https://orcid.org/0000-0001-7528-3736
Figures
Abstract
Background and objective: Practical identifiability analysis, i.e., ascertaining whether a model property can be determined from given data, is central to model-based data analysis in biomedicine. The main approaches used today all require that coverage of the parameter space be exhaustive, which is usually impossible. An alternative could be using structural identifiability methods, since they do not need such coverage. However, current structural methods are unsuited for practical identifiability analysis, since they assume that all higher-order derivatives of the measured variables are available. Herein, we provide new definitions and methods that allow for this assumption to be relaxed. Methods and results: We introduce the concept of -identifiability, which differs from previous definitions in that it assumes that only the first
derivatives of the measurement signal yi are available. This new type of identifiability can be determined using our new algorithms, as is demonstrated by applications to various published biomedical models. Our methods allow for identifiability of not only parameters, but of any model property, i.e., observability. These new results provide further strengthening of conclusions made in previous analysis of these models. For the first time, we can quantify the impact of the assumption that all derivatives are available in specific examples. If one, e.g., assumes that only up to third order derivatives, instead of all derivatives, are available, the number of identifiable parameters drops from 17 to 1 for a Drosophila model, and from 21 to 6 for an NF-
B model. In both models, the previously obtained identifiability is present only if at least 20 derivatives of all measurement signals are available. Conclusion: Our results demonstrate that the assumption regarding availability of derivatives made in traditional structural identifiability analysis requires a big overestimation regarding the number of parameters that can be estimated. Our new methods and algorithms allow for this assumption to be relaxed, bringing structural identifiability methodology one step closer to practical identifiability analysis.
Citation: Thompson P, Andersson BJ, Sundqvist N, Cedersund G (2025) A new method for a priori practical identifiability. PLoS One 20(7): e0327593. https://doi.org/10.1371/journal.pone.0327593
Editor: Fucai Lin, Minnan Normal University, CHINA
Received: December 27, 2024; Accepted: June 17, 2025; Published: July 17, 2025
Copyright: © 2025 Thompson et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting information files.
Funding: The following grants were awarded to Gunnar Cedersund: 2015–2021 Swedish Fund for Research without Animal Experiments 1 250 000 SEK (Main applicant) 2015–2020 CENIIT, “Multi-level modelling for improved drug development”, 4 500 000 (Main applicant) 2018–2021 AstraZeneca postdoc 2 716 000 SEK. (Main applicant) 2018–2022 SSF, 7 500 000 SEK, of which 4 000 000 SEK goes to my group (Co-applicant) 2018–2022 H2020, “PRECISE4Q”, ca 60 MSEK, of which 4 110 000 SEK goes to my group (Co-applicant) 2019–2022 VR-M, “Knowledge-driven drug development”, 3 200 000 SEK (Main applicant) 2019–2022 VR-NT, “M4Health – a foundation for general AI in healthcare”, 4 301 500 SEK (Main applicant) 2020–2028 Knowledge Foundation, X-HiDE, 120 MSEK, plan is to fund 20% guest professorship position for me (co-applicant) 2020–2021 SciLifeLab and Wallenberg Foundation (KAW). 800 000 SEK (Main applicant). 2020–2021 ELLIIT, “Usable digital twins in healthcare”, 2 000 000 SEK (Main applicant) 2021–2023 VINNOVA, “Digital twins in healthcare”, 3 000 000 SEK (Main applicant) The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript
Competing interests: The authors have declared that no competing interests exist.
1 Background
Biology and medicine involve the study of a myriad of time-dependent variables, which cross-talk with each other in complex networks. To deal with this complexity, one often makes use of mechanistic mathematical models, often formulated using ordinary differential equations (ODEs). These ODEs represent hypotheses regarding the biological mechanisms that can explain given biological data [9]. The model-based data analysis of such real data revolves around a correct uncertainty analysis of parameters and predictions: to what degree can one determine the parameters and predictions of interest? This uncertainty analysis is formally called identifiability analysis if it deals with parameter uncertainty, and observability analysis if it deals with other model properties such as states and composite variables [10]. Both observability and identifiability analysis can be done in many different ways, but they are generally subdivided into two types of approaches: structural and practical.
Practical identifiability analysis revolves around the specific data one has collected in a specific situation [10]. In other words, practical identifiability analysis looks at the real situation, where the data has noise, and where the experiment might not have been done in an optimal way to generate information about all parameters. Historically, one has often used sensitivity-based methods to study practical identifiability, such as analysis of the Fisher Information Matrix [35], and the covariance of the parameters [11]. In the last 10 years, these methods have been increasingly replaced by Markov Chain Monte Carlo (MCMC) and profile-likelihood based methods [10,11,25]. The benefit of these more modern methods is that they study global properties, and do not have to assume that the system is identifiable to assess identifiability [10,35]. These methods are highly useful and flexible, but they still have limitations. Most importantly, they rely on the exhaustive coverage of the entire parameter space, either using the MCMC sampling or using the optimization step in profile likelihood. For large models, this optimization step is difficult and time-consuming even for an expert in the field [26,30]. This difficulty will always leave the analysis results with an inherent doubt: if the coverage of the parameter space fails, the degree of identifiability (the calculated confidence intervals) is not correct.
Structural identifiability analysis has historically been the mirror image of practical identifiability analysis. Structural methods normally do not take a specific data set or the noise level into account. Instead, structural identifiability analysis only considers the form of the equations, including the measurement equations. Furthermore, structural identifiability analysis does not rely on an exhaustive coverage of the entire parameter space. Instead, structural identifiability analysis makes use of powerful theories from differential algebra and differential geometry to investigate whether there are structural limits for whether a parameter or a model property ever can be determined from a given measurement. Methods have been developed to determine which states and parameters are locally identifiable [29] and globally identifiable [6,15], to discover parameter combinations that are identifiable [19,20,23,24,30], and to suggest modifications in the modeling process based on identifiability [4,14]. In the last two decades, powerful methods have been developed that can do such an analysis also for relatively large biological models, featuring 50 to 100 states and parameters [4].
Because structural identifiability methods now can be applied to real, large models, it would be highly interesting if they also could perform practical identifiability analysis, where the limitations of a specific dataset are taken into account. One prominent such assumption is that current methods assume that all derivatives of all measurement signals can be estimated from the data. In practical situations, with real data, this assumption is never fulfilled. It is therefore a critical flaw that we are lacking definitions and methods to analyze the consequences of this unfulfilled assumption.
In summary, there is a need to develop new methods that combine the strengths of the two approaches: a method that does not rely on overly idealistic assumptions regarding the real situations, but that also does not rely on optimization and coverage of the entire parameter space. However, such methods have not yet been proposed. Herein, we introduce a new type of practical and structural identifiability, which puts an upper limit on how many times each measurement signal can be differentiated. We also introduce new algorithms that can calculate this new type of identifiability, and apply these to a series of published models. For all studied examples, traditional structural identifiability methods have widely overestimated the number of identifiable parameters: with a maximum of 3 derivatives available, more than 90% of the previously identifiable parameters lose their identifiability.
We present a new method for determining structural identifiability in a more realistic setting, where only a given finite number of derivatives are available for each measurement signal. While this is novel, there are a few related methods that should be mentioned. Most structural identifiability methods assume that all derivatives of outputs are measurable and all derivatives of all outputs are sufficiently varying. The latter condition is not always satisfied in practical modeling as, for example, a constant function may be used as an input, and this may lead to incorrect implementation of a model. In [32] a method for determining how many derivatives of the inputs must be non-zero was given, and an implementation was presented in [33]. In [4] an implementation was given for determining minimal sets of outputs that make a system structurally locally identifiable.
