Structural identifiability

{{Small|Source}}{{Cite journal |last1=Raue |first1=A. |last2=Karlsson |first2=J. |last3=Saccomani |first3=M. P. |last4=Jirstrand |first4=M. |last5=Timmer |first5=J. |date=2014-05-15 |title=Comparison of approaches for parameter identifiability analysis of biological systems |journal=Bioinformatics |language=en |volume=30 |issue=10 |pages=1440–1448 |pmid=24463185 |doi=10.1093/bioinformatics/btu006 |doi-access=free |s2cid=10052322 |issn=1367-4803}}

Consider a linear time-invariant system with the following state-space representation:

$\backslash begin\{align\}$

\dot{x}_1(t) &=-\theta_1 x_1, \\

\dot{x}_2(t) &=\theta_1 x_1, \\

y(t) &= \theta_2 x_2,

\end{align}

and with initial conditions given by $x\_1(0)\; =\; \backslash theta\_3$ and $x\_2(0)\; =\; 0$. The solution of the output $y$ is

$y(t)=\; \backslash theta\_2\; \backslash theta\_3\; e^\{-\backslash theta\_1\; t\}\; \backslash left(\; e^\{\backslash theta\_1\; t\}-1\; \backslash right),$

which implies that the parameters $\backslash theta\_2$ and $\backslash theta\_3$ are not structurally identifiable. For instance, the parameters $\backslash theta\_1\; =\; 1,\; \backslash theta\_2\; =\; 1,\; \backslash theta\_3\; =\; 1$ generates the same output as the parameters $\backslash theta\_1\; =\; 1,\; \backslash theta\_2\; =\; 2,\; \backslash theta\_3\; =\; 0.5$.

{{Small|Source}}{{Cite journal |last=Villaverde |first=Alejandro F. |date=2019-01-01 |title=Observability and Structural Identifiability of Nonlinear Biological Systems |journal=Complexity |language=en |volume=2019 |pages=1–12 |doi=10.1155/2019/8497093 |issn=1076-2787|doi-access=free |arxiv=1812.04525 }}

A model of a possible glucose homeostasis mechanism is given by the differential equations{{Cite journal |last1=Karin |first1=Omer |last2=Swisa |first2=Avital |last3=Glaser |first3=Benjamin |last4=Dor |first4=Yuval |last5=Alon |first5=Uri |date=2016 |title=Dynamical compensation in physiological circuits |journal=Molecular Systems Biology |language=en |volume=12 |issue=11 |pages=886 |doi=10.15252/msb.20167216 |issn=1744-4292 |pmc=5147051 |pmid=27875241}}

$\backslash begin\{aligned\}$

& \dot{G}=u(0)+u-(c+s_\mathrm{i} \, I) G, \\

& \dot{\beta}=\beta \left(\frac{1.4583 \cdot 10^{-5}}{1+\left(\frac{8.4}{G}\right)^{1.7}}-\frac{1.7361 \cdot 10^{-5}}{1+\left(\frac{G}{8.4}\right)^{8.5}}\right), \\

& \dot{I}=p \, \beta \, \frac{G^2}{\alpha^2+G^2}-\gamma \, I,

\end{aligned}

where (c, s_i, p, α, γ) are parameters of the system, and the states are the plasma glucose concentration G, the plasma insulin concentration I, and the beta-cell functional mass β. It is possible to show that the parameters p and s_i are not structurally identifiable: any numerical choice of parameters p and s_i that have the same product ps_i are indistinguishable.

Structural identifiability is assessed by analyzing the dynamical equations of the system, and does not take into account possible noises in the measurement of the output. In contrast, practical non-identifiability also takes noises into account.{{Cite journal |last1=Raue |first1=A. |last2=Kreutz |first2=C. |last3=Maiwald |first3=T. |last4=Bachmann |first4=J. |last5=Schilling |first5=M. |last6=Klingmüller |first6=U. |last7=Timmer |first7=J. |date=2009-08-01 |title=Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood |journal=Bioinformatics |language=en |volume=25 |issue=15 |pages=1923–1929 |doi=10.1093/bioinformatics/btp358 |issn=1460-2059 |pmid=19505944|doi-access=free |url=https://opus.bibliothek.uni-augsburg.de/opus4/files/113236/113236.pdf }}

The notion of structurally identifiable is closely related to observability, which refers to the capacity of inferring the state of the system by measuring the trajectories of the system output. It is also closely related to data informativity, which refers to the proper selection of inputs that enables the inference of the unknown parameters.{{Cite journal |last1=Van Waarde |first1=Henk J. |last2=Eising |first2=Jaap |last3=Camlibel |first3=M. Kanat |last4=Trentelman |first4=Harry L. |date=2023 |title=The Informativity Approach: To Data-Driven Analysis and Control |url=https://ieeexplore.ieee.org/document/10317631 |journal=IEEE Control Systems |volume=43 |issue=6 |pages=32–66 |doi=10.1109/MCS.2023.3310305 |issn=1066-033X|arxiv=2302.10488 |s2cid=257050367 }}{{Cite book |last1=Gevers |first1=Michel |last2=Bazanella |first2=Alexandre S. |last3=Coutinho |first3=Daniel F. |last4=Dasgupta |first4=Soura |title=52nd IEEE Conference on Decision and Control |chapter=Identifiability and excitation of polynomial systems |date=2013 |chapter-url=https://ieeexplore.ieee.org/document/6760547 |location=Firenze |publisher=IEEE |pages=4278–4283 |doi=10.1109/CDC.2013.6760547 |isbn=978-1-4673-5717-3 |s2cid=7796419}}

