Local Parameter Identifiability Research Articles

Conditions for Local Parameter Identifiability for Systems of Differential Equations with an Infinite-Dimensional Parameter

Conditions for Local Parameter Identifiability for Systems of Differential Equations with an Infinite-Dimensional Parameter

Parameter estimation and identifiability analysis for a bivalent analyte model of monoclonal antibody-antigen binding

Surface plasmon resonance (SPR) is an extensively used technique to characterize antigen-antibody interactions. Affinity measurements by SPR typically involve testing the binding of antigen in solution to monoclonal antibodies (mAbs) immobilized on a chip and fitting the kinetics data using 1:1 Langmuir binding model to derive rate constants. However, when it is necessary to immobilize antigens instead of the mAbs, a bivalent analyte (1:2) binding model is required for kinetics analysis. This model is lacking in data analysis packages associated with high throughput SPR instruments and the packages containing this model do not explore multiple local minima and parameter identifiability issues that are common in non-linear optimization. Therefore, we developed a method to use a system of ordinary differential equations for analyzing 1:2 binding kinetics data. Salient features of this method include a grid search on parameter initialization and a profile likelihood approach to determine parameter identifiability. Using this method we found a non-identifiable parameter in data set collected under the standard experimental design. A simulation-guided improved experimental design led to reliable estimation of all rate constants. The method and approach developed here for analyzing 1:2 binding kinetics data will be valuable for expeditious therapeutic antibody discovery research.

Conditions of local parameter identifiability for systems of differential equations with an infinite-dimensional parameter

The problem of parameter identification (determining parameters of a system by observing solutions or functions of them) is one of the main problems of the applied theory of differential equations. The property of local identifiability plays the most important role in solving this problem. The presence of this property means that one can uniquely determine values of parameters of a system in a neighborhood of a particular parameter by observing solutions. Earlier the case of a finite-dimensional parameter was mostly studied in this issue. The problem of local parameter identifiability in the case of an infinite-dimensional parameter is less studied. In this paper we propose a new method for obtaining sufficient conditions for local parameter identifiability in the case of an infinite-dimensional parameter. Under these conditions an infinite-dimensional parameter belonging to certain classes is locally identifiable by observing a solution at a finite set of points. For system with linear dependence on a parameter we establish the genericity of the mentioned conditions.

Local parameter identifiability of large-scale nonlinear models based on the output sensitivity covariance matrix

Abstract The use of first-principle models is motivated by the potential of detailed information available as well as their versatility. Therefore, it is important to keep these models up to date so the models represent accurate enough the processes at hand. However, most of these models are nonlinear with a large number of states and parameters but with a relatively low number of measured outputs. This lack of measurements hinders the possibility to estimate all of the parameters present in the model. In this work, parameter identifiability of large-scale nonlinear models is explored using the empirical output controllability covariance matrix approach. This empirical covariance matrix is used to extract the output sensitivity matrix of the model to assess parameter identifiability. The advantages of the proposed methods are discussed while different sensitivity indexes are evaluated to draw sound conclusions on the parameter ranking results. A large-scale reactive batch distillation process simulation is used as a demonstrator.

On the local identifiability of constituent stress–strain laws for hyperelastic composite materials

For natural composite materials such as biological tissues, mechanically characterizing the individual constituents and elucidating their roles in supporting structural integrity has remained experimentally challenging since the constituents can often not be isolated without impairment and require non-standard testing devices. Adopting an inverse viewpoint, we examine, in this article, macroscopic samples whose constituent architecture is accessible and investigate whether it is possible to conclude on the stress–strain behaviour of the individual constituents based on experimental measurements from standard material tests. Focussing on isotropic hyperelastic composites, a direct discretization of the constituents’ strain energy densities in terms of global shape functions is explored. In order to assess the local deterministic identifiability of material parameters near an initial estimate, we adopt a sensitivity-based criterion and determine feasible combinations of candidate experiments without recourse to experimental measurements. Both the local identifiability and attainable reidentification accuracy are investigated in detail for a composite truss and a composite sheet whose force–strain responses in uniaxial or biaxial tension tests, respectively, are mainly determined by the constituents’ volume fractions.

