Assessing local structural identifiability for environmental models

The local structural identifiability problem is investigated for the general case and demonstrated for a well-known microbial degradation model that includes 13 unknown parameters and 3 additional states. We address the identifiability question using a novel algorithm that can be used for large models with many parameters to be identified. A key ingredient in the analysis is the application of a singular value decomposition of the normalized parametric output sensitivity matrix that is obtained through a simple model integration. The SVD results are further analysed and verified in a complementary symbolic computation. It is especially the swiftness and accuracy of the suggested method that we consider to be a substantial advantage in comparison to existing methods for a structural identifiability analysis. The method also opens, in a natural way, the analysis of (parametric) uncertainty in general, and this is demonstrated in more detail in the results section.     

Indistinguishability and identifiability of kinetic models for the MurC reaction in peptidoglycan biosynthesis

An important question in Systems Biology is the design of experiments that enable discrimination between two (or more) competing chemical pathway models or biological mechanisms. In this paper analysis is performed between two different models describing the kinetic mechanism of a three-substrate three-product reaction, namely the MurC reaction in the cytoplasmic phase of peptidoglycan biosynthesis. One model involves ordered substrate binding and ordered release of the three products; the competing model also assumes ordered substrate binding, but with fast release of the three products. The two versions are shown to be distinguishable; however, if standard quasi-steady-state assumptions are made distinguishability cannot be determined. Once model structure uniqueness is ensured the experimenter must determine if it is possible to successfully recover rate constant values given the experiment observations, a process known as structural identifiability. Structural identifiability analysis is carried out for both models to determine which of the unknown reaction parameters can be determined uniquely, or otherwise, from the ideal system outputs. This structural analysis forms an integrated step towards the modelling of the full pathway of the cytoplasmic phase of peptidoglycan biosynthesis.     

Structure simplification of dynamic process models

Lumped process models derived from first engineering principles are usually too detailed for control purposes where only the major dynamic characteristics of the system should be captured. Two common steps of simplifying dynamic process models, the steady-state variable removal and the variable lumping simplification steps are investigated in this paper, in order to show if they preserve the key properties: the structural controllability, observability and stability of the models. In order to enable the formal analysis, these simplification steps are represented as context sensitive graph transformations acting on the structure graphs of the dynamic process models. It is shown that the simplification transformations above preserve the structural controllability and observability of process models. But only the steady-state variable removal transformation has been found not to destroy their structural stability. The variable lumping structure simplification transformation is further specialized to the case of cascade process models. It is shown that the inverse of this transformation does exist in this case, and both transformations preserve structural controllability and observability.     

Identifiability study of an aquatic ecosystem model

An example of ecosystem modelization is presented and used to underline the problems in this area of system analysis. The model construction is analysed. The parameters identification requires the test of a priori identifiability of a complex non-linear model (structural identifiability) ; it then calls for the choice of a good identification method ns the input signals of an ecosystem cannot be manipulated, to guarantee a posteriori identifiability. This method is applied to data collected on an alpine lake     

On the structural output distinguishability of parametric models, and its relations with structural identifiability

In this note the notion of output distinguishability is extended to parametric models with different structures. Methods are suggested for testing such models for structural output distinguishability. The possible relations of structural output distinguishability with structural identifiability and other structural properties are investigated.     

DAISY: a new software tool to test global identifiability of biological and physiological systems

A priori global identifiability is a structural property of biological and physiological models. It is considered a prerequisite for well-posed estimation, since it concerns the possibility of recovering uniquely the unknown model parameters from measured input-output data, under ideal conditions (noise-free observations and error-free model structure). Of course, determining if the parameters can be uniquely recovered from observed data is essential before investing resources, time and effort in performing actual biomedical experiments. Many interesting biological models are nonlinear but identifiability analysis for nonlinear system turns out to be a difficult mathematical problem. Different methods have been proposed in the literature to test identifiability of nonlinear models but, to the best of our knowledge, so far no software tools have been proposed for automatically checking identifiability of nonlinear models. In this paper, we describe a software tool implementing a differential algebra algorithm to perform parameter identifiability analysis for (linear and) nonlinear dynamic models described by polynomial or rational equations. Our goal is to provide the biological investigator a completely automatized software, requiring minimum prior knowledge of mathematical modelling and no in-depth understanding of the mathematical tools. The DAISY (Differential Algebra for Identifiability of SYstems) software will potentially be useful in biological modelling studies, especially in physiology and clinical medicine, where research experiments are particularly expensive and/or difficult to perform. Practical examples of use of the software tool DAISY are presented. DAISY is available at the web site http://www.dei.unipd.it/~pia/.     

