Necessary Conditions For Identifiability Research Articles

Identifiable Paths and Cycles in Linear Compartmental Models.

We introduce a class of linear compartmental models called identifiable path/cycle models which have the property that all of the monomial functions of parameters associated to the directed cycles and paths from input compartments to output compartments are identifiable and give sufficient conditions to obtain an identifiable path/cycle model. Removing leaks, we then show how one can obtain a locally identifiable model from an identifiable path/cycle model. These identifiable path/cycle models yield the only identifiable models with certain conditions on their graph structure and thus we provide necessary and sufficient conditions for identifiable models with certain graph properties. A sufficient condition based on the graph structure of the model is also provided so that one can test if a model is an identifiable path/cycle model by examining the graph itself. We also provide some necessary conditions for identifiability based on graph structure. Our proofs use algebraic and combinatorial techniques.

Nonlinear mixed effects modeling and warping for functional data using B-splines

In functional data the interest is to find a global mean pattern, but also to capture the individual curve differences in phase and amplitude. This can be done conveniently by building in random effects on two levels: in the warping functions to account for individual phase variations; and in the linear structure to deal with individual amplitude variations. Via an appropriate choice of the warping function and B-spline approximations, estimation in the nonlinear mixed effects functional model is feasible, and does not require any prior knowledge on landmarks for the functional data. Sufficient and necessary conditions for identifiability of the flexible model are provided. A theoretical study is conducted: we establish asymptotic normality and consistency of the estimators of the registration and amplitude models, convergence of the iterative process, and consistency of the final estimator provided by the iterative process. The finite-sample performance of the proposed estimation procedure is investigated in a simulation study, which includes comparisons with existing methods. The added value of the developed method is further illustrated via the analysis of a real data example.

Inverse-system method for identification of damping-rate functions in non-Markovian quantum systems

Identification of complicated quantum environments lies in the core of quantum engineering, which systematically constructs an environment model with the aim of accurate control of quantum systems. In this paper, we present an inverse-system method to identify damping rate functions which describe non-Markovian environments in time-convolution-less master equations. To access information on the environment, we couple a finite-level quantum system to the environment and measure time traces of local observables of the system. By using sufficient measurement results, an algorithm is designed, which can simultaneously estimate multiple damping rate functions for different dissipative channels. Further, we show that identifiability for the damping rate functions corresponds to the invertibility of the system and a necessary condition for identifiability is also given. The effectiveness of our method is shown in examples of an atom and three-spin-chain non-Markovian systems.

Sufficient and Necessary Conditions for the Identifiability of the Q-matrix

Restricted latent class models (RLCMs) have recently gained prominence in educational assessment, psychiatric evaluation, and medical diagnosis. Different from conventional latent class models, restrictions on the RLCM model parameters are imposed by a design matrix to respect practitioners' scientific assumptions. The design matrix, called $Q$-matrix in the cognitive diagnosis literature, is usually constructed by practitioners and domain experts, yet it is subjective and could be misspecified. To address this problem, researchers have proposed to estimate the $Q$-matrix from data. On the other hand, the fundamental learnability issue of the $Q$-matrix and model parameters remains underexplored and existing studies often impose stronger than needed or even impractical conditions. This paper proposes sufficient and necessary conditions for joint identifiability of the $Q$-matrix and the RLCM model parameters under different types of RLCMs. The developed identifiability conditions only depend on the design matrix and are easy to verify in practice.

The Sufficient and Necessary Condition for the Identifiability and Estimability of the DINA Model.

Cognitive diagnosis models (CDMs) are useful statistical tools in cognitive diagnosis assessment. However, as many other latent variable models, the CDMs often suffer from the non-identifiability issue. This work gives the sufficient and necessary condition for identifiability of the basic DINA model, which not only addresses the open problem in Xu and Zhang (Psychometrika 81:625-649, 2016) on the minimal requirement for identifiability, but also sheds light on the study of more general CDMs, which often cover DINA as a submodel. Moreover, we show the identifiability condition ensures the consistent estimation of the model parameters. From a practical perspective, the identifiability condition only depends on the Q-matrix structure and is easy to verify, which would provide a guideline for designing statistically valid and estimable cognitive diagnosis tests.

Identifiability for Blind Source Separation of Multiple Finite Alphabet Linear Mixtures

We give under weak assumptions a complete combinatorial characterization of identifiability for linear mixtures of finite alphabet sources, with unknown mixing weights and unknown source signals, but known alphabet. This is based on a detailed treatment of the case of a single linear mixture. Notably, our identifiability analysis applies also to the case of unknown number of sources. We provide sufficient and necessary conditions for identifiability and give a simple sufficient criterion together with an explicit construction to determine the weights and the source signals for deterministic data by taking advantage of the hierarchical structure within the possible mixture values. We show that the probability of identifiability is related to the distribution of a hitting time and converges exponentially fast to one when the underlying sources come from a discrete Markov process. Finally, we explore our theoretical results in a simulation study. Our work extends and clarifies the scope of scenarios for which blind source separation becomes meaningful.