2 Results and discussion
2.1 Problem: There is an upper limit for how many derivatives can be estimated from biological data
There is usually an upper limit for the number of derivatives of an output signal that can be estimated from data. This is illustrated in the following example.
Example 2.1. Consider the system
where x1 and x2 are state variables, is a constant parameter, and y is an observable output variable; that is y is measurable by experiments but x1, x2 and
are unknown. Suppose that, unbeknownst to the observer,
, x1(0) = 10, and x2(0) = 0. Consequently the true signal of y is given by
. However, in practice, the measured signal will not show a smooth curve but rather a discrete time series that also has added noise. An illustration of this measurement signal can be found in Fig 1A, where 10 realizations of the function
, where the second term represents normally distributed noise with mean 0 and standard deviation 2, are depicted alongside the true function y(t) (Fig 1A). The number of derivatives that can accurately be estimated can be evaluated by deriving the function of a polynomial interpolation of these data points and comparing these estimated derivatives to the derivatives of the true function (Fig 1B). More specifically, this example shows a 6th degree polynomial interpolation for t-values ranging from 0 to 6 in increments of 0.5. Subsequently, the 0th to the 5th derivatives of these interpolations are then compared to the derivatives of the true function y(t). The number of derivatives that can accurately be estimated is dependent on the magnitude of noise in the system. More specifically, the magnitude of the added noise increases the number of accurately estimated derivatives declines (Fig 1C). To illustrate this dependency, the example above was repeated with the added noise sigma ranging from 0.1 to 10 in increments of 0.1 (
) and with 1000 data realizations generated at each noise level. Fig 1C shows the average number of derivatives that could accurately be estimated at each noise level. A derivative is considered accurate if the square root of the sum of squared residuals does not exceed the amplitude of y. From this figure, it is clear that the number of derivatives that can accurately be estimated diminishes rapidly as the signal-to-noise ratio approaches 1.
A: Examples of 10 different realizations of noisy data (colored lines) and the true function (black line) B: Illustrations of the
,
, and
derivatives of the function y(t) with examples of a poorly estimated derivative (red line) and an accurately estimated derivative (green line). C: The relationship between the average number of accurately estimated derivatives and the magnitude of the added noise
/the signal-to-noise ratio. The bars show the mean number of accurately estimated derivatives at each noise level and the error bars show the standard error of the mean (SEM).
The following example illustrates how this lack of availability of all derivatives of y is associated with an unrealistic assessment of whether parameters are identifiable.
Example 2.2. Consider the system
where x1, x2, and x3 are (unobserved) state variables, and
are constant parameters, and y is an (observed) output variable.
Running the algorithm from [29] shows that x1, x2, x3, , and
are locally identifiable. Let us attempt to estimate
from observation of y. A relation among
and y over
is
. Thus we can estimate
by using our observed values of
and
at some time t0:
In practice, the measured signal y may be too noisy to give an accurate estimation of . In such a situation,
is not identifiable from the given data, even though a structural analysis says that it is. Suppose now that we know that our measurements are good enough to give reliable estimates of
but of no higher derivative of y. To use structural methods to determine the degree of identifiability in such a situation, we need new definitions and methods.
Remark 2.3. Our approach assumes measurements of several derivatives of the output signal are made at a single time. In practice it is more common to measure the 0-th derivative of the output signal at several times. Having measurements with some uncertainty of is equivalent to having measurements with some uncertainty of
. Hence this aspect of our approach does not limit its generality.
For Example 2.2, one can estimate using the relation
together with measurements of y and
at three different times. However knowledge of
at three times can be used to estimate
at a single time.
2.2 A new definition for practical and structural identifiability: 
-identifiable
We now provide an intuitively straightforward definition that is useful for understanding the new methods. In Sect 5, we provide a precise definition, together with analytical proofs of all stated method properties.
Let be a non-negative integer and let n, m, and r be positive integers. Let
,
,
, and
. Let
and
be tuples of rational functions in x, u, and
over
. This setup determines a class of systems of ODEs with initial conditions, or a model class:
where x are state variables, which usually correspond to time-varying concentrations or physical properties; are the initial values of the state variables;
are constant parameters, e.g. corresponding to rate constants or volumes; u are input variables, which are observed and typically controlled by the experimentalist; and y are output variables, which are observed experimentally. With these notations in place, the new definition is roughly given by:
Definition 2.4 (roughly stated). Let be non-negative integers. Rational expression
is said to be
-identifiable if perfect knowledge of the first
derivatives of yi at a particular time allows us to determine the value of h at that point in time, up to a finite set.
This definition, which is more technically stated in Definition 5.4, provides the essence of the new definition: given only a limited set of derivatives of the different measurement signals, can we determine the parameter to be locally identifiable, i.e. as belonging to a finite set? This definition is analogous to traditional definitions of structural identifiability, with the only addition being that we now only have access to a finite set of derivatives. In other words, in traditional structural identifiability analysis, no such constraints on the number of derivatives available is present. This new definition is closer to practical identifiability since, with real biological data, one can only estimate a few derivatives from the measurement data. Note that in practice it is more common to measure the signal at multiple times. We explain in Remark 2.3 that this does not limit the generality of our approach.
2.3 Solution: Our new algorithms for calculating practical structural identifiability
For most models, it would be difficult to prove the -identifiability of a given parameter directly from the definition. We provide Algorithm 1, which can do this using only straightforward algebraic computations. The algorithm applies to any rational combination of states and parameters, of which an individual parameter is a special case.
Algorithm 1 Determine whether a given element of is
-identifiable.
Input : Equations
non-negative integers
rational expression
Output : “Yes” if h is -identifiable
“No” if h is not -identifiable Step 1: Write the matrix
Step 2: Compute the rank of J.
Step 3: Write the matrix Jh, obtained by adding the row
to J. Then calculate the rank of Jh.
If , conclude that h is not
-identifiable. (Output “No”).
If , conclude that h is
-identifiable. (Output “Yes”).
In the case where one is interested in just an individual parameter or state, one can perform Algorithm 2, where one column is removed instead of a row added, resulting in a smaller matrix.
Algorithm 2 Determine whether a given element of is
-identifiable.
Input : Equations
non-negative integers
choice of (resp.
)
Output : “Yes” if (resp. xi) is
-identifiable
“No” if (resp. xi) is not
-identifiable Step 1: Write the matrix J as in Algorithm 1.
Step 2: Compute the rank of J.
Step 3: Remove the column corresponding to (resp. xi) and compute the rank of the resulting matrix.
If the rank has decreased, conclude that (resp. xi) is
-identifiable. (Output “Yes”)
If the rank is the same, conclude that (resp. xi) is not
-identifiable. (Output “No”)
The precise definition of -identifiability is given in Definition 5.4, and we prove that Algorithm 1 and 2 are correct in Sect 5.2.
The symbolic calculations can be impractically slow for all but the smallest models. Therefore we have also provided probabilistic analogs of these algorithms that are much faster in practice. These are presented and their correctness is proven in Sect 5.3.
2.4 Intuitive understanding of the new algorithms
We first discuss Algorithm 1. Each row of J represents how an output, or one of its derivatives, varies with respect to small changes in each state and parameter. If the last row of Jh is a linear combination of the other rows, then the variation of h can be accounted for as a combination of the variations of the output derivatives. This, in turn, means that it seems like h can be expressed in terms of the known signals, i.e. that h is identifiable. Conversely, if the last row of Jh is linearly independent of the other rows, then the variation of h cannot be explained in terms of that of the available derivatives of the outputs, and one should not expect to be able to determine h from the available outputs and their derivatives.
