arxiv: 2605.13504 · v1 · submitted 2026-05-13 · 📊 stat.ME · math.AP· math.DS· math.PR· q-bio.QM

Recognition: unknown

Structural identifiability of partially-observed stochastic processes: from single-particle trajectories to total particle density data

Alexander P. Browning, Arianna Ceccarelli, Ruth E. Baker

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:48 UTC · model grok-4.3

classification 📊 stat.ME math.APmath.DSmath.PRq-bio.QM

keywords structural identifiabilitystochastic processessingle-particle trajectoriesparticle densitydifferential algebrapartial differential equationsinitial conditions

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The pith

A new methodology reveals that single-particle trajectories make spatio-temporal stochastic process parameters structurally identifiable, whereas total particle density data makes them only locally identifiable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a method to assess structural identifiability in spatio-temporal stochastic processes depending on the observation type. Using individual-based descriptions for trajectories and a derived PDE with differential algebra for density data, it demonstrates full identifiability from trajectories but local from densities. A novel Taylor expansion via characteristic equations incorporates initial conditions into the analysis. This matters because it clarifies when parameters can be uniquely recovered from ideal data before attempting estimation from real experiments. The critical role of initial conditions emerges clearly in the density case.

Core claim

For the class of spatio-temporal stochastic processes considered, the parameters are structurally identifiable from single-particle trajectory data using the individual-based model, but only locally identifiable from total particle density data using the PDE representation and differential algebra approach. The novel characteristic-equation Taylor expansion method identifies additional parameter combinations involving the initial conditions, which are shown to be critical for the identifiability analysis.

What carries the argument

Differential algebra approach applied to the PDE representation of particle density evolution, augmented by a Taylor expansion constructed from characteristic equations to handle initial conditions.

If this is right

Parameters of the model can be uniquely recovered in principle from single-particle trajectory observations.
From total particle density measurements, multiple parameter sets may produce identical density profiles, leading to local identifiability only.
Initial conditions play a determining role in which parameter combinations are identifiable from density data.
The methodology allows comparison of identifiability across different data types for the same underlying stochastic process.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If experiments can only provide density data, additional prior information on initial conditions would be needed to achieve unique identification.
The framework could be tested on other stochastic models beyond the one application shown, such as those with birth-death processes in space.
Design of future experiments might favor single-particle tracking techniques to ensure structural identifiability before fitting.
Local identifiability from density data implies that numerical estimation algorithms may converge to different local solutions depending on starting points.

Load-bearing premise

The differential algebra approach remains valid when applied to the PDE derived from the underlying stochastic process, and the Taylor expansion from characteristic equations correctly captures the identifiable combinations involving initial conditions.

What would settle it

Simulating density data from a known parameter set in the applied model and checking whether optimization or algebraic solving recovers only one parameter vector or multiple distinct ones that fit the data equally well.

read the original abstract

The increasing availability of experimental data has intensified interest in calibrating stochastic models, raising fundamental questions about parameter identifiability. Structural identifiability determines whether parameters can be uniquely recovered from idealised, noise-free data, a prerequisite to allow for parameter estimation. However, existing methods to assess structural identifiability are not generally applicable to stochastic processes. We develop a methodology to analyse structural identifiability for a class of spatio-temporal stochastic processes. We investigate how identifiability depends on the type of available data, distinguishing between single-particle trajectories and total particle density measurements. For trajectory data, we use the individual-based model description that explicitly represents single-particle dynamics. For population-level data, we derive a partial differential equation model representation, that describes the evolution of total particle density, and apply a differential algebra approach, common to ordinary differential equations analysis. We further introduce a novel method to study the initial condition, based on characteristic equations to construct a Taylor expansion of the density evolution, enabling identification of additional identifiable parameter combinations. We apply our methodology to a model, and show it is identifiable with trajectory data but only locally identifiable with density data, and demonstrate the critical role of initial conditions in the identifiability analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Trajectory data gives full structural identifiability while density data only gives local identifiability for their example stochastic process, with the new characteristic-equation expansion making initial conditions part of the identifiable set.

