Identifiability for complex models with algebra equations

[Zhe-Jessica]

Thanks very much for the package. It is a very useful tool for checking identifiability before parameter estimation. I went through all the examples and tested it on a simple model and the package worked pretty well. However, I noticed that I cannot include any algebra equations when I define the model, otherwise errors occur. Only differential equations and output equations are allowed to be included. For many models, the differential equations become onerous after substituting all variables based on algebra equations. Furthermore, some variables are not easily substituted by solving all algebra equations. Is it possible to include algebra equations when defining the model?

[pogudingleb]

Thank you for using the software and providing feedback!

For the moment, algebraic constraints are not allowed (in full generality this is a complicated even in theory). If you would like to avoid substituting by hand, you can use the interface via ModelingToolkit and use SymbolicUtils for performing a substitution. Here is an example of the code

using StructuralIdentifiability
using ModelingToolkit, SymbolicUtils

@parameters a
@variables t, x1(t), x2(t), z(t)
D = Differential(t)

eqs = [D(x1) ~ z^2, D(x2) ~ a * z + x1]

subst = Dict(z => x2 + a)

eqs_transformed = [substitute(e, subst) for e in eqs]

ode = ODESystem(eqs_transformed, t, name = :dae)

println(assess_identifiability(ode, measured_quantities = [x1]))

If the substitution is not possible, then typically one should change the coordinates somehow to make it possible but this depends on the system at hand. If you have a particular system in mind, feel free to post it here or send to me (gleb.pogudin@polytechnique.edu).

[Zhe-Jessica]

Thank you very much for your quick reply. The interface SymbolicUtils you mentioned helps a lot. I tried it on the model I used and noticed a new problem.

I am unsure if StructuralIdentifiability includes the case where the model has parameters in the exponential function. For example, if I change the example code subst = Dict(z => x2 + a) to subst = Dict(z => x2 + exp(a)), I will get an error ERROR: ArgumentError: Function exp is not supported after I run assess_identifiability function. The model which I tried to check has parameters in exponential function, so I noticed this probelm.

Thanks again for your help. Look forward to your reply.

[pogudingleb]

Indeed, the package works only with rational dynamics.
However, it is often possible to recast a non rational system (e.g. with exponentials) to a rational one by a coordinate transformation. For example, see section A.2 and A.3 here.
In the specific example you have given (when both a and exp(a) appear) this is problematic though. Feel free to show me the particular model you are interested in, maybe can suggest an appropriate transformation.

[Zhe-Jessica]

Thanks for helping. Maybe Taylor expansion can be used to approximate the exponential function in this case.

I uploaded the .jl file which included the model and my test code. The file is in the link below. Please let me know if you have question about the model.
https://github.com/Zhe-Jessica/-Identifiability-for-a-generator-model/blob/main/3_Github_Issue.jl

[pogudingleb]

Thank you for sharing the model. In fact, while you have both UAx and exp(0.133 * UAx) in your expressions, the UAx cancels everywhere. For example (using your code)

julia> subst_gen[Tac]
(0.017674089784376106UAx*(1 - exp(0.0133UAx))) / (0.004413919737283497UAx - 0.017674089784376106UAx*exp(0.0133UAx)) + (-0.0133UAx*Tah(t)) / (0.004413919737283497UAx - 0.017674089784376106UAx*exp(0.0133UAx))
julia> simplify(subst_gen[Tac])
(0.017674089784376106 - 0.017674089784376106exp(0.0133UAx) - 0.0133Tah(t)) / (0.004413919737283497 - 0.017674089784376106exp(0.0133UAx))

Therefore, you can just replace exp(0.0133 * UAx) by some new parameters and eliminate any special functions.

[Zhe-Jessica]

Good to learn the simplify function. Thank you very much for helping. I can check the identifiability of the model now.