A similar explanation lies behind Algorithm 2. The column corresponding to is dependent on the other columns if and only if the variation in the outputs and their derivatives is fully explained by variation in the other parameters and states, which is true if and only if
cannot be determined from numerical knowledge of the outputs and their derivatives, i.e.
is not identifiable.
2.5 Examples
2.5.1 Algorithm 1 determines identifiability of rational quantities, which is important in hypothesis testing.
Algorithm 1 (and its faster probabilistic version Algorithm 3) can be used to analyze so-called core predictions (see [10]). A core prediction is a well-determined property, i.e. a model prediction with a small uncertainty. Such core predictions are therefore often tested in future experiments, and are a central part of model-based hypothesis testing. Algorithm 1 can test whether or not such a model property really can be well-determined, given the available data. We illustrate this new possibility in the following two examples.
Example 2.5. The following equations give a model of insulin (ins) binding and activation of insulin receptor (IR). This model was originally presented in [7], and is explained more in detail therein and in the supplementary materials.
This model was one of several hypotheses tested in the article. To draw one of the conclusions in that article, the authors looked at the proportion of insulin receptor bound to an internal membrane:
It was predicted that propi should be between .55 and .80. Despite the fact that there are four outputs, none of the seven states (insulin receptor or substrate quantities at different locations) are identifiable, so it is not immediately obvious whether propi can be estimated from the measurements. Algorithm 3 with for all i shows that this quantity is indeed single-experiment locally identifiable (see Definition 5.8 and Corollary 5.10). The authors were able to experimentally determine the proportion to be
. Thus they were able to reject the model, which was the conclusion in that step in the paper.
In summary, our analysis shows that the proportion of receptors that are bound to an internal membrane indeed is an identifiable model property. This new result holds from a structural point of view, without assuming that all derivatives are known. These results also show this without relying on optimization or exhaustive parameter sampling, as is done using the traditional practical identifiability analysis done in the original papers [7,10]. This new result thus strengthens and confirms the conclusion drawn in the original paper.
Example 2.6. The equations below constitute the model of the interaction between liver and pancreatic cells from [8].
There are seven states, , two outputs,
, and no inputs. Numerical values were inserted for parameters whose values were known from operating specifications or estimated from the literature. The remaining parameters, here denoted by
, were to be determined from experimental data. One quantity of interest was
representing hepatic insulin sensitivity (see [8, Fig. 4 Plot E]). Algorithm 3 tells us that this quantity is indeed identifiable. Moreover, by using different values of we find that it is (6,5)-identifiable.
2.5.2 Identifiability tends to increase with increased order of observability.
It is typically the case that as we increase the number of derivatives that can be reliably estimated, the number of identifiable parameters increases, often dramatically. Therefore assuming that all derivatives are available can give a large overestimate of the number of identifiable parameters. How large this overestimate is can now be determined by our new algorithms. We illustrate this capability in the following two examples.
Example 2.7 (Drosophila period protein). The following equations constitute the model that was used in [13] to model Drosophila period protein.
A local structural identifiability analysis done in [29, p. 737] shows that all states and parameters other than M, ,
, Km, and ks are identifiable. However, Algorithm 4 shows that only PN is identifiable using 19 or fewer derivatives of y1. If in addition we observe M, it was shown in [29] that all states and parameters are identifiable. However Algorithm 4 shows this is the case only if at least 16 derivatives of the outputs are available. Fig 2 shows the number of states and parameters that are identifiable as a function of the maximum number of derivatives of each output that are available; for example
means that exactly
are available for all i.
The horizontal axis shows k: . The vertical axis shows the number of identifiable states and parameters in the model. Blue circles represent the case with only one output:
. Red circles represent the case with two outputs:
, y2 = M.
Blue circles represent output set and red circles represent output set
.
Example 2.8 (NF-B). We consider a model of NF-
B regulatory module, as presented below. This model was first introduced in [18], and an identifiability analysis was done in [17]. The model has 15 states, 28 parameters, 4 outputs, and 0 inputs.
The analysis in [17] shows that 21 states and parameters are identifiable, assuming that all derivatives of the four outputs can be estimated. However, using Algorithm 4, we found that if no more than 26 derivatives are available, then only 9 states and parameters are identifiable. Further analysis is shown in Fig 3.
The horizontal axis shows :
. The vertical axis shows the number of identifiable states and parameters in the model.
2.6 Limitations and questions for further research
The definitions and algorithms presented here naturally propose paths for continued research. Our approach requires the modeler to have already determined or assumed the maximum derivatives that can be estimated reliably. This raises the following questions:
- Given a situation where the noise level is fixed, how can we determine the orders of the highest derivatives that can be reliably used to identify parameter values?
- Conversely, if one has decided what parameters and states must be identifiable for an experiment to be useful, how can one determine the maximum acceptable noise level so that the derivatives necessary for identification are known precisely enough?
3 Theory
The definition of -identifiability and the justification of our results are made rigorous in Sect 5. We give a brief summary of this theory here.
The notion of identifiability involves many subtleties. Our precise definition is a type of generic local single-experiment identifiability. This is done in terms of uniqueness (up to a finite set) of parameter and state values given known values of the output derivatives at a certain point in time. We show that if all derivatives of all outputs are available, then our definition is equivalent to the notion of local identifiability given in [15, Def. 2.5].
Our definition of -identifiability is intended to capture the property that a modeler really wants to ascertain, but it is often difficult to verify directly. Therefore we show that this property is equivalent to a property involving ranks of matrices, which is straightforward to check using existing software. This is in fact the property alluded to in Algorithm 1.
Although this rank property is straightforward to check, it involves computing ranks of symbolic matrices, which can cost the modeler time. Therefore, to alleviate this problem, we present probabilistic algorithms that substitute random integers for the variables and compute modulo a random prime (Algorithms 3 and 5.3.1). These are modifications of the method described in [29]. We present detailed probabilistic statements that a reader can use to create an implementation without investigating the proofs. We present detailed proofs that the reader can check without excessive pencil and paper computations.
Moreover, our work addresses an oversight in [29] surrounding the probability that a rational expression vanishes when evaluated at a random tuple of integers modulo a random prime. The author assumes that the denominator does not vanish, but then gives a bound for the probability that the numerator does not vanish regardless of this condition. We give a bound on the probability that the numerator does not vanish given that the denominator does not vanish.
4 Conclusion
Practical identifiability analysis lies at the heart of model-based data analysis. State-of-the-art numerical methods, such as profile likelihood and MCMC, suffer from the limitation that they require an exhaustive coverage of the parameter space, which is not possible for large and realistic models. This coverage is not needed for structural methods, but these methods traditionally assume that one can determine all derivatives of the measurements y. Such derivatives are not available in practice, but the consequences of this unfulfilled assumption have not been examined, because corresponding methods and algorithms are missing. Herein, we present a new definition, -identifability, which restricts the analysis to those derivatives of y that are available. We also present new algorithms that can determine such identifiability in practice. Applications to previously published models demonstrate that we can determine not only identifiability of parameters, but observability of any model property (Examples 2.5 and 2.6). These results allow us to strengthen previous conclusions. For instance, the previous rejection of a specific model for insulin signaling was based on the model-guided experiments measuring the amount of internalized and dephosphorylated insulin receptors (Example 2.5), and we now show that this model property indeed can be identifiable, using a methodology that does not require coverage of the entire parameter space. Our analysis also shows that the number of parameters identifiable according to traditional structural methods are widely overestimated. If one, e.g., assumes that only up to third order derivatives, instead of all derivatives, are available, the number of identifiable parameters drops from 17 to 1 for the Drosophila model (Fig 2), and from 21 to 4 for an NF-
B model (Fig 3). In both models, the previously obtained identifiability is present only if at least 20 derivatives of all measurement signals are available. Our results bring us one step closer to a structural approach for practical identifiability analysis.
