}$) of the parameter symmetries. This involves solving a system of linear equations derived from the invariance of the output ODEs.
3. Universal Invariants: Calculate the universal parameter invariants by solving the linear partial differential equation $X^\theta(I) = 0$ using the method of characteristics. These invariants represent the locally identifiable parameter combinations.

3. Key Contributions

• Bridging the Gap: The paper establishes a rigorous mathematical link between the standard differential algebra approach and Lie symmetry methods. It proves that the coefficients extracted in the differential algebra approach are, in fact, universal parameter invariants.
• Generalization of Identifiability: The framework generalizes previous work (e.g., Castro and de Boer) which focused primarily on scaling symmetries. The authors show that relying solely on scalings can be misleading and that the full class of parameter symmetries must be considered.
• Local vs. Global Distinction: The method explicitly distinguishes between local and global identifiability.
  • The differential algebra approach typically yields globally identifiable quantities (often polynomial combinations).
  • The parameter symmetry approach yields locally identifiable quantities (differential invariants).
  • The paper demonstrates that globally identifiable quantities are always functions of locally identifiable (universal) invariants.
• Algorithmic Innovation: The introduction of the CaLinInv recipe provides a systematic, automated pathway to compute local structural identifiability and the specific parameter transformations (symmetries) that cause non-identifiability.

4. Results and Applications

The authors validated their approach using four examples:

1. Toy Model (Decoupled ODEs):

   • Result: Confirmed that $\lambda$ and $\kappa_1 + \kappa_2$ are identifiable.
   • Insight: The symmetry approach identified the specific transformation $\Gamma: (\kappa_1, \kappa_2) \to (\kappa_1+\epsilon, \kappa_2-\epsilon)$ that preserves the output, explaining why individual $\kappa$ values are non-identifiable.

2. Linear Coupled Model:

   • Result: Demonstrated a case where local and global identifiability differ. Parameters $a$ and $c$ are locally identifiable (distinct values in a small neighborhood yield different outputs) but not globally identifiable (two distinct sets of $\{a, c\}$ yield the same output globally).
   • Insight: The symmetry method correctly identified $a$ and $c$ as locally identifiable invariants, whereas the differential algebra approach identified only the sum and product (global invariants).

3. Glucose-Insulin Model:

   • Result: Analyzed a non-autonomous model with 5 parameters.
   • Findings: Identified $p_1, p_3, V_p$ as locally identifiable. Found that $p_2$ and $p_4$ are non-identifiable individually, but their product $p_2 p_4$ is a universal invariant.
   • Symmetry: Explicitly derived the scaling symmetry $\Gamma: (p_2, p_4) \to (p_2 e^\epsilon, p_4 e^{-\epsilon})$.

4. Epidemiological SEI Model (Tuberculosis):

   • Result: Applied to a 3-state, 9-parameter model.
   • Findings: Identified 7 universal parameter invariants (e.g., $\mu_I, \mu_S, \delta+\mu_E, \beta/k_I$).
   • Visualization: The authors visualized the action of the parameter symmetries in 2D and 3D subspaces of the 9D parameter space, showing curves of indistinguishable model behaviors.

5. Significance

• Theoretical Unification: The paper resolves a long-standing ambiguity by proving that structural identifiability is fundamentally a problem of finding differential invariants of parameter symmetries.
• Enhanced Insight: Unlike the differential algebra approach, which provides what is identifiable, the symmetry approach provides how parameters can be transformed to produce identical outputs. This is crucial for experimental design and understanding model degeneracy.
• Computational Feasibility: The CaLinInv recipe reduces the problem of finding symmetries to solving linear systems (via Gaussian elimination) and first-order PDEs, making it amenable to automation using symbolic solvers (e.g., SymPy).
• Future Directions: The authors suggest that this framework can be extended to Partial Differential Equations (PDEs) and that the CaLinInv recipe can be integrated into existing software tools to provide a more comprehensive analysis of mechanistic models.

In summary, this work re-frames structural identifiability from an algebraic coefficient-extraction problem to a geometric symmetry problem, offering a more robust and informative tool for analyzing complex biological and physical models.