The (lack of) structural identifiability is also important in the context of dynamical compensation of physiological control systems. These systems should ensure a precise dynamical response despite variations in certain parameters.{{Cite journal |last1=Karin |first1=Omer |last2=Swisa |first2=Avital |last3=Glaser |first3=Benjamin |last4=Dor |first4=Yuval |last5=Alon |first5=Uri |date=2016 |title=Dynamical compensation in physiological circuits |journal=Molecular Systems Biology |language=en |volume=12 |issue=11 |pages=886 |doi=10.15252/msb.20167216 |issn=1744-4292 |pmc=5147051 |pmid=27875241}}{{Cite journal |last=Sontag |first=Eduardo D. |date=2017-04-06 |editor-last=Komarova |editor-first=Natalia L. |title=Dynamic compensation, parameter identifiability, and equivariances |journal=PLOS Computational Biology |language=en |volume=13 |issue=4 |pages=e1005447 |doi=10.1371/journal.pcbi.1005447 |issn=1553-7358 |pmc=5398758 |pmid=28384175|bibcode=2017PLSCB..13E5447S |doi-access=free }} In other words, while in the field of systems identification, unidentifiability is considered a negative property, in the context of dynamical compensation, unidentifiability becomes a desirable property.

Identifiability also appears in the context of inverse optimal control. Here, one assumes that the data comes from a solution of an optimal control problem with unknown parameters in the objective function. Here, identifiability refers to the possibility of infering the parameters present in the objective function by using the measured data.{{Cite journal |last1=Zhang |first1=Han |last2=Ringh |first2=Axel |date=2024 |title=Inverse optimal control for averaged cost per stage linear quadratic regulators |url=https://linkinghub.elsevier.com/retrieve/pii/S0167691123002050 |journal=Systems & Control Letters |language=en |volume=183 |pages=105658 |doi=10.1016/j.sysconle.2023.105658|arxiv=2305.15332 }}

There exist many software that can be used for analyzing the identifiability of a system, including non-linear systems:{{Cite journal|last1=Barreiro |first1=Xabier Rey |last2=Villaverde |first2=Alejandro F. |date=2023-01-31 |title=Benchmarking tools for a priori identifiability analysis |url=https://doi.org/10.1093/bioinformatics/btad065 |journal=Bioinformatics |language=en |volume=39 |issue=2 |pages=btad065 |doi=10.1093/bioinformatics/btad065 |pmid=36721336 |pmc=9913045 |issn=1367-4811}}

• PottersWheel: MATLAB toolbox that uses profile likelihood for structural and practical identifiability analysis.
• STRIKE-GOLDD: MATLAB toolbox for structural identifiability analysis.{{Cite arXiv |eprint=2207.07346 |class=eess.SY |first1=Sandra |last1=Díaz-Seoane |first2=Xabier |last2=Rey-Barreiro |title=STRIKE-GOLDD 4.0: user-friendly, efficient analysis of structural identifiability and observability |date=2022-07-15 |last3=Villaverde |first3=Alejandro F.}}
• [https://github.com/SciML/StructuralIdentifiability.jl StructuralIdentifiability.jl]: Julia library for assessing structural parameter identifiability.{{Cite journal |last1=Dong |first1=Ruiwen |last2=Goodbrake |first2=Christian |last3=Harrington |first3=Heather A. |last4=Pogudin |first4=Gleb |date=2023-03-31 |title=Differential Elimination for Dynamical Models via Projections with Applications to Structural Identifiability |url=https://epubs.siam.org/doi/10.1137/22M1469067 |journal=SIAM Journal on Applied Algebra and Geometry |language=en |volume=7 |issue=1 |pages=194–235 |doi=10.1137/22M1469067 |arxiv=2111.00991 |s2cid=245650629 |issn=2470-6566}}
• [https://github.com/insysbio/LikelihoodProfiler.jl LikelihoodProfiler.jl]: Julia library for practical identifiability analysis.{{Cite journal |last1=Borisov |first1=Ivan |last2=Metelkin |first2=Evgeny |date=2020 |editor-last=Beard |editor-first=Daniel A. |title=Confidence intervals by constrained optimization—An algorithm and software package for practical identifiability analysis in systems biology |journal=PLOS Computational Biology |language=en |volume=16 |issue=12 |pages=e1008495 |doi=10.1371/journal.pcbi.1008495 |doi-access=free |issn=1553-7358 |pmc=7785248 |pmid=33347435}}

• System identification
• Observability
• Model order reduction
• Adaptive control

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Category:Dynamical systems