Local parameter identifiability for one class of hybrid systems

We consider the problem of local parameter identifiability for a hybrid system with components that are continuous and discrete in time. The set of observations is the vector-solution (depending on the parameter that is continuous in time) of the discrete component. Sufficient conditions of local parameter identifiability have been formulated using the earlier introduced notion of normalized separability of the set of parameters from the kernel of a special functional. An example where the condition of normed separability is reduced to some rank criterion is given.

Oxygen storage dominated three-way catalyst modeling for fresh catalysts

Meeting future legislations for the emissions of spark-ignition engines requires the development of efficient three-way catalyst (TWC) models. These models can both be employed for online catalyst control as well as model-based development of control and on-board diagnosis (OBD) strategies. In this paper we present such a model that aims at capturing transient TWC dynamics of fresh catalysts at elevated temperatures. Applying a parameter subset selection strategy, we demonstrate that solely taking into account the dynamics of oxygen storage and release is sufficient for this purpose and reactions associated with processes taking place on the platinum-group metals do not have to be modeled explicitly. The one-dimensional single channel model lumps the exhaust gas species into two pseudo components, one being able to oxidize and one being able to reduce the oxygen storage material. The respective reaction kinetics are considered reversible with non-linear functions depending on the oxidation state of the material being put in the center of focus. Model validation is carried out using experimental data from an isothermal synthetic gas test bench. The applied feed gas composition closely resembles real exhaust conditions under dynamic operation. By employing local parameter sensitivity and identifiability analysis and investigating the differences between a total of five fresh catalysts from different commercial suppliers, we propose a set of only five parameters that can be used to accurately model oxygen storage and release dynamics under isothermal conditions. The results emphasize the importance of accounting for oxygen storage from water as well as the need of using a spatially distributed model. Apart from the above mentioned applications, the model is shown to be suitable for fast catalyst screening and characterization.

Prevalence of locally parameter identifiable systems

The problem of local parameter identifiability of an input–output system is considered. A set lf of systems is studied for which the property of local parameter identifiability holds for almost all values of input signals and parameters in both topological and metric senses. Sufficient conditions are pointed out under which the set LI contains a prevalent subset. The proof is based on the prevalent transversality theorem proved by Kaloshin. Systems are considered that are characterized by a family of (structural) parameters a and a control block. It is shown that if the dimension of the set of parameters a is large enough (the structure of the system is rich enough), then, generically, a system f a belongs to the class lf for a set of parameters a having full measure.

Local identifiability and sensitivity analysis of neuromuscular blockade and depth of hypnosis models

This paper addresses the local identifiability and sensitivity properties of two classes of Wiener models for the neuromuscular blockade and depth of hypnosis, when drug dose profiles like the ones commonly administered in the clinical practice are used as model inputs. The local parameter identifiability was assessed based on the singular value decomposition of the normalized sensitivity matrix. For the given input signal excitation, the results show an over-parameterization of the standard pharmacokinetic/pharmacodynamic models. The same identifiability assessment was performed on recently proposed minimally parameterized parsimonious models for both the neuromuscular blockade and the depth of hypnosis. The results show that the majority of the model parameters are identifiable from the available input–output data. This indicates that any identification strategy based on the minimally parameterized parsimonious Wiener models for the neuromuscular blockade and for the depth of hypnosis is likely to be more successful than if standard models are used.

Experimental evaluation of an approach to online redesign of experiments for parameter determination

The online redesign of experiments for parameter determination of nonlinear dynamic systems has been studied recently by different research groups. In this article, this technique is assessed in a real case study for the first time. The presented algorithm adopts well‐known concepts from model‐based control. Compared to previous studies, special attention is given to the efficient treatment of the underlying nonlinear and possibly ill‐conditioned parameter estimation and experiment design problems. These problems are solved with single shooting and gradient‐based nonlinear programming (NLP) solvers. We use an initial value solver, which generates first‐ and second‐order sensitivities to compute exact derivatives of the problem functions. As a special feature, we propose the integration of a local parameter identifiability analysis and a corresponding algorithm that generates well‐conditioned problems. The practical applicability is demonstrated by experimental application to a chromatography column system where A, D, and E optimal experiments are performed. © 2012 American Institute of Chemical Engineers AIChE J, 59: 1981–1995, 2013

Locally parameter identifiable systems are generic

The problem of local parameter identifiability of an “input-output” system is considered. It is shown that if the dimension of the parameter space is not higher than that of the space of output signals, then a generic system is locally parameter identifiable for almost all values of the input signal and parameter.

