18 research outputs found
On the existence of identifiable reparametrizations for linear compartment models
The parameters of a linear compartment model are usually estimated from
experimental input-output data. A problem arises when infinitely many parameter
values can yield the same result; such a model is called unidentifiable. In
this case, one can search for an identifiable reparametrization of the model: a
map which reduces the number of parameters, such that the reduced model is
identifiable. We study a specific class of models which are known to be
unidentifiable. Using algebraic geometry and graph theory, we translate a
criterion given by Meshkat and Sullivant for the existence of an identifiable
scaling reparametrization to a new criterion based on the rank of a weighted
adjacency matrix of a certain bipartite graph. This allows us to derive several
new constructions to obtain graphs with an identifiable scaling
reparametrization. Using these constructions, a large subclass of such graphs
is obtained. Finally, we present a procedure of subdividing or deleting edges
to ensure that a model has an identifiable scaling reparametrization
Identifiable reparametrizations of linear compartment models
Identifiability concerns finding which unknown parameters of a model can be
quantified from given input-output data. Many linear ODE models, used in
systems biology and pharmacokinetics, are unidentifiable, which means that
parameters can take on an infinite number of values and yet yield the same
input-output data. We use commutative algebra and graph theory to study a
particular class of unidentifiable models and find conditions to obtain
identifiable scaling reparametrizations of these models. Our main result is
that the existence of an identifiable scaling reparametrization is equivalent
to the existence of a scaling reparametrization by monomial functions. We also
provide partial results beginning to classify graphs which possess an
identifiable scaling reparametrization.Comment: 5 figure
Identifiability results for several classes of linear compartment models
Identifiability concerns finding which unknown parameters of a model can be
estimated from given input-output data. If some subset of the parameters of a
model cannot be determined given input-output data, then we say the model is
unidentifiable. In past work we identified a class of models, that we call
identifiable cycle models, which are not identifiable but have the simplest
possible identifiable functions (so-called monomial cycles). Here we show how
to modify identifiable cycle models by adding inputs, adding outputs, or
removing leaks, in such a way that we obtain an identifiable model. We also
prove a constructive result on how to combine identifiable models, each
corresponding to strongly connected graphs, into a larger identifiable model.
We apply these theoretical results to several real-world biological models from
physiology, cell biology, and ecology.Comment: 7 figure
Identifiability of linear compartmental tree models
A foundational question in the theory of linear compartmental models is how
to assess whether a model is identifiable -- that is, whether parameter values
can be inferred from noiseless data -- directly from the combinatorics of the
model. We completely answer this question for those models (with one input and
one output) in which the underlying graph is a bidirectional tree. Such models
include two families of models appearing often in biological applications:
catenary and mammillary models. Our proofs are enabled by two supporting
results, which are interesting in their own right. First, we give the first
general formula for the coefficients of input-output equations (certain
equations that can be used to determine identifiability). Second, we prove that
identifiability is preserved when a model is enlarged in specific ways
involving adding a new compartment with a bidirected edge to an existing
compartment.Comment: 32 page
Input-output equations and identifiability of linear ODE models
Structural identifiability is a property of a differential model with
parameters that allows for the parameters to be determined from the model
equations in the absence of noise. The method of input-output equations is one
method for verifying structural identifiability. This method stands out in its
importance because the additional insights it provides can be used to analyze
and improve models. However, its complete theoretical grounds and applicability
are still to be established. A subtlety and key for this method to work
correctly is knowing whether the coefficients of these equations are
identifiable.
In this paper, to address this, we prove identifiability of the coefficients
of input-output equations for types of differential models that often appear in
practice, such as linear models with one output and linear compartment models
in which, from each compartment, one can reach either a leak or an input. This
shows that checking identifiability via input-output equations for these models
is legitimate and, as we prove, that the field of identifiable functions is
generated by the coefficients of the input-output equations. Finally, we
exploit a connection between input-output equations and the transfer function
matrix to show that, for a linear compartment model with an input and strongly
connected graph, the field of all identifiable functions is generated by the
coefficients of the transfer function matrix even if the initial conditions are
generic
Computing all identifiable functions of parameters for ODE models
Parameter identifiability is a structural property of an ODE model for
recovering the values of parameters from the data (i.e., from the input and
output variables). This property is a prerequisite for meaningful parameter
identification in practice. In the presence of nonidentifiability, it is
important to find all functions of the parameters that are identifiable. The
existing algorithms check whether a given function of parameters is
identifiable or, under the solvability condition, find all identifiable
functions. However, this solvability condition is not always satisfied, which
presents a challenge. Our first main result is an algorithm that computes all
identifiable functions without any additional assumptions, which is the first
such algorithm as far as we know. Our second main result concerns the
identifiability from multiple experiments (with generically different inputs
and initial conditions among the experiments). For this problem, we prove that
the set of functions identifiable from multiple experiments is what would
actually be computed by input-output equation-based algorithms (whether or not
the solvability condition is fulfilled), which was not known before. We give an
algorithm that not only finds these functions but also provides an upper bound
for the number of experiments to be performed to identify these functions. We
provide an implementation of the presented algorithms
Parameter identifiability and input-output equations
Structural parameter identifiability is a property of a differential model
with parameters that allows for the parameters to be determined from the model
equations in the absence of noise. One of the standard approaches to assessing
this problem is via input-output equations and, in particular, characteristic
sets of differential ideals. The precise relation between identifiability and
input-output identifiability is subtle. The goal of this note is to clarify
this relation. The main results are:
1) identifiability implies input-output identifiability;
2) these notions coincide if the model does not have rational first
integrals;
3) the field of input-output identifiable functions is generated by the
coefficients of a "minimal" characteristic set of the corresponding
differential ideal.