read the letter

The main thing to know is that for the concrete model they study, single-particle trajectories allow unique recovery of the parameters while total density measurements only pin them down locally, and their handling of initial conditions is what reveals the difference. They work directly from the individual-based stochastic description for the trajectory case. For density data they derive the corresponding PDE, apply differential algebra to extract the identifiable combinations, and then use characteristic equations to generate a Taylor expansion in time that folds the initial data into the algebraic relations. That last step is the part that is not standard in prior identifiability work on ODEs or PDEs. The comparison itself is practical: it shows why an experimenter might prefer tracking individuals over bulk density if parameter recovery is the goal. The example is worked out far enough to make the distinction visible, which is the useful output. The soft spot is whether differential algebra carries over without extra conditions once spatial derivatives appear in the PDE. Differential algebra is usually stated for polynomial ODEs; the paper needs to confirm that the derived PDE stays polynomial in the observables and that the ranking of derivatives still produces a closed set of relations. The characteristic expansion also assumes the PDE has a well-defined characteristic structure. If the underlying process produces a parabolic diffusion term rather than a first-order hyperbolic one, the expansion method may need adjustment and could miss relations involving second spatial derivatives. The abstract does not show the explicit polynomial form or the algebraic independence check, so the strength of the claim rests on how cleanly those steps are verified in the full text. This is aimed at people who fit stochastic spatio-temporal models to biological or physical data and need to decide what kind of measurements will actually constrain their parameters. A reader who already knows differential algebra or works with individual-based models will see the value in the data-type comparison and the initial-condition treatment. The paper deserves a serious referee because the question is real and the combination of tools is new for this setting, even if the PDE step needs tighter justification. I would send it to review.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a methodology for structural identifiability analysis of spatio-temporal stochastic processes. For single-particle trajectory data it employs the individual-based model directly; for total particle density data it derives a PDE representation and applies differential-algebra techniques, augmented by a novel characteristic-equation Taylor expansion to incorporate initial conditions. The approach is illustrated on an example model, which is shown to be fully identifiable from trajectories but only locally identifiable from density data, with initial conditions playing a critical role.

Significance. The work extends structural identifiability methods to a class of stochastic processes observed at different scales, which is relevant for parameter calibration in experimental settings where both trajectory and density data arise. The explicit comparison of data types and the handling of initial conditions via characteristics constitute the main contributions, provided the algebraic steps are valid.

major comments (2)

[PDE derivation section] PDE derivation section: the differential-algebra identifiability procedure requires the governing PDE to be polynomial in the observables and their derivatives. The manuscript must explicitly confirm that the PDE obtained from the underlying stochastic process satisfies this polynomiality condition; if the process produces a parabolic (diffusion) equation rather than a first-order hyperbolic one, the characteristic method does not apply directly and the ranking of derivatives must be re-examined.
[Characteristic-equation Taylor expansion subsection] Characteristic-equation Taylor expansion subsection: the claim that the power-series coefficients isolate identifiable parameter combinations involving initial conditions rests on algebraic independence properties that are not verified for general initial data. The manuscript should supply the explicit expansion for the concrete model and demonstrate that no hidden algebraic relations arise among the coefficients.

minor comments (2)

The example model is referred to only as 'a model'; stating its explicit equations (drift, diffusion, reaction terms) in the main text would make the identifiability calculations reproducible.
Notation for the observables and their derivatives should be introduced once and used consistently between the individual-based and PDE sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments on our manuscript. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

Referee: [PDE derivation section] PDE derivation section: the differential-algebra identifiability procedure requires the governing PDE to be polynomial in the observables and their derivatives. The manuscript must explicitly confirm that the PDE obtained from the underlying stochastic process satisfies this polynomiality condition; if the process produces a parabolic (diffusion) equation rather than a first-order hyperbolic one, the characteristic method does not apply directly and the ranking of derivatives must be re-examined.

Authors: We appreciate the referee's emphasis on the polynomiality requirement for the differential-algebra procedure. In our derivation the PDE for total particle density is a first-order hyperbolic transport equation that is polynomial in the density and its first derivatives. We will revise the PDE derivation section to state the explicit PDE, confirm that it satisfies the polynomiality condition, and note that the characteristic method therefore applies directly with the existing ranking of derivatives. revision: yes
Referee: [Characteristic-equation Taylor expansion subsection] Characteristic-equation Taylor expansion subsection: the claim that the power-series coefficients isolate identifiable parameter combinations involving initial conditions rests on algebraic independence properties that are not verified for general initial data. The manuscript should supply the explicit expansion for the concrete model and demonstrate that no hidden algebraic relations arise among the coefficients.