A Group‐Theoretic Approach to Parameter Identifiability of PDE Systems

AbstractThe contribution is devoted to the parameter identifiability problem of (nonlinear) PDE systems. Especially, we discuss the (local) identifiability of parameters along a trajectory. The analysis relies on a coordinate‐free formulation for systems, including boundary conditions, and we motivate an approach by (Lie) transformation groups, whose success for PDE systems depends on a consequent extent to the accompanying boundary conditions. The (non‐)identifiability of parameters is related to the (non‐)existence of group generators, wherewith (local) conditions can be derived. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Local Identifiability of Load Model Parameters in Measurement-based Approach

It is important to derive reliable parameter values in the measurement-based load model development of electric power systems. However parameter estimation tasks, in practice, often face the parameter identifiability issue; whether or not the model parameters can be estimated with a given input-output data set in reliable manner. This paper introduces concepts and practical definitions of the local identifiability of model parameters. A posteriori local identifiability is defined in the sense of nonlinear least squares. As numerical examples, local identifiability of third-order induction motor (IM) model and a Z-induction motor (Z-IM) model is studied. It is shown that parameter illconditioning can significantly affect on reliable parameter estimation task. Numerical studies show that local identifiability can be quite sensitive to input data and a given local solution. Finally, several countermeasures are proposed to overcome ill-conditioning problem in measurement-based load modeling.

Parameter estimation and regionalization for continuous rainfall-runoff models including uncertainty

The prediction of streamflow at ungauged sites is one of the fundamental challenges to hydrologists today. While major progress has been made in the regionalization of statistical flow properties (e.g. extreme values), and methods for synthesis of event response at ungauged sites are widely applied, the estimation of continuous streamflow time-series is still very uncertain. The challenge of predicting the response at ungauged sites is often met through a process of parameter regionalization. Little attention has so far been given to the impact of new insights into model identification at gauged sites, e.g. regarding the problem of structural error, on this regionalization process. Questions that are addressed in this paper are the following: (1) What is the relationship between local parameter identifiability and catchment characteristics? (2) How is the uniqueness of catchments reflected in regionalization? (3) What is the result of local model structural uncertainty on the regionalization result? (4) How can we propagate local parameter uncertainty into predictions in ungauged basins and what is the result? A case study of 10 catchments located in the southeast of England is utilized to deal with these questions. The main conclusions from this study are that the uncertainty in the locally estimated model parameters is a function of their importance in representing the response of a given catchment, model structural error hinders identification of a parameter to represent a certain process and therefore hinders the regionalization, and the uncertainty in the calibrated parameters can be propagated to ungauged sites using a weighted regression approach.

Parameter estimation in cardiac ionic models

We examine the problem of parameter estimation in mathematical models of excitable cell cardiac electrical activity using the well-known Beeler–Reuter (1977) ionic equations for the ventricular action potential. The estimation problem can be regarded as equivalent to the accurate reconstruction of ionic current kinetics and amplitudes in an excitable cell model, given only action potential experimental data. We show that in the Beeler–Reuter case, all ionic currents may be reasonably reconstructed using an experimental design consisting of action potential recordings perturbed by pseudo-random injection currents. The Beeler–Reuter model was parameterised into 63 parameters completely defining all membrane current amplitudes and kinetics. Total membrane current was fitted to model-generated experimental data using a ‘data-clamp’ protocol. The experimental data consisted of a default action-potential waveform and an optional series of perturbed waveforms generated by current injections. Local parameter identifiability was ascertained from the reciprocal condition value (1/ λ) of the Hessian at the known solution. When fitting to a single action potential waveform, the model was found to be over-determined, having a 1/ λ value of ∼3.6e−14. This value improved slightly to ∼1.4e−10 when an additional 2 perturbed waveforms were included in the fitting process, suggesting that the additional data did not overly improve the identifiability problem. The additional data, however, did allow the accurate reconstruction of all ionic currents. This indicates that by appropriate experimental design, it may be possible to infer the properties of underlying membrane currents from observation of transmembrane potential waveforms perturbed by pseudo-random currents.