We expect that some of these facts may be known to the experts in the area,
but we are not aware of any articles in which these facts are stated precisely
and rigorously proved.Comment: arXiv admin note: substantial text overlap with arXiv:1910.0396
A Differential Algebra Approach to Commuting Polynomial Vector Fields and to Parameter Identifiability in ODE Models
In the first part, we study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field. One motivating factor is that we can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. We first show that a linear vector field admits a full complement of commuting vector fields. Then we study a type of planar vector field for which there exists an upper bound on the degree of a commuting polynomial vector field. Finally, we turn our attention to conservative Newton systems, which form a special class of Hamiltonian systems, and show the following result. Let f be in K[x], where K is a field of characteristic zero, and d be the derivation that corresponds to the differential equation x\u27\u27 = f(x) in a standard way. We show that if the degree of f is at least 2, then any K-derivation commuting with d is equal to d multiplied by a conserved quantity. For example, the classical elliptic equation x\u27\u27 = 6x^2 + a, where a is a complex number, falls into this category.
In the second part, we study structural identifiability of parameterized ordinary differential equation models of physical systems, for example, systems arising in biology and medicine. A parameter is said to be structurally identifiable if its numerical value can be determined from perfect observation of the observable variables in the model. Structural identifiability is necessary for practical identifiability. We study structural identifiability via differential algebra. In particular, we use characteristic sets. A system of ODEs can be viewed as a set of differential polynomials in a differential ring, and the consequences of this system form a differential ideal. This differential ideal can be described by a finite set of differential equations called a characteristic set. The technique of studying identifiability via a set of special equations, sometimes called “input-output” equations, has been in use for the past thirty years. However it is still a challenge to provide rigorous justification for some conclusions that have been drawn in published studies. Our main result is on linear systems, which are a topic of current interest. We show that for a linear system of ODEs with one output, the coefficients of a monic characteristic set are identifiable. This result is then generalized, with additional hypotheses, to nonlinear systems
AutoRepar: a method to obtain identifiable and observable reparameterizations of dynamic models with mechanistic insights
Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGMechanistic dynamic models of biological systems allow for a quantitative and systematic interpretation of data and the generation of testable hypotheses. However, these models are often over-parameterized, leading to nonidentifiability and nonobservability, that is, the impossibility of inferring their parameters and state variables. The lack of structural identifiability and observability (SIO) compromises a model's ability to make predictions and provide insight. Here we present a methodology, AutoRepar, that corrects SIO deficiencies of nonlinear ODE models automatically, yielding reparameterized models that are structurally identifiable and observable. The reparameterization preserves the mechanistic meaning of selected variables, and has the exact same dynamics and input-output mapping as the original model. We implement AutoRepar as an extension of the STRIKE-GOLDD software toolbox for SIO analysis, applying it to several models from the literature to demonstrate its ability to repair their structural deficiencies. AutoRepar increases the applicability of mechanistic models, enabling them to provide reliable information about their parameters and dynamics.Consejo Superior de Investigaciones Científicas https://doi.org/10.13039/501100003339 | Ref. PIE 202070E062MCIN/AEI/10.13039/501100011033 | Ref. RYC-2019-027537-IMCIN/AEI/10.13039/501100011033 | Ref. PID2020-113992RA-I00MCIN/AEI/ 10.13039/501100011033 | Ref. PID2020-117271RB-C2MCIN/AEI/ 10.13039/501100011033 | Ref. DPI2017-82896-C2-2-RXunta de Galicia | Ref. ED431F 2021/00