5 Appendix: Technical details
5.1 Notation and definitions
The notion of identifiability involves many subtleties. This subsection provides the setup necessary for rigorous treatment.
We repeat the setup described previously. Let be a non-negative integer and let n, m, and r be positive integers. Let
,
,
, and
. Let
and
be tuples of rational functions in x, u, and
over
. This setup determines a class of systems of ODEs with initial conditions, or a model class:
Fix non-negative integer , positive integers n, m, and r, n-tuple f and m-tuple g of elements in
for the rest of this section. The symbol
denotes the set of non-negative integers.
Notation 5.1. (Differential Algebraic Setting)
- (a) A differential ring
is a ring together with a derivation, that is, a function
satisfying
and
. All rings are assumed to be commutative and with multiplicative identity. For
and
,
denotes the image of a after k applications of
. The set
will be called the set of derivatives of a.
- (b) Let
be a differential ring and let
be a set of indeterminates. Then
denotes the polynomial ring in infinitely many indeterminates
with derivation
extended by
.
- (c) Let
be a differential subring of
and let
. Then
denotes the smallest subring of S2 containing S1 and the derivatives of the elements of Z.
- (d) Let
be a differential ring that is a subring of a differential integral domain
and let
. Then
denotes the fraction field of S1{Z}. This is a differential ring under the extension of
via the quotient rule
.
- (e) For any differential ring containing any elements of
we assume
maps each such element to 0.
- (f) Let
be a differential ring. A differential ideal I is an ideal satisfying
. For subset
, we denote by [A] the smallest differential ideal containing A.
- (g) For an ideal I and element a in a ring S, we denote
. This set is also an ideal in S.
- (h) Write
and
as fractions of elements of
, let
be the LCM of the denominators, and let
and
be such that
and
. Note that Q is unique up to multiplication by an element of
and choice of Q will not affect our definitions and results. We define the differential ideal of
as
. This is a prime (differential) ideal (cf [15, Lemma 3.2]) and thus
is a differential ring that is an integral domain.
- (i) For
, we denote by
the subset
of R. We define
.
Notation 5.2.1 (Analytic Setting)
- (a) Let
. Then
denotes the set of all functions that are complex analytic in some neighborhood of t = t0.
- (b) Let
. Then
denotes the complement to the set where at least one of the denominators of f or g vanishes at t = t0.
- (c) For
and
, let
.
- (d) For
and
, let
and
denote the unique solution of the initial value problem
(cf. [1, Theorem 2.2.2]). For appropriate integers j and k, let
denote the k-th derivative of the j-th component of
.
- (e) For
, a subset
is called Zariski open if there exists P in
such that U is the complement to the zero set of P.
- (f) For
and
, a subset
is called Zariski open if there exists P in
such that
- (g) For
,
, and
or
, the set of all nonempty Zariski open subsets of W will be denoted by
.
- (h) A subset
is called codiscrete if for any distinct
there exist disjoint open sets
and
such that
and
.
Notation 5.3. Let , let
, and let
.
- (a) Substituting
for
in h gives a function of one variable. The image of h under this substitution will be denoted by
.
- (b) The connected component containing t1 of the intersection of the domains of
,
,
and
will be denoted by
.
Definition 5.4. Let . Let
. The expression h is said to be
-identifiable if
where
One can interpret this definition as follows: If at some time t0, not belonging to a certain discrete “singular” set, we know the exact values of , then if we consider all possible values of the state variables and parameters at t0 that produce these values of
, and then we consider the set of values of h obtained by evaluating h at these values, this set is finite.
5.1.1 Examples to illustrate the definition.
Example 5.5. Consider the system
We show that is (1,0)-identifiable. Note that n = 1 and m = 2. Since two parameters occur we can take
to be any integer at least 2. Although no inputs occur the definitions require that we take r to be at least 1. Set
and r = 1. Let
and let
. Note that since there are no denominators we have
for all t0. Fix
. Let
and fix
. It is trivial to obtain a formula for the solution of this system, and we have
Let , and note that
For , the condition
gives
By the definition of , it follows that no side of the first or third of these equations is 0. Dividing the second equation by the product of the first and third proves that
. Since
, we see that
and thus has cardinality 1. We conclude that
is (1,0)-identifiable.
We now show that is not (0,0)-identifiable. Fix
and
. Let
. Fix codiscrete V, and let
. Using the first and third equations of (4), we see that for all
the tuple
satisfies the condition
for
. Therefore
and the right-hand side is infinite. We conclude that is not (0,0)-identifiable.
The next example includes an input and demonstrates why sometimes a discrete set must be excluded from V.
Example 5.6. Consider the system
where f2 is some rational function.
We have n = 2 and m = 2. Take and r = 1. We show that x2 is (0,0)-identifiable. We have
. Let
and let
. Let
. Note that because of the way U was defined,
is not the zero function. Let V be the complement of the vanishing of
, which is necessarily codiscrete since
and
are analytic. Let
. If
is such that
for
then it follows that
Since we know that
and
are defined and since
we know that
. Thus
and we conclude that x2 is (0,0)-identifiable.
5.1.2 
-identifiable implies single-experiment locally identifiable and the converse is true for sufficiently large 
.
Many notions of identifiability are used in the literature. Some of these are stated and compared in [5]. In general, Definition 5.4 is not equivalent to any published definition as far as we are aware.
Definition 5.4 can be viewed as a type of single-experiment generic local identifiability. It is generic because there is a (possibly empty) set of numerical values of parameters and input functions for which h cannot be recovered but if the true values of the parameters and inputs lie outside this set h can be recovered. It is local because we allow h to lie in a finite set. One could change “is finite” to “equals one” to create a definition of a globally -identifiable function, however we do not address this in the current work. It is “single-experiment” because it refers to only one instance of the model. This is in contrast to a multi-experiment approach where one observes multiple instances of the model, usually with the same equation parameters but different initial conditions.
Since the equations of are rational in x,
, and u, there exist relations among sufficiently high derivatives of y and hence it is not meaningful to consider
beyond a certain order. This is made precise by the following proposition.
Proposition 5.7. Let and
. For each
let
be the greatest non-negative integer such that the set
is not algebraic over
. Then h is
-identifiable if and only if h is
-identifiable. Moreover,
.
Proof: Since the transcendence degree of over
is
, it must be that
is algebraic over
. Hence such a
exists and moreover
.
For the direction, note that if
are such that
for all i, it follows from the definition that h is b-identifiable implies h is a-identifiable.
We now address the direction. Suppose h is
-identifiable. By Proposition 5.12, h is algebraic over
. For any i, by writing an algebraic dependence of
over
and differentiating (noting that the field has characteristic 0), we see that for all
the element
is algebraic over
. It follows that h is algebraic over
. By Proposition 5.12 h is
-identifiable.
