On Role of S-Synchronizability and Excitation Constancy in Structural Identifiability Problem of Nonlinear Systems

A class of dynamical systems with a single nonlinearity considered. The S-synchronizability concept of input introduced. It is shown that S-synchronizability is a condition for the structural identifiability of a nonlinear system. The decisionmaking on structural identifiability based on the properties analysis for a special class of geometric frameworks. Geometric frameworks reflect properties of the nonlinear dynamic system. Requirements for the model allowed us to obtain a geometric structure based on the input and output data considered. The constant excitation effect of input on the structural identifiability of the system is studied. The constant excitation effect of input studied on the structural identifiability of the system. Nonfulfillment the constant excitation condition gives a nonsignificant geometric framework. Various types of structural identifiability based on structure analysis considered. The concept of d-optimality described properties of the geometric structure introduced. Conditions for non-identifiability of nonlinear system structure obtained if the d-optimality of the geometric framework does not hold for the given properties of the input. Methods for estimating identifiability of the system and determining the identifiability area under uncertainty proposed. The proposed approach is generalized to the system having two nonlinearities. Conditions for partial structural identifiability obtained. Structural identifiability features of this class systems noted. The method for estimating the structure of the system proposed when the condition of structural identifiability satisfied. It has shown how the phase portrait used to estimate the system non-identifiability. A method proposed for constructing the structural identifiability domain of the system. Proposed methods and procedures are applied to study systems with Bouc-Wen hysteresis and two nonlinearity.

S-synchronization and Excitation Constancy in Structural Identifiability Problem of Nonlinear Systems

An approach to analysis the structural identifiability (SI) of nonlinear dynamical systems under uncertainty was proposed. S-synchronizability condition of an input is the basis for the structural identifiability estimation of the nonlinear system. A method for obtaining a set containing information about the nonlinear part of the system wasproposed. The decision on SI of the system was based on the analysis of geometric frameworks reflected the state of the system nonlinear part. Geometric frameworks were defined on the specified set. Conditions for structural indistinguishability of geometric frameworks and local identifiability of the nonlinear part were obtained. It shown that a non-S-synchronizing input gives an insignificant geometric framework. This input is a sign of structural non-identifiability of the nonlinear system. The method for estimating the structural identifiability of the nonlinear system was proposed. We show that the structural identifiability is the basis for structural identification of the system. The structural identifiability degree was introduced, and the method of its estimation was proposed.

A Gamma-Type Distribution with Applications

This article introduces a new probability distribution capable of modeling positive data that present different levels of asymmetry and high levels of kurtosis. A slashed quasi-gamma random variable is defined as the quotient of independent random variables, a generalized gamma is the numerator, and a power of a standard uniform variable is the denominator. The result is a new three-parameter distribution (scale, shape, and kurtosis) that does not present the identifiability problem presented by the generalized gamma distribution. Maximum likelihood (ML) estimation is implemented for parameter estimation. The results of two real data applications revealed a good performance in real settings.

Constraints, the identifiability problem and the forecasting of mortality

Models of mortality often require constraints in order that parameters may be estimated uniquely. It is not difficult to find references in the literature to the “identifiability problem”, and papers often give arguments to justify the choice of particular constraint systems designed to deal with this problem. Many of these models are generalised linear models, and it is known that the fitted values (of mortality) in such models are identifiable, i.e., invariant with respect to the choice of constraint systems. We show that for a wide class of forecasting models, namely ARIMA $(p,\delta, q)$ models with a fitted mean and $\delta = 1$ or 2, identifiability extends to the forecast values of mortality; this extended identifiability continues to hold when some model terms are smoothed. The results are illustrated with data on UK males from the Office for National Statistics for the age-period model, the age-period-cohort model, the age-period-cohort-improvements model of the Continuous Mortality Investigation and the Lee–Carter model.

