Source-linked AI summary
Observability and Structural Identifiability of Nonlinear Biological Systems
Alejandro F. Villaverde
TL;DR
Biological models require state and parameter inference from outputs, but feasible measurements can be limited and structural identifiability does not ensure accurate estimates from real data. This review presents a differential-geometry approach to analysing observability and structural identifiability, and discusses recent results and open questions, including how inputs affect identifiability.
Problem
Method
The review synthesizes differential-geometry methods for jointly analysing observability and structural identifiability in nonlinear models, including extensions for time-varying inputs.
Source: §1 · p. 2, §4.3 · p. 8Results
Some models unidentifiable under a constant input become identifiable with a continuous time-varying input, and extended Lie derivatives can determine the input derivatives required.
Source: §4.3 · pp. 8–9Takeaways & Limitations
Input choice can matter for structural identifiability, and extended Lie derivatives can support experimental design by determining required input variation.
Source: §4.3 · p. 8, §5 · p. 11The review treats structural local identifiability; global identifiability requires other approaches.
Source: §3.3 · p. 6
Abstract
from arXiv · showhide
Observability is a modelling property that describes the possibility of inferring the internal state of a system from observations of its output. A related property, structural identifiability, refers to the theoretical possibility of determining the parameter values from the output. In fact, structural identifiability becomes a particular case of observability if the parameters are considered as constant state variables. It is possible to simultaneously analyse the observability and structural identifiability of a model using the conceptual tools of differential geometry. Many complex biological processes can be described by systems of nonlinear ordinary differential equations, and can therefore be analysed with this approach. The purpose of this review article is threefold: (I) to serve as a tutorial on observability and structural identifiability of nonlinear systems, using the differential geometry approach for their analysis; (II) to review recent advances in the field; and (III) to identify open problems and suggest new avenues for research in this area.
1 Introduction
Observability concerns whether system states can be inferred from outputs, while structural identifiability concerns whether parameters can be determined; treating parameters as constant states makes the latter a special case of the former. This review focuses on differential-geometry methods for applying both concepts to biological models, alongside recent developments and open questions.
1 Introduction
- Observability asks whether internal states can be inferred from outputs; structural identifiability asks whether parameters can be determined, and is a special case of observability when parameters are constant states.Both properties depend on the model equations and output definition, not measurement frequency or accuracy.Source: §1 · p. 1
- Biological applications often face tighter limits on feasible measurements than engineering applications, making parameter identification more challenging.This motivates deeper study of structural-identifiability methods in biological modelling.Source: §1 · p. 1
- Observability and structural identifiability are central to system identification, with prior reviews and books covering biological identifiability and nonlinear mechanical systems.The introduction points readers to references on structural and practical identifiability and related concepts.Source: §1 · p. 2
- The review focuses on differential-geometry tools for biological models, surveys recent developments, and identifies open problems and research directions.Other symbolic, semi-numerical, and numerical approaches are noted, but comparing them is outside the review’s scope.Source: §1 · p. 2
2 Motivation: implications of unobservability and unidentifiability in biological models
The section shows how unidentifiability can leave states unresolved and compromise biological interpretation, while distinguishing structural from practical identifiability. A glucose-insulin example illustrates the issue, and the section outlines responses such as measuring additional states or simplifying the model.
2 Motivation: implications of unobservability and unidentifiability in biological models
- Identifiability and observability analyses can be difficult even for small systems, yet their deficiencies can compromise biological insight.Which property matters most depends on whether a model is used to infer parameters or predict unmeasured states.Source: §2 · p. 2
- In the βIG glucose-insulin model, measuring glucose and β-cell mass leaves p and si structurally unidentifiable, while insulin remains unobservable.Halving si and doubling p preserves the outputs; knowing either parameter can overcome this limitation.Source: §2 · p. 2, Figure 1 · p. 3
- A cardiac troponin T model was revised to remove redundancies after structural unidentifiability was recognized as a potential source of wrong conclusions.The revised model was equivalent to the original and structurally identifiable.Source: §2 · p. 3
- Structural unidentifiability can coexist with observable states, so both properties should be analysed to reduce the risk of false conclusions.In the example, replacing c with c1 + c2 makes the new parameters unidentifiable while I remains observable.Source: §2 · p. 3
- Structural identifiability does not guarantee accurate parameter estimates: insufficient or poor-quality data can leave a model practically unidentifiable.Cancer-chemotherapy models illustrate the risk of incorrect cell-cycle distributions and suboptimal therapies.Source: §2 · p. 3
- More calibration data may address practical unidentifiability, but cannot remove structural unidentifiability unless the model output or structure changes.Trying to fix structural unidentifiability by denser or longer sampling can waste time and resources.Source: §2 · p. 4
- When parameter estimates are the goal, possible responses include direct parameter measurements, measuring additional states, or reducing the number of parameters.Parameters can be reduced by fixing values from the literature or merging unidentifiable parameters into an identifiable one.Source: §2 · p. 4
- A structurally unidentifiable model may still serve state-inference goals if the states of interest remain observable.If states are not observable, the section notes that remedies can be similar to those for structural unidentifiability.Source: §2 · p. 4
3 Background: observability and structural identifiability
The section develops observability and structural identifiability from state distinguishability, then presents rank-based tests for linear and nonlinear systems. It treats parameter identifiability as observability and illustrates the analysis with a nonlinear model whose states and parameters are not all recoverable.