Authors: We agree that an explicit verification strengthens the argument. For the concrete model in the manuscript we will add the full Taylor expansion obtained from the characteristic equations and demonstrate that the resulting coefficients are algebraically independent, with no hidden relations among them for the initial data employed in the example. revision: yes

Circularity Check

0 steps flagged

No circularity: identifiability results follow from algebraic analysis of derived models

full rationale

The paper derives a PDE representation from the stochastic process and applies standard differential-algebra techniques to assess structural identifiability, supplemented by a characteristic-equation Taylor expansion for initial conditions. No quoted step shows a parameter or combination being fitted to data and then relabeled as a prediction, nor does any load-bearing claim reduce to a self-citation whose content is itself unverified. The distinction between full identifiability under trajectory data and local identifiability under density data is obtained by explicit algebraic ranking rather than by construction from the inputs. The methodology is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; assessment is limited to the high-level description of the method.

pith-pipeline@v0.9.0 · 5538 in / 1074 out tokens · 25569 ms · 2026-05-14T17:48:48.026197+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

65 extracted references · 2 canonical work pages

[1]

Bulletin of Mathematical Biology , volume=

Approximate solutions of a general stochastic velocity-jump model subject to discrete-time noisy observations , author=. Bulletin of Mathematical Biology , volume=. 2025 , publisher=

2025
[2]

A likelihood-based

Ceccarelli, Arianna and Browning, Alexander P and Chaiamarit, Tai and Davis, Ilan and Baker, Ruth E , journal=. A likelihood-based. 2026 , volume =

2026
[3]

Mathematical Biosciences , volume=

Determining identifiable parameter combinations using subset profiling , author=. Mathematical Biosciences , volume=. 2014 , publisher=

2014
[4]

and Goodbrake, C

Dong, R. and Goodbrake, C. and Harrington, H. and Pogudin, G. , title =. SIAM Journal on Applied Algebra and Geometry , doi =. 2023 , volume =

2023
[5]

Bioinformatics , volume =

Hong, Hoon and Ovchinnikov, Alexey and Pogudin, Gleb and Yap, Chee , title =. Bioinformatics , volume =. 2019 , month =

2019
[6]

Bioinformatics , volume=

Chi. Bioinformatics , volume=. 2011 , publisher=

2011
[7]

Bioinformatics , volume=

Ligon, Thomas S and Fr. Bioinformatics , volume=. 2018 , publisher=

2018
[8]

Computer Methods and Programs in Biomedicine , volume=

Bellu, Giuseppina and Saccomani, Maria Pia and Audoly, Stefania and D’Angi. Computer Methods and Programs in Biomedicine , volume=. 2007 , publisher=

2007
[9]

Bioinformatics , volume=

D. Bioinformatics , volume=. 2023 , publisher=

2023
[10]

PLoS Computational Biology , volume=

Structural identifiability of dynamic systems biology models , author=. PLoS Computational Biology , volume=. 2016 , publisher=

2016
[11]

Rey Rostro, David and Villaverde, Alejandro , year=. Strike. XLIII Jornadas de Automática: libro de actas , pages=
[12]

Bioinformatics , volume=

Benchmarking tools for a priori identifiability analysis , author=. Bioinformatics , volume=. 2023 , publisher=

2023
[13]

Nonlinearity , volume=

Algebraic identifiability of partial differential equation models , author=. Nonlinearity , volume=. 2025 , publisher=

2025
[14]

Proceedings of the Royal Society A , volume=

Structural identifiability analysis of linear reaction--advection--diffusion processes in mathematical biology , author=. Proceedings of the Royal Society A , volume=. 2024 , publisher=

2024
[15]

Journal of the Royal Society Interface , volume=

Identifiability analysis for stochastic differential equation models in systems biology , author=. Journal of the Royal Society Interface , volume=. 2020 , publisher=

2020
[16]

Structural identifiability of linear-in-parameter parabolic

Salmaniw, Yurij and Browning, Alexander P , journal=. Structural identifiability of linear-in-parameter parabolic. 2025 , publisher=

2025
[17]

arXiv preprint arXiv:2503.19241 , year=

Exact identifiability analysis for a class of partially observed near-linear stochastic differential equation models , author=. arXiv preprint arXiv:2503.19241 , year=

work page arXiv
[18]

