Higher-order Lie symmetries in identifiability and predictability analysis of dynamic models

Merkt, Benjamin; Timmer, Jens; Kaschek, Daniel

doi:10.1103/physreve.92.012920

Cited by 45 publications

(58 citation statements)

References 11 publications

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“…To analyze structural identifiability, most approaches are based on differential algebra (Ljung and Glad, 1994; Bellu et al, 2007). Based on Lie algebra, a recent study (Merkt et al, 2015) argued that structural non-identifiablity results from symmetries in differential equations with time-varying measurements. In case of a structurally non-identifiable model, one can resolve this by reducing or rebuilding the model.…”

Section: Introductionmentioning

confidence: 99%

Estimation and Identifiability of Model Parameters in Human Nociceptive Processing Using Yes-No Detection Responses to Electrocutaneous Stimulation

et al. 2016

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Healthy or pathological states of nociceptive subsystems determine different stimulus-response relations measured from quantitative sensory testing. In turn, stimulus-response measurements may be used to assess these states. In a recently developed computational model, six model parameters characterize activation of nerve endings and spinal neurons. However, both model nonlinearity and limited information in yes-no detection responses to electrocutaneous stimuli challenge to estimate model parameters. Here, we address the question whether and how one can overcome these difficulties for reliable parameter estimation. First, we fit the computational model to experimental stimulus-response pairs by maximizing the likelihood. To evaluate the balance between model fit and complexity, i.e., the number of model parameters, we evaluate the Bayesian Information Criterion. We find that the computational model is better than a conventional logistic model regarding the balance. Second, our theoretical analysis suggests to vary the pulse width among applied stimuli as a necessary condition to prevent structural non-identifiability. In addition, the numerically implemented profile likelihood approach reveals structural and practical non-identifiability. Our model-based approach with integration of psychophysical measurements can be useful for a reliable assessment of states of the nociceptive system.

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Section: Introductionmentioning

confidence: 99%

Estimation and Identifiability of Model Parameters in Human Nociceptive Processing Using Yes-No Detection Responses to Electrocutaneous Stimulation

et al. 2016

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“…The core functionality is extended by two symbolic methods implemented in Python and interfaced via the rPython package: identifiability and observability analysis based on Lie-group symmetries (Merkt, Timmer, and Kaschek 2015) and steady-state constraints for parameter estimation (Rosenblatt, Timmer, and Kaschek 2016).…”

Section: Introductionmentioning

confidence: 99%

Dynamic Modeling, Parameter Estimation and Uncertainty Analysis in 𝗥

Kaschek

Mader

Fehling-Kaschek

et al. 2016

Preprint

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In a wide variety of research fields, dynamic modeling is employed as an instrument to learn and understand complex systems. The differential equations involved in this process are usually non-linear and depend on many parameters whose values decide upon the characteristics of the emergent system. The inverse problem, i.e. the inference or estimation of parameter values from observed data, is of interest from two points of view. First, the existence point of view, dealing with the question whether the system is able to reproduce the observed dynamics for any parameter values. Second, the identifiability point of view, investigating invariance of the prediction under change of parameter values, as well as the quantification of parameter uncertainty.In this paper, we present the R package dMod providing a framework for dealing with the inverse problem in dynamic systems. The particularity of the approach taken by dMod is to provide and propagate accurate derivatives computed from symbolic expressions wherever possible. This derivative information highly supports the convergence of optimization routines and enhances their numerical stability, a requirement for the applicability of sofisticated uncertainty analysis methods. Computational efficiency is achieved by automatic generation and execution of C code. The framework is object oriented (S3) and provides a variety of functions to set up dynamic models, observation functions and parameter transformations for multi-conditional parameter estimation.The key elements of the framework and the methodology implemented in dMod are highlighted by an application on a three-compartment transporter model.

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“…Then taking Y = X 1 we find that [X, A] lies in the set in the proposition for each A ∈ Θ G . The set of such X form a Lie algebra containing the diagonal matrices and hence spanned by some subset of the elementary matrices E ij , and this Lie algebra can be easily determined from the graph G. This Lie algebra captures Lie point symmetries of the ODE, as in [MTK15]. But the set in the proposition is often larger than (the image of) this algebra, and indeed does not correspond to any Lie algebra acting on the parameter space.…”

Section: Reformulating the Dimension Criterionmentioning

confidence: 99%

“…This can be done using a reparametrization of the original system, which reduces it to a model having a parameter space of lower dimension. A procedure for finding such a reparametrization has been discussed for the differential algebra approach [LG94,MED09,MAD11], for the Taylor series approach [EC00], and for the similarity transformation approach [CG98,MTK15]. In [MTK15,YEC09] it is observed that non-identifiability of a system of ODE's is often due to Lie point symmetries.…”

Section: Previous Workmentioning

confidence: 99%

On the Existence of Identifiable Reparametrizations for Linear Compartment Models

Baaijens¹,

Draisma

2016

SIAM J. Appl. Math.

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Abstract. The parameters of a linear compartment model are usually estimated from experimental input-output data. A problem arises when infinitely many parameter values can yield the same result; such a model is called unidentifiable. In this case, one can search for an identifiable reparametrization of the model: a map which reduces the number of parameters, such that the reduced model is identifiable. We study a specific class of models which are known to be unidentifiable. Using algebraic geometry and graph theory, we translate a criterion given by Meshkat and Sullivant for the existence of an identifiable scaling reparametrization to a new criterion based on the rank of a weighted adjacency matrix of a certain bipartite graph. This allows us to derive several new constructions to obtain graphs with an identifiable scaling reparametrization. Using these constructions, a large subclass of such graphs is obtained. Finally, we present a procedure of subdividing or deleting edges to ensure that a model has an identifiable scaling reparametrization.

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Higher-order Lie symmetries in identifiability and predictability analysis of dynamic models

Cited by 45 publications

References 11 publications

Estimation and Identifiability of Model Parameters in Human Nociceptive Processing Using Yes-No Detection Responses to Electrocutaneous Stimulation

Estimation and Identifiability of Model Parameters in Human Nociceptive Processing Using Yes-No Detection Responses to Electrocutaneous Stimulation

Dynamic Modeling, Parameter Estimation and Uncertainty Analysis in 𝗥

On the Existence of Identifiable Reparametrizations for Linear Compartment Models

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