On the parameter combinations that matter and on those that do not: data-driven studies of parameter (non)identifiability
Nikolaos EvangelouNoah J. WichrowskiGeorge A. KevrekidisFelix DietrichMahdi KooshkbaghiSarah McFannIoannis G. Kevrekidis
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We present a data-driven approach to characterizing nonidentifiability of a model's parameters and illustrate it through dynamic as well as steady kinetic models. By employing Diffusion Maps and their extensions, we discover the minimal combinations of parameters required to characterize the output behavior of a chemical system: a set of effective parameters for the model. Furthermore, we introduce and use a Conformal Autoencoder Neural Network technique, as well as a kernel-based Jointly Smooth Function technique, to disentangle the redundant parameter combinations that do not affect the output behavior from the ones that do. We discuss the interpretability of our data-driven effective parameters, and demonstrate the utility of the approach both for behavior prediction and parameter estimation. In the latter task, it becomes important to describe level sets in parameter space that are consistent with a particular output behavior. We validate our approach on a model of multisite phosphorylation, where a reduced set of effective parameters (nonlinear combinations of the physical ones) has previously been established analytically.Keywords:
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Large environmental simulation models are usually overparameterized with respect to given sets of observations. This results in poorly identifiable or nonidentifiable model parameters. For small models, plots of sensitivity functions have proven to be useful for the analysis of parameter identifiability. For models with many parameters, however, near‐linear dependence of sensitivity functions can no longer be assessed graphically. In this paper a systematic approach for tackling the parameter identifiability problem of large models based on local sensitivity analysis is presented. The calculation of two identifiability measures that are easy to handle and interpret is suggested. The first accounts for the sensitivity of model results to single parameters, and the second accounts for the degree of near‐linear dependence of sensitivity functions of parameter subsets. It is shown how these measures provide identifiability diagnosis for parameter subsets, how they are able to guide the selection of identifiable parameter subsets for parameter estimation, and how they facilitate the interpretation of the correlation matrix of the parameter estimate with respect to parameter identifiability. In addition, we show how potential bias of the parameter estimates, due to a priori fixing of some of the parameters, can be analyzed. Finally, two case studies are presented in order to illustrate the suggested approach.
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In experiments on biological systems one often cannot measure all state variables (compartments). Given a particular experiment of that type, a basic kinetic parameter may have no effect on the observations; such a parameter is an insensible parameter for that experiment. A parameter may influence the observations and not be uniquely determinable; such a parameter is nonidentifiable for that experiment. Only identifiable parameters can be estimated uniquely, by that experiment. I review the basic theory to check identifiability for a nominal value of a parameter (local identifiability), and present some examples of problems that may arise in estimation. ( Journal of Parenteral and Enteral Nutrition 15: 55S‐59S, 1991)
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This short paper considers the parameter identification problem of general discrete-time, nonlinear, multiple input-multiple output dynamic systems with Gaussian, white distributed measurement errors. Knowledge of the system parameterization is assumed to be known. Regions of constrained maximum likelihood (CML) parameter identifiability are established. A computation procedure employing interval arithmetic is proposed for finding explicit regions of parameter identifiability for the case of linear systems.
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Issues of parameter identifiability of routinely used pharmacodynamics models are considered in this paper. The structural identifiability of 16 commonly applied pharmacodynamic model structures was analysed analytically, using the input-output approach. Both fixed-effects versions (non-population, no between-subject variability) and mixed-effects versions (population, including between-subject variability) of each model structure were analysed. All models were found to be structurally globally identifiable under conditions of fixing either one of two particular parameters. Furthermore, an example was constructed to illustrate the importance of sufficient data quality and show that structural identifiability is a prerequisite, but not a guarantee, for successful parameter estimation and practical parameter identifiability. This analysis was performed by generating artificial data of varying quality to a structurally identifiable model with known true parameter values, followed by re-estimation of the parameter values. In addition, to show the benefit of including structural identifiability as part of model development, a case study was performed applying an unidentifiable model to real experimental data. This case study shows how performing such an analysis prior to parameter estimation can improve the parameter estimation process and model performance. Finally, an unidentifiable model was fitted to simulated data using multiple initial parameter values, resulting in highly different estimated uncertainties. This example shows that although the standard errors of the parameter estimates often indicate a structural identifiability issue, reasonably "good" standard errors may sometimes mask unidentifiability issues.
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Interpretability of epidemiological models is a key consideration, especially when these models are used in a public health setting. Interpretability is strongly linked to the identifiability of the underlying model parameters, i.e., the ability to estimate parameter values with high confidence given observations. In this paper, we define three separate notions of identifiability that explore the different roles played by the model definition, the loss function, the fitting methodology, and the quality and quantity of data. We define an epidemiological compartmental model framework in which we highlight these non-identifiability issues and their mitigation.
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This paper considers the parameter identification problem of general discrete-time, nonlinear, multiple-input/multiple-output dynamic systems with Gaussian-white distributed measurement errors. Knowledge of the system parameterization is assumed to be known. Regions of constrained maximum likelihood (CML) parameter identifiability are established. A computation procedure employing interval arithmetic is proposed for finding explicit regions of parameter identifiability for the case of linear systems. It is shown that if the vector of true parameters is locally CML identifiable, then with probability one, the vector of true parameters is a unique maximal point of the maximum likelihood function in the region of parameter identifiability and the CML estimation sequence will converge to the true parameters.
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Interpretability of epidemiological models is a key consideration, especially when these models are used in a public health setting. Interpretability is strongly linked to the identifiability of the underlying model parameters, i.e., the ability to estimate parameter values with high confidence given observations. In this paper, we define three separate notions of identifiability that explore the different roles played by the model definition, the loss function, the fitting methodology, and the quality and quantity of data. We define an epidemiological compartmental model framework in which we highlight these non-identifiability issues and their mitigation.
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One of the challenging tasks in systems biology is parameter estimation in nonlinear dynamic models. A biological model usually contains a large number of correlated parameters leading to non-identifiability problems. Although many approaches have been developed to address both structural and practical non-identifiability problems, very few studies have been made to systematically investigate parameter correlations. In this study we present an approach that is able to identify both pairwise parameter correlations and higher order interrelationships among parameters in nonlinear dynamic models. Correlations are interpreted as surfaces in the subspaces of correlated parameters. Based on the correlation information obtained in this way both structural and practical non-identifiability can be clarified. Moreover, it can be concluded from the correlation analysis that a minimum number of data sets with different inputs for experimental design are needed to relieve the parameter correlations, which corresponds to the maximum number of correlated parameters among the correlation groups. The information of pairwise and higher order interrelationships among parameters in biological models gives a deeper insight into the cause of non-identifiability problems. The result of our correlation analysis provides a necessary condition for experimental design in order to acquire suitable measurement data for unique parameter estimation.
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Reliable simulations of hydrological models require that model parameters are precisely identified. In constraining model parameters to small ranges, high parameter identifiability is achieved. In this study, it is investigated how precisely model parameters can be constrained in relation to a set of contrasting performance criteria. For this, model simulations with identical parameter samplings are carried out with a hydrological model (SWAT) applied to three contrasting catchments in Germany (lowland, mid-range mountains, alpine regions). Ten performance criteria including statistical metrics and signature measures are calculated for each model simulation. Based on the parameter identifiability that is computed separately for each performance criterion, model parameters are constrained to smaller ranges individually for each catchment. An iterative repetition of model simulations with successively constrained parameter ranges leads to more precise parameter identifiability and improves model performance. Based on these results, a more consistent handling of model parameters is achieved for model calibration.
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