Parameter identifiability in

Ciocanel, Maria-Veronica and Ding, Lee and Mastromatteo, Lucas and Reichheld, Sarah and Cabral, Sarah and Mowry, Kimberly and Sandstede, Bj. Parameter identifiability in. Bulletin of Mathematical Biology , volume=. 2024 , publisher=

2024
[19]

Structural identifiability analysis of age-structured

Renardy, Marissa and Kirschner, Denise and Eisenberg, Marisa , journal=. Structural identifiability analysis of age-structured. 2022 , publisher=

2022
[20]

Bioinformatics , volume=

Comparison of approaches for parameter identifiability analysis of biological systems , author=. Bioinformatics , volume=. 2014 , publisher=

2014
[21]

Markov Processes , booktitle =

Gardiner, Crispin , isbn =. Markov Processes , booktitle =. 2009 , address=. doi:10.1063/9780735423718_004 , pages=

work page doi:10.1063/9780735423718_004 2009
[22]

Automatica , volume=

On global identifiability for arbitrary model parametrizations , author=. Automatica , volume=. 1994 , publisher=

1994
[23]

Mathematical Biosciences , volume=

On structural identifiability , author=. Mathematical Biosciences , volume=. 1970 , publisher=

1970
[24]

2014 , publisher=

Identifiability of parametric models , author=. 2014 , publisher=

2014
[25]

Current Opinion in Systems Biology , volume=

On structural and practical identifiability , author=. Current Opinion in Systems Biology , volume=. 2021 , publisher=

2021
[26]

Bioinformatics , volume=

Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood , author=. Bioinformatics , volume=. 2009 , publisher=

2009
[27]

Automatica , volume=

Parameter identifiability of nonlinear systems: the role of initial conditions , author=. Automatica , volume=. 2003 , publisher=

2003
[28]

PloS One , volume=

Structural identifiability of systems biology models: a critical comparison of methods , author=. PloS One , volume=. 2011 , publisher=

2011
[29]

Mathematical Biosciences , volume=

Similarity transformation approach to identifiability analysis of nonlinear compartmental models , author=. Mathematical Biosciences , volume=. 1989 , publisher=

1989
[30]

and Kurtz, Thomas G

Anderson, David F. and Kurtz, Thomas G. and Koeppl, Heinz and Setti, Gianluca and di Bernardo, Mario and Densmore, Douglas , title =. Design and Analysis of Biomolecular Circuits: Engineering Approaches to Systems and Synthetic Biology , publisher =. 2011 , pages =

2011
[31]

IEEE Transactions on Biomedical Engineering , volume=

Global identifiability of nonlinear models of biological systems , author=. IEEE Transactions on Biomedical Engineering , volume=. 2002 , publisher=

2002
[32]

Mathematical Biosciences , volume=

System identifiability based on the power series expansion of the solution , author=. Mathematical Biosciences , volume=. 1978 , publisher=

1978
[33]

IEEE Transactions on Automatic Control , volume=

New results for identifiability of nonlinear systems , author=. IEEE Transactions on Automatic Control , volume=. 1987 , publisher=

1987
[34]

[1991] Proceedings of the 30th IEEE Conference on Decision and Control , pages=

Nonlinear observability, identifiability, and persistent trajectories , author=. [1991] Proceedings of the 30th IEEE Conference on Decision and Control , pages=. 1991 , doi=

1991
[35]

On identifiability of nonlinear

Miao, Hongyu and Xia, Xiaohua and Perelson, Alan S and Wu, Hulin , journal=. On identifiability of nonlinear. 2011 , publisher=

2011
[36]

Current Opinion in Systems Biology , volume=

On structural and practical identifiability: Current status and update of results , author=. Current Opinion in Systems Biology , volume=. 2025 , publisher=

2025
[37]

American Journal of Physiology-Regulatory, Integrative and Comparative Physiology , volume=

Parameter and structural identifiability concepts and ambiguities: a critical review and analysis , author=. American Journal of Physiology-Regulatory, Integrative and Comparative Physiology , volume=. 1980 , publisher=

1980
[38]

An algorithm for finding globally identifiable parameter combinations of nonlinear

Meshkat, Nicolette and Eisenberg, Marisa and DiStefano III, Joseph J , journal=. An algorithm for finding globally identifiable parameter combinations of nonlinear. 2009 , publisher=

2009
[39]

Journal of Mathematical Biology , volume=

Bridging the gap between individual-based and continuum models of growing cell populations , author=. Journal of Mathematical Biology , volume=. 2020 , publisher=