Definition 5.8. Let . The expression h is said to be single-experiment locally identifiable (SELI) if
where
For , Definition 5.8 is equivalent to the definition of local identifiability given in Definition 2.5 of [15] (generalization to multiple inputs is asserted in Remark 2.2). In [15, Prop. 3.4 (a)
(c)], it was shown that for
, h is SELI if and only if h is algebraic over
. We extend this result to arbitrary
.
Proposition 5.9. Let . Then h is SELI if and only if h is algebraic over
.
Proof: Let be the system obtained by adding to
the equations
,
, and
. Note that
in
is the projection of
onto all coordinates but xn + 1. We divide the proof into the following three steps: h is
-SELI
is
-SELI
is algebraic over
h is algebraic over
.
We show h is -SELI
is
-SELI. Suppose h is
-SELI. Let
and U be as required by the definition. Define
. Let
. We verify that
. Since
, we have
where . It follows that
where is considered a subset of
. Thus
and we know that because h is
-SELI. This completes the first direction. Now suppose xn + 1 is
-SELI. Let
and U be as required by the definition. Define
to be the projection of
onto all coordinates other than xn + 1. Let
. Let
be such that
. Such a
exists because of the way
was defined. Now
Hence
and we know that because xn + 1 is
-SELI.
From [15, Prop. 3.4 (a) (c)], we have that xn + 1 is
-SELI
is algebraic over
.
We now show that h is algebraic over is algebraic over
. Suppose h is algebraic over
. Then
is algebraic over
. We now address the other direction. Let
and let
. Suppose xn + 1 is algebraic over
. Then
is algebraic over
. Let
be the minimal polynomial of h over
. Suppose ym + 1 appears in some ai. Let
be the differential field automorphism on
such that
is fixed pointwise,
, and
. Now
is a non-zero polynomial of lower degree with coefficients in
that has h as a root. Thus we have a contradiction. Therefore all the ai lie in
and h is algebraic over
.
This leads to the main result of this section.
Corollary 5.10. Let and
.
- h is
-identifiable
h is SELI.
- If
, then h is SELI
h is
-identifiable.
Proof: Suppose h is -identifiable. Then by Proposition 5.12 we know h is algebraic over
. It follows that h is algebraic over
and then by Proposition 5.9 we have that h is SELI.
Suppose that . Then by the arguments presented in Proposition 5.7 we have
. Suppose h is SELI. By Proposition 5.9 we have that h is algebraic over
, which equals
. By Proposition 5.12 we have that h is
-identifiable.
5.2 Proof of algorithms
The definition of -identifiability is stated in terms of analytic functions. The Proposition 5.12 gives a correspondence between the analytic property and an algebraic property. Its proof will use the following lemma.
Lemma 5.11. Let , and let
. The map
gives a -algebra homomorphism
, where S is the ring
. Moreover, under this map
.
Proof: Denote the stated map by . Since
satisfy all the relations among
and no denominator of S is sent to 0, we see that
is a homomorphism on S. Now
, so
. The existence and uniqueness theorem guarantees the existence and uniqueness of
, and
is given by
.
Proposition 5.12. Let . Expression
is
-identifiable if and only if h is algebraic over the subfield
of Frac(R).
Proof: Part 1: Assume that h is algebraic over
. Consider the minimal polynomial of h over
. Clearing denominators, we obtain the polynomial
, where each
and
. By [16, Corollary 6.6], there exist
and
such that for all
it holds that
(see Notation 5.3) is not the zero function. Fix such
and U and choose a
. Let V be the complement of the vanishing of
. Since
is analytic about 0, we know that V is codiscrete.
Fix . Let
,
be such that
and
. Since each ai belongs to
, it follows from Lemma 5.11 that
for all i. Applying Lemma 5.11 to the equation
, we find that
. Thus
is a root of the non-zero polynomial
. Noting that
we conclude that
is finite.
Part 2: Assume that h is not algebraic over
and that h is
-identifiable. We will give a proof by contradiction. The proof can be divided into four steps:
Step 0 Label the rings that will be used in the proof.
Step 1 Choose . This will be done in terms of the non-vanishing of minimal polynomials over fraction fields of intermediate rings. Fix V and t0 and note that Lemma 5.11 with
gives a ring homomorphism
.
Step 2 Show that the set is infinite. This will involve careful extension of ring homomorphisms.
Step 3 Verify that is infinite by noting that each
corresponds to a tuple
.
We begin the proof.
Step 0 Label as
. Without loss of generality, let k be such that
if and only if bj is algebraic over
. Let R2 equal
and let F2 be its field of fractions. Let
be such that
is a transcendence basis for
over F2, relabeling if necessary. Let
and let F3 be its field of fractions. Now
are each algebraic over
. For
let Pj(Z) be the minimal polynomial of bj over F4 multiplied by the LCM of the denominators, where F4 is viewed as the field of fractions of
. For each
, let
be the leading coefficient of Pj. Write
, where
. Let
be the minimal polynomial of
over F4 multiplied by the LCM of the denominators. (Note that if
then
.) Let
and
be its leading and constant coefficients, respectively.
Step 1 Let and U be as guaranteed by the definition of
-identifiability. Consider the subset of
consisting of all
such that for all
the expressions Lj,
, and
evaluated at
and then t = 0 are not zero. This subset is non-empty because it is the intersection of finitely many non-empty Zariski open sets. Hence its intersection with
is non-empty. Fix such a
. Let V be as required by the definition of
-identifiability, let
(recall Notation 5.3), and let
be the
-algebra homomorphism from R[h] to
given by applying the ring homomorphism described in Lemma 5.11 with
.
Step 2 We now show that there are infinitely many such that there exists a
-algebra homomorphism
such that
,
, and
.
Define on R2 by
. For each j, define
to be the element obtained by evaluating the coefficients (in R2) of Lj via
; define the analogous expressions with Lj replaced by
and
. By the way
was chosen, we have that
,
, and all
are non-zero. Using Lemma 5.13 we can extend
to
in a way that makes neither
nor
nor any
equal to zero. Choose such an extension and call this, somewhat abusing notation,
. Since h is not algebraic over F3, by Lemma 5.13
can be extended by mapping h to any element of
. Now there are infinitely many c such that
. Fix such a c for the remainder of the proof and extend
to R4 by letting
.
We now extend to
. By the preceding discussion, the leading and constant coefficients of
are non-zero, so by Lemma 5.13 we can extend
to
so that
.
Next we extend to
. Observe that the minimal polynomial
of
over
is a factor of
. Therefore
is a non-zero factor of
. By Lemma 5.13 we can extend
to
.
The argument from the preceding paragraph can be repeated to show that can be extended to
. Applying this several more times, we see that we can extend
to R[h].
Let . We have shown that
is infinite.
Step 3 We will now show that is infinite. We have
By Lemma 5.11 it holds that and
. Hence
Since for each we have
, the first conjunct is satisfied. By the definition of T, the second conjunct is satisfied for all
. Hence
We showed in Step 2 that the right hand side is infinite, and thus is infinite. Therefore h is not
-identifiable, contradicting our assumption.
Lemma 5.13. Let W be a (possibly infinite) set of indeterminates and let S be a -subalgebra of the field of rational functions
. Let
be a
-algebra homomorphism. Let
.
Suppose f is not algebraic over . Then
can be extended to S[f] by mapping f to any element of
.
Suppose f is algebraic over . Let P(Z) be the minimal polynomial of f over
multiplied by the LCM of the denominators. Write
, where
. If
, then
can be extended to S[f]. If furthermore
, then in this extension
.
Proof: Suppose f is not algebraic over . Our result follows from [2, p. 99].