Structural identifiability of non-linear systems using linear/non-linear splitting

In this paper, the local state space isomorphism theorem for non-linear systems is used to analyse structural identifiability. In particular it is shown that, under certain restrictions, it is possible to perform a linear/non-linear splitting of the analysis. The relatively straightforward linear analysis then restricts the class of local diffeomorphic transformations as given by the non-linear state space isomorphism theorem. This, in turn, leads to possible simplifications to the subsequent non-linear analysis by providing an efficient means for calculating the local state diffeomorphism.     

An approach to the problem of linear system identifiability through the investigation of structural properties of the connection,injection and observation matrices

The data of the linear system identificaility problem are stated, with an emphasis on the difficulties met in solving this problem by current methods, in the case of complex models.Based on the investigation of structural properties of connection, injection and observation matrices, the proposed method overcomes some of these difficulties. It leads to a necessary conditions for highly accurate identifiability, and to a necessary and sufficient condition for local identifiability.A typical application illustrates the potential of this method, although further investigations are still required, especially for making the method fully programmable.     

Structural identifiability analyses of candidate models for in vitro Pitavastatin hepatic uptake

In this paper a review of the application of four different techniques (a version of the similarity transformation approach for autonomous uncontrolled systems, a non-differential input/output observable normal form approach, the characteristic set differential algebra and a recent algebraic input/output relationship approach) to determine the structural identifiability of certain in vitro nonlinear pharmacokinetic models is provided. The Organic Anion Transporting Polypeptide (OATP) substrate, Pitavastatin, is used as a probe on freshly isolated animal and human hepatocytes. Candidate pharmacokinetic non-linear compartmental models have been derived to characterise the uptake process of Pitavastatin. As a prerequisite to parameter estimation, structural identifiability analyses are performed to establish that all unknown parameters can be identified from the experimental observations available.     

Structural identifiability of linear compartmental systems

In biology and mathematics compartmental systems are frequently used. System identification of systems based on physical laws often involves parameter estimation. Before parameter estimation can take place, we have to examine whether the parameters are structurally identifiable. In this paper tests for the structural identifiability of linear compartmental systems are proposed. The method is based on the similarity transformation approach. New contributions in the theory are the conditions for structural identifiability of structured positive linear systems. In addition, structural identifiability from the Markov parameters is extended to structural identifiability from the input-output data, in which the initial condition is (partially) unknown and nonnegligible. Finally, conditions are presented for structural identifiability of a sampled continuous-time linear dynamic system     

统计机器学习中参数可辨识性研究及其关键问题

参数可辨识性研究在统计机器学习中具有重要的理论意义和应用价值.参数可辨识性是关于模型参数能否被惟一确定的性质.在包含物理参数的学习模型中,可辨识性不仅是物理参数获得正确估计的前提条件,更重要的是,它反映了学习机器中由参数决定的物理特征.为扩展到未来类人智能机器研究的考察视角,我们将学习模型纳入"知识与数据共同驱动模型"的框架中讨论.在此框架下,我们提出两个关键问题.第一是参数可辨识性准则问题.该问题考察与可辨识性密切相关的各种判断准则,其中知识驱动子模型与数据驱动子模型的耦合方式为参数可辨识性问题提供了新的研究空间.第二是参数可辨识性与机器学习理论和应用相关联的研究.该研究包括可辨识性对参数估计、模型选择、学习算法、学习动态过程、奇异学习理论、贝叶斯推断等内容的深刻影响.     

Identification and Identifiability of Unknown Parameters in Multibody Dynamic Systems

Assessing design changes in mechanical systems from simulationresults requires both accurate dynamic models and accurate values forparameters in the models. Model parameters are often unavailable ordifficult to measure. This study details an identification procedure fordetermining optimal values for unknown or estimated model parametersfrom experimental test data. The resulting optimization problem issolved by Levenberg–Marquardt methods. Partial derivative matricesneeded for the optimization are computed through sensitivity analysis.The sensitivity equations to be solved are generated analytically.Unfortunately, not all parameters can be uniquely determined using anidentification procedure. An issue of parameter identifiability remains.Since a global identifiability test is impractical for even the simplestmodels, a local identifiability test is developed. Two examples areprovided. The first example highlights the test for parameteridentifiability, while the second shows the usefulness of parameteridentification by determining vehicle suspension parameters fromexperimentally measured data.     

Structured parametric epidemic models

A stage-structured model for a theoretical epidemic process that incorporates immature, susceptible and infectious individuals in independent stages is formulated. In this analysis, an input interpreted as a birth function is considered. The structural identifiability is studied using the Markov parameters. Then, the unknown parameters are uniquely determined by the output structure corresponding to an observation of infection. Two different birth functions are considered: the linear case and the Beverton–Holt type to analyse the structured epidemic model. Some conditions on the parameters to obtain non-zero disease-free equilibrium points are given. The identifiability of the parameters allows us to determine uniquely the basic reproduction number ?₀ and the stability of the model in the equilibrium is studied using ?₀ in terms of the model parameters.     