3 Background: observability and structural identifiability
- Observability concerns distinguishing internal states from output histories; unlike reconstructability, it uses present and future measurements rather than present and past measurements.States are indistinguishable when their output trajectories coincide for all t ≥ t0.Source: §3 · p. 4
3.1 Observability of linear systems
- For linear time-invariant systems, the observability rank condition is necessary and sufficient for complete observability: the observability matrix must have rank n.The matrix is formed from the output matrix and its successive products with the system matrix.Source: §3.1 · p. 5
3.2 Observability of nonlinear systems
- For nonlinear systems with constant input, full rank of the Lie-derivative observability matrix is sufficient for local observability, but is not strictly necessary or a guarantee of global observability.The condition can miss indistinguishable distant states.Source: §3.2 · p. 6
3.3 Structural local identifiability as observability
- Structural identifiability is treated as a particular case of observability by augmenting the state with constant parameters; the resulting analysis addresses local, not global, identifiability.The paper notes that global structural identifiability requires other approaches.Source: §3.3 · p. 6
3.4 Example: observability and structural identifiability analysis of a nonlinear model
- The example identifies insulin I as unobservable, p and si as structurally unidentifiable, and γ, c, and α as locally identifiable; only si·p and si·I are identifiable.The augmented matrix reaches rank 7, but remains non-full-rank when built with six Lie derivatives.Source: §3.4 · p. 7
4 Recent developments
Recent work develops computational tools and examines how initial conditions, inputs, model symmetries, and parameter sensitivity shape observability and structural identifiability. It also distinguishes structural unidentifiability from related concepts such as sloppiness and dynamical compensation.
4.1 Computational implementations of the rank conditions
- STRIKE-GOLDD supports partial analyses and model decomposition for rank-condition calculations, while sum-of-squares rank assessment can become costly as problem size grows.The Exact Arithmetic Rank method offers a numerical rank alternative for rational systems.Source: §4.1 · p. 8
4.2 Accessibility and the role of initial conditions
- Rank conditions hold for almost all state and parameter values, but specific initial conditions can reduce identifiability; loss of accessibility is one possible cause, not the only one.A method for finding potentially problematic initial-condition vectors scales poorly with system size.Source: §4.2 · p. 8
4.3 The role of inputs
- For known differentiable time-varying inputs, extended Lie derivatives extend the differential-geometric procedure, and their nonzero input-derivative terms can restore full rank.The method can also guide experimental design by identifying how many nonzero input derivatives are required.Source: §4.3 · pp. 8–9
- In the two-compartment example, constant input leaves parameters structurally unidentifiable, whereas a ramp input makes them structurally identifiable.The rank test gives rank 5 under the constant-input procedure, with x2 observable but all parameters structurally unidentifiable; extended Lie derivatives give rank 6 with five derivatives for the ramp case.Source: §4.3 · p. 9, Figure 2 · p. 9
- Unknown time-varying inputs can be treated as disturbances; an extended observability rank condition checks system observability, but not the inputs themselves in its published form.The approach augments the state with the unknown input and its derivatives.Source: §4.3 · p. 9
4.4 Model symmetries and identifiable combinations
- Methods for finding identifiable parameter combinations include similarity transformations, differential algebra, and Lie-symmetry approaches, but no generally applicable automatic procedure exists.STRIKE-GOLDD uses a null-space calculation to derive identifiable combinations.Source: §4.4 · p. 10
4.5 Sloppiness, dynamical compensation, and structural identifiability
- Sloppiness describes outputs sensitive to stiff parameters but largely insensitive to sloppy ones; it relates to practical identifiability and does not imply structural unidentifiability.The account links sloppiness to a gap between large and small Fisher information matrix eigenvalues.Source: §4.5 · p. 10
- Dynamical compensation concerns a circuit maintaining its dynamic behaviour after a parameter change and is distinct from structural unidentifiability under its more detailed definition.The original output-invariance definition was equivalent to structural unidentifiability; an alternative definition was proposed to distinguish the phenomena.Source: §4.5 · pp. 10–11
5 Open problems and future directions
Despite decades of use and advances that have increased applicability, the differential geometry approach has unresolved limits and open questions concerning inputs, computational demands, and predictive accuracy.
5 Open problems and future directions
- Despite decades of use and advances increasing applicability, the approach yields only local results; differential algebra can provide global structural identifiability results.The review suggests differential algebra could inspire or be hybridized with differential geometry techniques for global analyses.Source: §5 · p. 11
- Future implementations should reduce computational burden, potentially using parallelization and high-performance computing to support increasingly large biological models.The review presents these techniques as possible ways to facilitate applying the analyses to larger models.Source: §5 · p. 11
- Some systems unidentifiable under one constant input can become identifiable with time-varying or multiple constant inputs, but input equivalence and unmeasured-input analysis remain unresolved.Multiple constant inputs can be equivalent to one piecewise constant input.Source: §5 · p. 11
- The relationship between observability or identifiability and predictive accuracy remains unclear: their absence can lead to prediction errors, but does not necessarily do so.Further insight into requirements for accurate predictive modelling is identified as a valuable contribution.Source: §5 · p. 11