Suppose f is algebraic over and
. If
then the result is trivial. Assume f is not in
. By [2, Theorem 3.2 p. 347],
can be extended to S[f] or
. If the former is true the proof is complete. Suppose
can be extended to
. Writing
, we have that
. If
, then it follows that
, which contradicts our hypotheses. Thus
. Now we can extend
to the ring
, which is equal to the subring of
consisting of fractions with numerators in
and denominators in
(cf. [2, p. 346]). Since
, the element f belongs to
. By restricting
to S[f] we have an extension of
to S[f].
Still assuming f is algebraic over , suppose
. The image of f must satisfy
. Thus it is impossible that
. ◻
It is not always obvious whether a given field element is algebraic over a given subfield. The following proposition gives an equivalence that, for our purposes, reduces this question to the problem of checking the rank of a matrix with easily computable entries. We will use the following notation:
Notation 5.14. Let be elements of a
-algebra and let
be an algebraically independent set over
. We denote by
the matrix whose (i,j)-th entry is
. If M is such a matrix and
, then
denotes the result when the column corresponding to Z0 is removed. If this column does not appear in M then
is equal to M.
We will use the following algebraic fact, which generalizes [12, Thm. 2.3].
Proposition 5.15. Let each be algebraic over
, where
and all wi and zi are indeterminates. The elements
are algebraically independent over
if and only if the matrix
has rank equal to s.
Proof: Let and let
. Note that
where I is the identity matrix and A is an
matrix.
Now is algebraically independent over
iff
is algebraically independent over
iff (by [12, Thm. 2.3])
iff
.
Proposition 5.16. Let and let
. Then h is algebraic over
if and only if the matrix
has the same rank as
.
Proof: Note that a subset of is algebraically independent over
if and only if it is algebraically independent over
. Let T be a maximal subset of
that is algebraically independent over
. Then by Proposition 5.15 with
and
we have
.
Suppose h is algebraic over . Then
is algebraic over
. Then
by Proposition 5.15. Therefore
.
Suppose h is not algebraic over . Then
is not algebraic over
. Then
by Proposition 5.15.
Corollary 5.17. Algorithm 1 always terminates. The output is “Yes” if and only if h is -identifiable.
Proof: Termination is obvious. The other result follows from Proposition 5.12 and Proposition 5.16.
Proposition 5.18. Let and let J be as in Algorithm 2. Let
. Then z is algebraic over
if and only if
.
Proof: Without loss of generality we assume . By Proposition 5.16,
is algebraic over
if and only if
. Now
Corollary 5.19. Algorithm 2 always terminates. The output is “Yes” if and only if is
-identifiable.
Proof: Termination is obvious. The other result follows from Proposition 5.12 and Proposition 5.18.
5.3 Probabilistic method for improved speed
5.3.1 Presentation of algorithms and summary of results.
Algorithm 1 and 2 involve computing the rank of a matrix of rational expressions. It is usually much faster to insert random numbers for the variables and compute the rank of the resulting matrix. The disadvantage of this is that the numerical matrix may have lower rank than the symbolic. In [28] a method for doing this in the case where the coefficients of are integers with user-specified probability of success was given. We adapt that method to our algorithms.
For the rest of this section, assume the coefficients of f and g in (2) and h in Algorithm 1 are integers. Note that the entries of J belong to . Our strategy involves randomly choosing
non-negative integers and a prime number, and then evaluating the determinant of our matrix at these integers modulo the prime. Algorithm 3 implements this strategy on Algorithm 1.
Algorithm 3 Determines whether parameter combination h is -identifiable.
Input : Equations , where
Output : “Yes” if h is -identifiable with probability at least
“No” if h is not -identifiable with probability at least
Step 1a: Compute the least such that
.
Step 1b: Compute D and the least N as in Proposition 5.25 with .
Step 1c: Choose uniformly at random.
Step 1d: If Q(T) ≡ 0 mod p, repeat Step 1c. Otherwise continue to Step 1e.
Step 1e: Compute Jh(T) mod p.
Step 2: Compute and rank (J(T) mod p).
Step 3: If , output “Yes”. Otherwise output “No”.
Algorithm 4 Determines the -identifiable subset of
.
Input : Equations , where
Output : Subset of consisting of exactly the
-identifiable elements, with
probability at least
Step 1a: Compute the least such that
.
Step 1b: Compute D and the least N as in Proposition 5.25 with .
Step 1c: Choose uniformly at random.
Step 1d: If , repeat Step 1c. Otherwise continue to Step 1e.
Step 1e: Compute J(T) mod p.
Step 2: Compute .
Step 3: Let .
For
Compute .
If , add z to Sout.
End For
Step 4: Return Sout.
While Algorithm 2 determines whether an individual parameter is -identifiable, Algorithm 4 uses this concept to determine, with user-specified probability, all elements of
that are
-identifiable.
The main results on Algorithm 3 and Algorithm 4 are the following:
- The expected time to reach Step 1e is negligible. Once Step 1e is reached the algorithm is guaranteed to terminate. (Proposition 5.29)
- The probability that the output is correct is at least
. (Propositions 5.30 and 5.31)
- For Algorithm 3, if in Step 2
, then h is
-identifiable. For Algorithm 4, if in Step 2
, then every element of
is
-identifiable. (Proposition 5.32)
Remark 5.20. At the beginning of this section, we assumed n, m, and r are positive integers. As noted in Example 5.5, and r must be at least equal to the number of parameters and outputs, respectively, that appear in
, but can be chosen to be greater without changing identifiability results. In Algorithms 5.3.1 and 5.3.1 it is sufficient to use
and r equal to the number of parameters and inputs, respectively, appearing in the equations. In particular one can use r = 0 if no inputs appear and the main results on the algorithms are correct.
Remark 5.21. We have presented Algorithms 5.3.1 and 5.3.1 only for the case where f and g are not all elements of , since removing this restriction would require addressing special cases in several of the proofs. If
, then h is
-identifiable if and only if
. One could easily add a step at the beginning of either algorithm to accommodate this case.
Remark 5.22. Our algorithms do not specify the methods used to compute the ranks of matrices. One can use the state-of-the-art method for this to achieve maximum speed.
5.3.2 Proof of algorithms.
In Algorithms 5.3.1 and 5.3.1, a random tuple of integers T and a random prime p are chosen, we check that the denominator of a determinant evaluated at T does not vanish modulo p, and then calculate the rank of a matrix after evaluating at T modulo p. We show that this gives the correct results with user-specified probability. The main results on Algorithms 3 and 5.3.1 are Propositions 5.29, 5.30, 5.31, and 5.32. The theory is based on bounds on integer roots of polynomials with integer coefficients, as well as the distribution of the prime numbers.
First, we use Proposition 5.23, Lemma 5.24, and Lemma 5.26 to prove Proposition 5.25, which gives conditions on the sets from which we choose T and p so that the numerator of the determinant evaluated at T does not vanish modulo p with user-specified probability. This proposition is essentially a more precisely stated version of [28, Proposition 6], and the proof we give is outlined in [28].
Next, Proposition 5.28 shows that the probability that the numerator vanishes given that the denominator does not vanish can be specified by the user. Note that this is the true probability associated with the algorithm, since we must first check that our choice of (T,p) does not make the denominator vanish before proceeding. This issue is not addressed in similar algorithms (cf. [29, p. 739], [17]) and is non-trivial, as shown by Example 5.27.