Structural identifiability and observability of compartmental models of the COVID-19 pandemic

The recent coronavirus disease (COVID-19) outbreak has dramatically increased the public awareness and appreciation of the utility of dynamic models. At the same time, the dissemination of contradictory model predictions has highlighted their limitations. If some parameters and/or state variables of a model cannot be determined from output measurements, its ability to yield correct insights – as well as the possibility of controlling the system – may be compromised. Epidemic dynamics are commonly analysed using compartmental models, and many variations of such models have been used for analysing and predicting the evolution of the COVID-19 pandemic. In this paper we survey the different models proposed in the literature, assembling a list of 36 model structures and assessing their ability to provide reliable information. We address the problem using the control theoretic concepts of structural identifiability and observability. Since some parameters can vary during the course of an epidemic, we consider both the constant and time-varying parameter assumptions. We analyse the structural identifiability and observability of all of the models, considering all plausible choices of outputs and time-varying parameters, which leads us to analyse 255 different model versions. We classify the models according to their structural identifiability and observability under the different assumptions and discuss the implications of the results. We also illustrate with an example several alternative ways of remedying the lack of observability of a model. Our analyses provide guidelines for choosing the most informative model for each purpose, taking into account the available knowledge and measurements.     

State isomorphism approach to global identifiability of nonlinearsystems

Global deterministic identifiability of nonlinear systems is studied by constructing the family of local state isomorphisms that preserve the structure of the parametric system. The method is simplified for homogeneous systems, where such isomorphisms are shown to be linear, thereby reducing the identifiability problem to solving a set of algebraic equations. The known conditions for global identifiability in linear and bilinear systems are special cases of these results     

Structural identifiability and indistinguishability analyses of the Minimal Model and a Euglycemic Hyperinsulinemic Clamp model for glucose-insulin dynamics

Many mathematical models have been developed to describe glucose-insulin kinetics as a means of analysing the effective control of diabetes. This paper concentrates on the structural identifiability analysis of certain well-established mathematical models that have been developed to characterise glucose-insulin kinetics under different experimental scenarios. Such analysis is a pre-requisite to experiment design and parameter estimation and is applied for the first time to these models with the specific structures considered. The analysis is applied to a basic (original) form of the Minimal Model (MM) using the Taylor Series approach and a now well-accepted extended form of the MM by application of the Taylor Series approach and a form of the Similarity Transformation approach. Due to the established inappropriate nature of the MM with regard to glucose clamping experiments an alternative model describing the glucose-insulin dynamics during a Euglycemic Hyperinsulinemic Clamp (EIC) experiment was considered. Structural identifiability analysis of the EIC model is also performed using the Taylor Series approach and shows that, with glucose infusion as input alone, the model is structurally globally identifiable. Additional analysis demonstrates that the two different model forms are structurally distinguishable for observation of both glucose and insulin.     

The Side Transformation Pattern: Making Transforms Modular and Re-usable

The basic principles of meta-modelling are now well established for individual models. Activities such as the MOF QVT [QVT-Merge Group, “Revised submission for MOF 2.0 Query/Views/Transformations RFP (ad/2002-04-10)”, OMG Document ad/04-04-01, URL: http://www.omg.org/docs/ad/04-04-01.pdf] are now extending these principles to transformation between models. However, meta-model incompatibilities between transformations reduce opportunities for effective re-use, hindering wide scale adoption. We introduce a pattern, the Side Transformation Pattern, that arises naturally as transformations are made re-usable, and present a series of examples that show how its use can bring clarity and robustness to complex transformation problems.     

Sensitivity enhancing transformations for monitoring the process correlation structure

In this article we address the general problem of monitoring the process cross- and auto-correlation structure through the incorporation of information about its internal structure in a pre-processing stage, where sensitivity enhancing transformations are applied to collected data. We have found out that the sensitivity of the monitoring statistics based on partial or marginal correlations in detecting structural changes is directly related to the nominal levels of the correlation coefficients during normal operation conditions (NOC). The highest sensitivities are obtained when the process variables involved are uncorrelated, a situation that is hardly met in practice. However, not all transformations perform equally well in producing uncorrelated transformed variables with enhanced detection sensitivities. The most successful ones are based on the incorporation of the natural relationships connecting the process variables. In this context, a set of sensitivity enhancing transformations are proposed, which are based on a network reconstruction algorithm. These new transformations make use of fine structural information of the variables connectivity and therefore are able to improve the detection capability to local changes in correlation, leading to better performances when compared to current marginal-based methods, namely those based on latent variables models, such as PCA or PLS. Moreover, a novel monitoring statistic for the transformed variables variance proved to be very useful in the detection of structural changes resulting from model mismatch. This statistic allows for the detection of multiple structural changes within the same monitoring scheme and with higher detection performances when compared to the current methods.