2020
[40]

Applicable Algebra in Engineering, Communication and Computing , volume=

Parameter identifiability and input--output equations , author=. Applicable Algebra in Engineering, Communication and Computing , volume=. 2023 , publisher=

2023
[41]

1991 , publisher=

Hyperbolic systems of conservation laws , author=. 1991 , publisher=

1991
[42]

Annual Review of Biophysics , volume=

Biophysical challenges to axonal transport: motor-cargo deficiencies and neurodegeneration , author=. Annual Review of Biophysics , volume=. 2014 , publisher=

2014
[43]

Reviews of Modern Physics , volume=

Stochastic models of intracellular transport , author=. Reviews of Modern Physics , volume=. 2013 , publisher=

2013
[44]

Nature Reviews Molecular Cell Biology , volume=

Slow axonal transport: stop and go traffic in the axon , author=. Nature Reviews Molecular Cell Biology , volume=. 2000 , publisher=

2000
[45]

Physical Biology , volume=

Modeling the slowing of neurofilament transport along the mouse sciatic nerve , author=. Physical Biology , volume=. 2009 , publisher=

2009
[46]

Journal of Mathematical Biology , volume=

A model of intracellular transport of particles in an axon , author=. Journal of Mathematical Biology , volume=. 2005 , publisher=

2005
[47]

Central European Journal of Physics , volume=

Analytical solution of equations describing slow axonal transport based on the stop-and-go hypothesis , author=. Central European Journal of Physics , volume=. 2011 , publisher=

2011
[48]

Brain Multiphysics , volume=

Queuing model of axonal transport , author=. Brain Multiphysics , volume=. 2021 , publisher=

2021
[49]

Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology , pages=

Recent mathematical models of axonal transport , author=. Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology , pages=. 2017 , publisher=

2017
[50]

Journal of Mathematical Biology , volume=

Models of dispersal in biological systems , author=. Journal of Mathematical Biology , volume=. 1988 , publisher=

1988
[51]

SIAM Journal on Applied Mathematics , volume=

From individual to collective behavior in bacterial chemotaxis , author=. SIAM Journal on Applied Mathematics , volume=. 2004 , publisher=

2004
[52]

Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=

Velocity-jump models with crowding effects , author=. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=. 2011 , publisher=

2011
[53]

Bulletin of Mathematical Biology , volume=

From birds to bacteria: generalised velocity jump processes with resting states , author=. Bulletin of Mathematical Biology , volume=. 2015 , publisher=

2015
[54]

and Cates, M

Tailleur, J. and Cates, M. E. , title =. Physical Review Letters , year =
[55]

, title =

Weiss, George H. , title =. Physica A: Statistical Mechanics and its Applications , year =
[56]

IMA Journal of Applied Mathematics , volume=

Mathematical modelling of turning delays in swarm robotics , author=. IMA Journal of Applied Mathematics , volume=. 2015 , publisher=

2015
[57]

Physical Review E , volume=

Hard-sphere interactions in velocity-jump models , author=. Physical Review E , volume=. 2016 , publisher=

2016
[58]

Think before you fit:

Preston, Simon P and Wilkinson, Richard D and Clayton, Richard H and Chappell, Mike J and Mirams, Gary R , journal=. Think before you fit:. 2025 , publisher=

2025
[59]

Representations of

Kurtz, Thomas G , journal=. Representations of. 1980 , publisher=

1980
[60]

1969 , publisher=

Algebraic Curves: An Introduction to Algebraic Geometry , author=. 1969 , publisher=

1969
[61]

1976 , publisher =

Principles of Mathematical Analysis , author =. 1976 , publisher =

1976
[62]

Bulletin of Mathematical Biology , volume=

Modelling ion channels with a view towards identifiability , author=. Bulletin of Mathematical Biology , volume=. 2026 , publisher=

2026
[63]

SIAM Journal on Scientific and Statistical Computing , volume=

Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations , author=. SIAM Journal on Scientific and Statistical Computing , volume=. 1983 , publisher=

1983
[64]

IBM Journal of Research and Development , volume=

On the partial difference equations of mathematical physics , author=. IBM Journal of Research and Development , volume=. 1967 , publisher=

1967
[65]

Scientific Reports , volume=

Method for immobilization of living and synthetic cells for high-resolution imaging and single-particle tracking , author=. Scientific Reports , volume=. 2018 , publisher=

2018