Finally, we prove statements about the algorithms that are directly relevant to helping the user interpret them. Although in principle, arbitrarily many instances of (T,p) may need to be chosen before finding one that does not make the denominator vanish, Proposition 5.29 shows that the expected time for a successful choice is negligible. It also asserts the algorithm’s termination after such a successful choice. Propositions 5.30 and 5.31 show that the algorithms produce the correct result with user-specified probability. Proposition 5.32 states that when the rank of the specialized matrix is full, the algorithms output the correct result with certainty.
Proposition 5.23 ([3] Prop. 98 p. 192). Let be a polynomial of total degree D over an integral domain A. Let
. If an element
is chosen from
uniformly at random, then
Lemma 5.24 ([34] Lemma 18.9 p. 525). Let be a nonempty finite set of prime numbers, let
, and let
. If p is chosen from S uniformly at random, then
Proposition 5.25. Let J (resp. Jh) be as in Algorithm 2 (resp. Algorithm 1) and suppose J (resp. Jh) has at least one non-zero entry. Let
- d1 = maximum degree of the numerators and denominators of
and
(resp.
,
and h)
, where
(resp.
)
- C be such that
,
where(resp.
)
such that
Let J0 be a square submatrix of J (resp. Jh) with rank equal to that of J (resp. Jh). Note that the numerator of lies in
. If a tuple of values T of the variables is chosen uniformly at random from
and p is chosen uniformly at random from S, then
where represents the specialization of the numerator of
at the chosen values.
Proof: We prove the main version first. The version for Jh will follow quickly from this.
Note that since J has at least one non-zero entry, d1 is well-defined and positive, and hence D is positive. Let W denote . We have that
By Proposition 5.23, we have that
We show that the degree of the numerator of any entry of J is no greater than . Fix
. We first show by induction that for
we can write
, where
. For the base case k = 0, recall from Notation 5.1 we have that
, and
. For the inductive hypothesis, fix
and assume
with
. Applying the quotient rule we have that
recalling from Notation 5.1 that each . Since
and
do not exceed (n + m)d1, we conclude the inductive step. Continuing the proof of the bound on the degrees of the numerators of the entries of J, note that for any
we can write
where
. So we can write
where
. In J0 the value of k does not exceed
so we conclude the argument.
Since J0 is square it has at most rows. Therefore
is no greater than D. Thus, we have
Assume that . Lemma 5.26 below shows that
. Our assumptions imply that
and hence
. From [34, Exercise 18.18] it follows that S is non-empty. Using Lemma 5.24 with M = W, we have
By [34, Exercise 18.18], we have . It follows that
Combining (5), (7), and (8), we have our result.
We now prove the version with Jh. Let be the system obtained by adding the equation ym + 1 = h to
and set
. Now Jh for
is equal to J for
. Our result follows from the main version of the proposition.
Lemma 5.26. In the setup of Proposition 5.25, .
Proof: For a polynomial p with integer coefficients, we shall define the height of p as . We will use the following properties ([28, Lemma 1]): For polynomials
in
variables, tuple of integers T0, and partial derivative
, the following hold:
,
,
, and
.
Fix and write
, we will prove by induction that for all
We noted earlier that , all
, and
are no greater than (n + m)d1. It follows from this and the product height property that
. For conciseness, we will use
and
to denote these degree and height bounds, respectively.
For the base case k = 0, note that . For the inductive hypothesis, fix
and suppose Ak satisfies (9). As shown in the (6), we have
Using the derivation and product height properties, as well as the bound , we have
By the sum height property we have that
By the product height property we have
By the sum height property we have
Noting that , we conclude the inductive step.
Now for , we have that
is equal to
. A bound on the height of the numerator is given by the right-hand side of (10). Since RHS of (10)
RHS of (11)
RHS of (12), we see that a bound on this height is also given by the RHS of (9) with k replaced by k + 1. Observing that the maximum value of k occurring for an entry in J0 is
and applying the evaluation height property, we have that the numerator of each entry of J0(T) is bounded by
Since our bounds assume that entries in the same row have the same denominator, we have that , where
is the matrix whose elements are the numerators of J0. Henceforth we assume J0 has polynomial entries with heights bounded by B when evaluated at T. Using Hadamard’s Theorem, we have that
, where
is the square root of the sum of the squares of the elements of the i-th column of J0(T). Hence
. Thus
Recalling that
we conclude the proof.
After a set of random integers and a random prime number is chosen, we must first check that no denominator vanishes modulo the prime number before evaluating the rank of the matrix. Thus the probability that the algorithm gives the correct answer is not simply the probability that p does not divide , but rather the probability that p does not divide
given that p does not divide
. If a random tuple T is chosen and used to evaluate two polynomials A and B, it is not necessarily the case that
, as the following example shows:
Example 5.27. Let k be a positive integer and let and
. If an integer T is chosen uniformly at random from
, then
and
.
The following proposition shows that the conditions used for Proposition 5.25 also give a bound on the conditional probability that is relevant to our algorithms.
Proposition 5.28. Let J (resp. Jh) be as in Algorithm 2 (resp. Algorithm 1) and suppose J (resp. Jh) has at least one non-zero entry. Let , D, S, and J0 be as in Proposition 5.25. If a tuple of values T of the variables is chosen uniformly at random from
and p is chosen uniformly at random from S, then
where represents the specialization of the numerator of
at the chosen values.
Proof: Let M denote the sample space . Let
Now
By Proposition 5.25 we have . We now prove the same bound holds for
. Now
, so using Proposition 5.23 we have
. By the first statement of [28, Prop. 3] with j = 0 we have that the height of Q(T) is no greater than
, which is no greater than
. Thus
and if
then by Lemma 5.24 we have
, which by [34, Exercise 18.18] is no greater than
. Thus
. Now we have
Proposition 5.29. Fix the input to Algorithm 3 (or Algorithm 4). (i) The probability that Step 1e will be reached in no more than k iterations of Step 1c is at least . (ii) When Step 1e is reached the algorithm is guaranteed to terminate.
Proof: (i) It was shown in the proof of Proposition 5.28 that , and based on the way
was chosen in Step 1a it is trivial to verify that this is no greater than
. Therefore for k independent choices the probability that
for at least one of them is at least
.
(ii) This is obvious.
Proposition 5.30. Fix the input to Algorithm 3. If h is -identifiable, the probability that the output is “Yes” is at least
. If h is not
-identifiable, the probability that the output is “No” is at least
.
Proof: Consider the following subsets of :
,
,
, and
. We have
Proposition 5.28 gives us that and
.
Suppose h is -identifiable. By Propositions 5.12 and 5.16, we have
. We show that
. Suppose
. Then
, and we deduce that
, so the algorithm outputs “Yes”.
Suppose h is not -identifiable. By Propositions 5.12 and 5.16, we have
. We show that
. Suppose
. Then
, so the algorithm outputs “No”.
Proposition 5.31. Fix the input to Algorithm 4. Let Sid denote the -identifiable subset of
and let Sout denote the output of Algorithm 4. The probability that
is at least
.
Proof: Because Sout depends on (T,p) we shall use the notation Sout(T,p). Consider the following subsets of :
and
. We have
By Proposition 5.28 we have . We show
. Suppose
. Let
. Suppose z is
-identifiable. By Propositions 5.12 and 5.16, we have
. Now
. Since the final row of Jz(T) must be
, it follows that
. Therefore
. Suppose z is not
-identifiable. It follows from Propositions 5.12 and 5.16 that
. Because
and the final row of Jz(T) is
we have that
. Now
. Therefore
.