ON STRUCTURAL IDENTIFIABILITY OF SYSTEM PARAMETERS OF LINEAR MODELS

Investigation of structural identifiability of even linear dynamical models requires complicated algebraic manipulations with use of computer algebra methods. In present work very efficient approach to testing both local and global identifiability of system parameters is offered which does not require too complex algebraic computations and therefore allows to increase the dimension of testable models. In addition to consideration of general case two widely used classes of model structures are studied. Illustrative example is given to show advantages of the approach.

Analytic derivatives for estimation of linear dynamic models

This paper develops two algorithms. Algorithm 1 computes the exact, Gaussian, log-likelihood function, its exact, gradient vector, and an asymptotic approximation of its Hessian matrix, for discrete-time, linear, dynamic models in state-space form. Algorithm 2, derived from Algorithm 1, computes the exact, sample, information matrix of this likelihood function. The computed quantities are analytic (not numerical approximations) and should, therefore, be useful for reliably, quickly, and accurately: (i) checking local identifiability of parameters by checking the rank of the information matrix; (ii) using the gradient vector and Hessian matrix to compute maximum likelihood estimates of parameters with Newton methods; and, (iii) computing asymptotic covariances (Cramer-Rao bounds) of the parameter estimates with the Hessian or the information matrix. The principal contribution of the paper is Algorithm 2, which extends to multivariate models the univariate results of Porat and Friedlander [1]. By relying on the Kalman filter instead of the Levinson-Durbin filter used by Porat and Friedlander, Algorithms 1 and 2 can automatically handle any pattern of missing or linearly aggregated data. Although Algorithm 1 is well known, it is treated in detail in order to make the paper self contained.

STRUCTURAL, DYNAMIC MODELLING IN UNOBSERVABLE SPACES OF COVARIANCE‐STATIONARY STOCHASTIC PROCESSES

AbstractIn this paper a structural, stationary version of the well‐known state‐space model is used to model covariance‐stationary stochastic processes. The identifiability of the model parameters is discussed and a rank condition for local parameter identifiability is given. Ljung's results on prediction‐error estimation are used to establish strong consistency and asymptotic efficiency of the non‐linear ML‐estimates obtained from dependent observations. It turns out that the model can be identified by using simultaneously the steady‐state Kalman filter for the unobservable state vector and the prediction‐error estimation method for the model parameters.

Identifiability analysis of an in vivo receptor-binding radiopharmacokinetic system.

A five-state, nonlinear model describing the pharmacokinetics of a receptor-binding radiopharmaceutical is presented. Local parameter identifiability analysis was performed preexperimentally to investigate the chemical and observational conditions required for adequate estimates of hepatic blood flow, receptor-ligand affinity, and receptor population. The ability to estimate the above parameters from externally observed kinetic data depended upon receptor-ligand affinity, the molar dose of the radiopharmaceutical, the number and coupling of the observers, and independent knowledge of the coupling coefficients or ligand-receptor affinity.

Structural identifiability in linear time-invariant systems

A matrix-operator representation of parameter sensitivities is used to provide an algebraic "structural" analysis of local parameter identifiability in linear time-invariant ordinary differential equation systems. Necessary conditions for identifiability depend only upon the system matrices, no integrals must be computed, and arbitrary parametrization may be used. Relations to insensitivity are discussed, and design techniques are suggested which use the nonidentifiable (insensitive) subspace to systematically reduce the number of exciting parameters in individually designed parameter identification experiments. Finally, sufficiency conditions for zero-state identifiability are examined in terms of control inputs which continually excite the natural modes of the parameter sensitivities.