Testing structural identifiability by a simple scaling method

Successful mathematical modeling of biological processes relies on the expertise of the modeler to capture the essential mechanisms in the process at hand and on the ability to extract useful information from empirical data. The very structure of the model limits the ability to infer numerical values for the parameters, a concept referred to as structural identifiability. Most of the available methods to test the structural identifiability of a model are either too complex mathematically for the general practitioner to be applied, or require involved calculations or numerical computation for complex non-linear models. In this work, we present a new analytical method to test structural identifiability of models based on ordinary differential equations, based on the invariance of the equations under the scaling transformation of its parameters. The method is based on rigorous mathematical results but it is easy and quick to apply, even to test the identifiability of sophisticated highly non-linear models. We illustrate our method by example and compare its performance with other existing methods in the literature.Author summaryTheoretical Biology is a useful approach to explain, generate hypotheses, or discriminate among competing theories. A well-formulated model has to be complex enough to capture the relevant mechanisms of the problem, and simple enough to be fitted to data. Structural identifiability tests aim to recognize, in advance, if the structure of the model allows parameter fitting even with unlimited high-quality data. Available methods require advanced mathematical skills, or are too costly for high-dimensional non-linear models. We propose an analytical method based on scale invariance of the equations. It provides definite answers to the structural identifiability problem while being simple enough to be performed in a few lines of calculations without any computational aid. It favorably compares with other existing methods.

Finding ranges of optimal transcript expression quantification in cases of non-identifiability

Current expression quantification methods suffer from a fundamental but under-characterized type of error: the most likely estimates for transcript abundances are not unique. Current quantification methods rely on probabilistic models, and the scenario where it admits multiple optimal solutions is called non-identifiability. This means multiple estimates of transcript abundances generate the observed RNA-seq reads equally likely, and the underlying true expression cannot be determined. The non-identifiability problem is further exacerbated when incompleteness of reference transcriptome and existence of unannotated transcripts are taken into consideration. State-of-the-art quantification methods usually output a single inferred set of abundances, and the accuracy of the single expression set is unknown compared to other equally optimal solutions. Obtaining the set of equally optimal solutions is necessary for evaluating and extending downstream analyses to take non-identifiability into account. We propose methods to compute the range of equally optimal estimates for the expression of each transcript, accounting for non-identifiability of the quantification model using several novel graph theoretical approaches. It works under two scenarios, one assuming the reference transcriptome is complete, another assuming incomplete reference and allowing for expression of unannotated transcripts. Our methods calculate a “confidence” range for each transcript, representing its possible abundance across equally optimal estimates. This range can be used for evaluating the reliability of detected differentially expressed (DE) transcripts, as a large overlap of confidence range between DE conditions indicates the prediction may be unreliable due to uncertainty. We observe that 5 out of 257 DE predictions are unreliable on an MCF10 cell line and 19 out of 3152 are unreliable on a CD8 T cell dataset. The source code can be found at https://github.com/Kingsford-Group/subgraphquant.

Quadratic Frequency Modulation Signals Parameter Estimation Based on Product High Order Ambiguity Function-Modified Integrated Cubic Phase Function

In inverse synthetic aperture radar (ISAR) imaging system for targets with complex motion, such as ships fluctuating with oceanic waves and high maneuvering airplanes, the multi-component quadratic frequency modulation (QFM) signals are more suitable model for azimuth echo signals. The quadratic chirp rate (QCR) and chirp rate (CR) cause the ISAR imaging defocus. Thus, it is important to estimate QCR and CR of multi-component QFM signals in ISAR imaging system. The conventional QFM signal parameter estimation algorithms suffer from the cross-term problem. To solve this problem, this paper proposes the product high order ambiguity function-modified integrated cubic phase function (PHAF-MICPF). The PHAF-MICPF employs phase differentiation operation with multi-scale factors and modified coherently integrated cubic phase function (MICPF) to transform the multi-component QFM signals into the time-quadratic chirp rate (T-QCR) domains. The cross-term suppression ability of the PHAF-MICPF is improved by multiplying different T-QCR domains that are related to different scale factors. Besides, the multiplication operation can improve the anti-noise performance and solve the identifiability problem. Compared with high order ambiguity function-integrated cubic phase function (HAF-ICPF), the simulation results verify that the PHAF-MICPF acquires better cross-term suppression ability, better anti-noise performance and solves the identifiability problem.