Proposition 5.32. Suppose that in Step 2 of Algorithm 3 (resp. Algorithm 4) . Then h is
-identifiable and the output is “Yes” (resp. every element of
is
-identifiable and the output is
).
Proof: We first prove the statement regarding Algorithm 3. We have , so
. It follows that
and by Propositions 5.12 and 5.16 we have that h is
-identifiable. Since
, it follows that
and the output will be “Yes”.
We now address Algorithm 4. By the preceding paragraph . Since for any
the matrix
has only
columns, it must be that
and hence by Propositions 5.12 and 5.18 each element of
is
-identifiable. Similarly,
and hence the output is
.
Supporting information
S1 File. Maple code for algorithm 4.
Currently contains entries for Drosophila period protein model.
https://doi.org/10.1371/journal.pone.0327593.s001
(PDF)
S2 File. Maple code for algorithm 3.
Currently contains entries for NF-B model.
https://doi.org/10.1371/journal.pone.0327593.s002
(MW)
S3 Table. Description of models.
Details of the models used in the paper.
https://doi.org/10.1371/journal.pone.0327593.s003
(MW)
References
- 1.
Hille E. Ordinary differential equations in the complex domain. Dover; 1997.
- 2.
Lang S. Algebra. 3rd ed. New York: Springer; 2002.
- 3.
Zippel F. Effective polynomial computation. Boston: Springer; 1993.
- 4. Anguelova M, Karlsson J, Jirstrand M. Minimal output sets for identifiability. Math Biosci. 2012;239(1):139–53. pmid:22609467
- 5. Anstett-Collin F, Denis-Vidal L, Millérioux G. A priori identifiability: an overview on definitions and approaches. Annu Rev Control. 2020;50:139–49.
- 6. Bellu G, Saccomani MP, Audoly S, D’Angiò L. DAISY: a new software tool to test global identifiability of biological and physiological systems. Comput Methods Programs Biomed. 2007;88(1):52–61. pmid:17707944
- 7. Brännmark C, Palmér R, Glad ST, Cedersund G, Strålfors P. Mass and information feedbacks through receptor endocytosis govern insulin signaling as revealed using a parameter-free modeling framework. J Biol Chem. 2010;285(26):20171–9. pmid:20421297
- 8. Casas B, Vilen L, Bauer S, Kanebratt K, Huldt CW, Magnusson L, et al. Integrated experimental-computational analysis of a liver-islet microphysiological system for human-centric diabetes research. bioRxiv. 2021.
- 9. Cedersund G, Roll J. Systems biology: model based evaluation and comparison of potential explanations for given biological data. FEBS J. 2009;276(4):903–22. pmid:19215297
- 10. Cedersund G. Conclusions via unique predictions obtained despite unidentifiability–new definitions and a general method. FEBS J. 2012;279(18):3513–27. pmid:22846178
- 11.
Cedersund G. Prediction uncertainty estimation despite unidentifiability: an overview of recent developments. Studies in mechanobiology, tissue engineering and biomaterials. Springer; 2015. p. 449–66. https://doi.org/10.1007/978-3-319-21296-8_17
- 12. Ehrenborg R, Rota G-C. Apolarity and canonical forms for homogeneous polynomials. Eur J Combinator. 1993;14(3):157–81.
- 13. Goldbeter A. A model for circadian oscillations in the Drosophila period protein (PER). Proc Biol Sci. 1995;261(1362):319–24. pmid:8587874
- 14. Gross E, Harrington H, Meshkat N, Shiu A. Linear compartmental models: input-output equations and operations that preserve identifiability. SIAM J Appl Math. 2019;79(4):1423–47.
- 15. Hong H, Ovchinnikov A, Pogudin G, Yap C. Global identifiability of differential models. Commun Pure Appl Math. 2020;73(9):1831–79.
- 16.
Hong H, Ovchinnikov A, Pogudin G, Yap C. Global identifiability of differential models. https://arxiv.org/abs/1801.08112
- 17. Karlsson J, Anguelova M, Jirstrand M. An efficient method for structural identifiability analysis of large dynamic systems*. IFAC Proc Vol. 2012;45(16):941–6.
- 18. Lipniacki T, Paszek P, Brasier ARAR, Luxon B, Kimmel M. Mathematical model of NF-kappaB regulatory module. J Theor Biol. 2004;228(2):195–215. pmid:15094015
- 19. Meshkat N, Kuo CE, DiStefano J 3rd. On finding and using identifiable parameter combinations in nonlinear dynamic systems biology models and COMBOS: a novel web implementation. PLoS One. 2014;9(10):e110261. pmid:25350289
- 20. Meshkat N, Sullivant S, Eisenberg M. Identifiability results for several classes of linear compartment models. Bull Math Biol. 2015;77(8):1620–51. pmid:26337290
- 21.
Milne JS. Fields and galois theory (v5.00). 2021. https://www.jmilne.org/math/CourseNotes/ft.html
- 22. Nowak K. Some elementary proofs of Puiseux’s theorems. Zeszyty Naukowe Uniwersytetu Jagiellon´skiego Universitatis Iagellonicae Acta Mathematica. 2000;1245.
- 23. Ovchinnikov A, Pogudin G, Thompson P. Parameter identifiability and input–output equations. Applicable algebra in engineering, communication and computing. 2021.
- 24. Ovchinnikov A, Pillay A, Pogudin G, Scanlon T. Computing all identifiable functions of parameters for ODE models. Syst Control Lett. 2021;157:105030.
- 25. Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingmüller U, et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics. 2009;25(15):1923–9. pmid:19505944
- 26. Raue A, Schilling M, Bachmann J, Matteson A, Schelker M, Kaschek D, et al. Lessons learned from quantitative dynamical modeling in systems biology. PLoS One. 2013;8(9):e74335. pmid:24098642
- 27.
Saccomani MP, Bellu G. DAISY: An efficient tool to test global identifiability. Some case studies. In: 2008 16th Mediterranean Conference on Control and Automation; 2008. p. 1723–8.
- 28.
Sedoglavic A. A probabilistic algorithm to test local algebraic observability in polynomial time. In: Proceedings of the 2001 international symposium on symbolic and algebraic computation. 2001. p. 309–17. https://doi.org/10.1145/384101.384143
- 29. Sedoglavic A. A probabilistic algorithm to test local algebraic observability in polynomial time. J Symbol Comput. 2002;33(5):735–55.
- 30. Villaverde AF, Barreiro A, Papachristodoulou A. Structural identifiability of dynamic systems biology models. PLoS Comput Biol. 2016;12(10):e1005153. pmid:27792726
- 31. Villaverde AF, Fröhlich F, Weindl D, Hasenauer J, Banga JR. Benchmarking optimization methods for parameter estimation in large kinetic models. Bioinformatics. 2019;35(5):830–8. pmid:30816929
- 32. Villaverde AF, Evans ND, Chappell MJ, Banga JR. Sufficiently exciting inputs for structurally identifiable systems biology models. IFAC-PapersOnLine. 2018;51(19):16–9.
- 33. Villaverde AF, Evans ND, Chappell MJ, Banga JR. Input-dependent structural identifiability of nonlinear systems. IEEE Control Syst Lett. 2019;3(2):272–7.
- 34.
von zur Gathen J, Gerhard J. Modern computer algebra. 3rd ed. Cambridge University Press; 2013.
- 35. Wieland F-G, Hauber AL, Rosenblatt M, Tönsing C, Timmer J. On structural and practical identifiability. Curr Opin Syst Biol. 2021;25:60–9.