Modular Identifiability Analysis

• Modular identifiability is the process of decomposing a mathematical model into independent modules that preserve the uniqueness of parameter estimation via input–output equivalence.
• This approach employs differential algebra methods, rescaling, and modular decomposition to analyze each module independently and combine results for global identifiability.
• It is applied in systems biology, control systems, and network science to streamline parameter estimation, optimize experimental design, and overcome computational challenges.

Modular identifiability refers to the analysis, preservation, and exploitation of identifiability properties under decomposition, transformation, or structural modification of mathematical models—typically those arising in systems biology, control, network science, and related fields. The central concept is that identifiability (uniqueness of parameter determination from observable data) is preserved across input–output equivalent representations and can often be analyzed, ensured, or restored by treating system modules independently and then composing their identifiability properties.

1. Input–Output Equivalence and Identifiability Preservation

A model’s identifiability is uniquely determined by its input–output map $\Phi: p \mapsto y$. If two models $M_1: \dot{x} = f(x, t, u, p),\, y = g(x, t, p)$ and $$M_2: \dot{\tilde{x}} = \tilde{f}(\tilde{x}, t, u, p),\, y = \tilde{g}(\tilde{x}, t, p)$$ are input–output equivalent—meaning they generate identical output trajectories for all inputs, initial conditions, and $p$—then they share identical identifiability structure. This is formalized in Lemma 2.3: for almost every choice of data, identical input–output sets $\mathcal{S}_1(y) = \mathcal{S}_2(y)$ guarantee that the structure (location, combinations) of identifiable and unidentifiable parameters is preserved.

This principle underlies all modular identifiability: modules or model fragments that interact via measured signals maintain their identifiability properties when embedded within a larger input–output equivalent model, provided the transformation or extension preserves the external signal structure.

$p_1, p_2 \in c^{-1}(c(p)) \iff \text{%%%%5%%%% and %%%%6%%%% have identical identifiability properties}$

where $c(p)$ is the coefficient map extracted from the input–output equations via characteristic set, substitution, or (differential) Gröbner bases.

2. Modular Decomposition and Aggregation of Identifiability

If a complex system can be decomposed into $n$ submodels or modules $M_1, \dots, M_n$, each of which is parameterized and coupled through measured quantities (e.g., known signals $u$, outputs $y$, time $t$, and structural parameters $p$), then, under prescribed coupling, the identifiability of the overall system is equivalent to the identifiability of the individual modules:

$\text{%%%%14%%%% is identifiable} \iff \text{each %%%%15%%%% is identifiable}$

(as in Propositions 2.7, 2.8). That is, if each module’s parameterization can be uniquely determined from its own input–output behavior (possibly conditioned on the signals couplings from other modules), and all inter-module signals are measured or known, then the composition preserves identifiability globally.

This property enables modular analysis strategies: one can analyze identifiability at the module level (often greatly simplifying calculations) and infer global identifiability from local results, rather than tackling the full coupled system directly.

3. Input–Output Equation Generation and Differential Algebra Methods

The input–output equations that encapsulate observable behavior are central to differential algebraic identifiability analysis. Traditionally, these are generated using the characteristic set method (a variant of Ritt’s pseudodivision), but the paper highlights a broader suite of techniques which are input–output equivalence preserving:

• Substitution and elimination (with care to not cancel vanishings on the solution manifold)
• Gröbner bases or differential Gröbner bases (e.g., via Mansfield’s approach)

In these techniques, parameters map to coefficients $c(p)$ in the input–output equations (e.g., monic differential polynomials in $y$ and $u$). The injectivity of the parameter-to-coefficient map is necessary and sufficient for identifiability of parameter combinations. Each module's input–output elimination, carried out locally, provides its own $c_i(p)$; global identifiability is obtained by concatenating these across modules and analyzing parameter mapping collectively. In single-output systems, this yields unique input–output equations per output; in multi-output systems, one may work with mutually autoreduced sets or module-wise elimination.

4. Transformations That Maintain Identifiability

Several classes of transformations leave identifiability unchanged due to preservation of input–output equivalence:

Non–first order ODE models: Any $n$th order ODE is convertible to a first-order system via standard variable change $x_1 = z, x_2 = \dot{z}, \dots, x_n = z^{(n-1)}$; because the transformation is bijective (modulo appropriate initial conditions), identifiability is fully preserved.

Nondimensionalization and Rescaling: When variables and/or parameters are rescaled, e.g., $\tilde{x} = \varphi(x, p)$, and the transformation reduces the number of effective parameters, then only invariant combinations (i.e., the $\varphi_i(p)$) remain identifiable. The parameter space contracts to combinations observable in the rescaled input–output map.

Model Reducibility: If state variables (e.g., $x_3$) are not connected to any measured output, parameters unique to those sub-systems are structurally unidentifiable. Dropping such reducible subsystems projects the identifiability map to the observable subspace, isolating relevant identifiable combinations.

The broad principle emerges: any mapping or transformation that preserves the external input–output behavior cannot alter the fundamental structure of the parameter identifiability map.

5. Modular Identifiability in Workflow: Practice and Applications

The modular approach is crucial for analyzing large-scale or complex models encountered in biomodeling, pharmacokinetics, cell signaling, or mechanistic engineering models. Modular identifiability practice proceeds as follows:

1. Decompose the system into modules interacting via measured or known signals.
2. Analyze each module’s identifiability independently—derive local input–output equations and characterize the identifiability of each module’s parameters or parameter combinations.
3. Aggregate the local identifiability results. If module coupling is via measured/known signals and transformations are input–output equivalent, then the identifiability structure of the global system is simply the direct sum (or product) of modular structures.
4. Preserve identifiability under any transformation (rescaling, reduction, or change of presentation) that leaves the global input–output behavior unchanged.

Such workflow allows modelers to localize identifiability issues, target experimental design or intervention (e.g., which module requires additional measurement), and avoid intractable global parameter searches.

6. Implementation Considerations: Algorithms and Limitations

Algorithmic approaches are informed by the modular perspective:

• Classical differential algebraic approaches can be applied module-wise, with characteristic set, substitution, or differential Gröbner bases computed per module.
• The method is computationally tractable for large systems provided module decompositions result in lower-dimensional subsystems.
• Care must be taken with transformations to ensure they are truly input–output equivalent. Failure to account for unobservable variable/parameter interactions or unmeasured couplings may lead to overoptimistic identifiability conclusions.
• In high-dimensional or nonlinear models, the calculation of input–output equations or their coefficients may be the computational bottleneck, but a modular approach often eliminates algebraic redundancy.
• Key limitations: identifiability can only be preserved under modular decomposition if all inter-module signals are known or measured; latent, unmeasured interconnections may confound the aggregation step.

7. Summary Table: Transformations and Modular Identifiability

Transformation Type	Effect on Identifiability	Example
Non–first order to first–order ODE	Preserved	$z^{(n)} = h(\dots) \mapsto x_i = z^{(i-1)}$
Variable/parameter rescaling (nondim.)	Only combinations (functions of parameters) survive	$x_2 = p_2 x_1, q = p_1 p_2 \implies$ ident. of $q$
Removing reducible submodels	Irrelevant parameters lost; ident. projected to observed space	Dropping $x_3$ unconnected to measured $y$
Modular decomposition & recomposition	Identifiability is inherited if modules coupled via known sigs.	As in $M_1,\dots,M_n$ such that input–output agrees

Modular identifiability is thus a foundational concept in structural modeling, enabling scalable, structurally transparent identifiability analysis, robust to transformations and model reorganizations that preserve input–output equivalence (Eisenberg, 2013).