Optimal Sample Complexity for Blind Gain and Phase Calibration

Blind gain and phase calibration (BGPC) is a structured bilinear inverse problem, which arises in many applications, including inverse rendering in computational relighting (albedo estimation with unknown lighting), blind phase and gain calibration in sensor array processing, and multichannel blind deconvolution. The fundamental question of the uniqueness of the solutions to such problems has been addressed only recently. In a previous paper, we proposed studying the identifiability in bilinear inverse problems up to transformation groups. In particular, we studied several special cases of blind gain and phase calibration, including the cases of subspace and joint sparsity models on the signals, and gave sufficient and necessary conditions for identifiability up to certain transformation groups. However, there were gaps between the sample complexities in the sufficient conditions and the necessary conditions. In this paper, under a mild assumption that the signals and models are generic, we bridge the gaps by deriving tight sufficient conditions with optimal sample complexities.

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.

A note on the difficulties associated with the analysis of capture–recapture experiments with heterogeneous capture probabilities

A variety of methods have been proposed to estimate the size of a closed population non-parametrically using capture–recapture data when the capture probabilities are heterogeneous. None of the methods have been proven to be unbiased in all circumstances and in simulations they tend to perform well in some cases but poorly in others. Here we examine random effects models and conclude that if the capture probabilities are heterogeneous then without the imposition of conditions the population size is not identifiable. A necessary condition for identifiability is given, with the derivation of a sufficient condition remaining an open problem.

On identifiability in the autopsy model of reliability theory

A coherent system is observed until it fails. At the instant of system failure, the set of failed components and the failure time of the system are noted. The failure times of the components are not known. We consider whether the component life distributions can be determined from the distributions of the observed data.Meilijson (1981) gave a condition on the structure of the system that was sufficient for the identifiability of the component distributions, under the assumption that the component life distributions are continuous and have common essential extrema. Nowik (1990) gave necessary and sufficient conditions for identifiability under the more restrictive condition that the component distributions have atoms at their common essential infimum and are mutually absolutely continuous. We give a necessary condition for identifiability, which we show to be equivalent to Nowik's condition, under the assumption that the distributions are continuous and strictly increasing. We derive a sufficient condition for identifiability, more general than Meilijson's, for the case in which the component distributions are assumed to be analytic. We also show that our necessary condition for identifiability is both necessary and sufficient when the component life distributions are assumed to belong to certain parametric families.

On identifiability in the autopsy model of reliability theory

A coherent system is observed until it fails. At the instant of system failure, the set of failed components and the failure time of the system are noted. The failure times of the components are not known. We consider whether the component life distributions can be determined from the distributions of the observed data. Meilijson (1981) gave a condition on the structure of the system that was sufficient for the identifiability of the component distributions, under the assumption that the component life distributions are continuous and have common essential extrema. Nowik (1990) gave necessary and sufficient conditions for identifiability under the more restrictive condition that the component distributions have atoms at their common essential infimum and are mutually absolutely continuous. We give a necessary condition for identifiability, which we show to be equivalent to Nowik's condition, under the assumption that the distributions are continuous and strictly increasing. We derive a sufficient condition for identifiability, more general than Meilijson's, for the case in which the component distributions are assumed to be analytic. We also show that our necessary condition for identifiability is both necessary and sufficient when the component life distributions are assumed to belong to certain parametric families.

The Identification Problem in Systems Nonlinear in the Variables

This paper examines the identifiability of the coefficients of a single equation in a simultaneous equation model which is nonlinear only in the variables. The concept of identifiability in this model is motivated and developed using the closely related concept of observational equivalence. This framework is then utilized to develop necessary and sufficient conditions for identifiability when the disturbances are required to be independent of the exogenous variables. The approach recommended by Fisher is shown to yield sufficient but not necessary conditions for identifiability. For several relatively common special cases the necessary and sufficient conditions are found to simplify to the familiar rank condition for identifiability in the linear model. THE SIMULTANEOUS EQUATION MODEL that is nonlinear only in the variables has enjoyed widespread application in economics. Such models are linear in the parameters and typically seem to be linear in the variables as well, when viewing a single equation. In many models the nonlinearity in the variables arises due to endogenous variables entering in different forms in different equations (logged and unlogged form, for example). In macroeconometric models nonlinearity in the variables arises when the model includes endogenous real, nominal, and price variables, which are nonlinearly related.2 Whatever the reason for the nonlinearity in the variables, it is important to determine the conditions under which the equations of such models can be identified.

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.