Branching Out into Structural Identifiability Analysis with Maple: Interactive Exploration of Uncontrolled Linear Time-Invariant Structures

Abstract

Suppose we wish to predict a physical system’s behaviour. We represent the system by model structure S (a set of related mathematical models defined by parametric relationships between variables), and parameter set $\varTheta$. Each parameter vector in $\varTheta$ corresponds to a completely specified model in S. We use S with system data in estimating the “true” (unknown) parameter vector. Inconveniently, S may approximate our data equally well for multiple parameter vectors. If we cannot distinguish between alternatives, we may be unable to use S in decision making. If so, our efforts in data collection and modelling are fruitless.

This outcome occurs when S is not structurally global identifiable (SGI). Fortunately, we can test various structure classes for SGI prior to data collection. A non-SGI result may inform a remedy to the problem.

We aim to assist SGI testing with suitable Maple 2020 procedures. We consider a class of “state-space” structure where a state-variable vector $\mathbf{x}$ is described by constant-coefficient, ordinary differential equations, and outputs depend linearly on $\mathbf{x}$. The “transfer function” approach is suitable here, and also for the “compartmental” subclass (mass is conserved).

Our use of Maple’s “Explore” permits an interactive consideration of a parent structure, and variants of this produced by user choices. Results of the SGI test may differ for different variants. Our approach may inform the interactive analysis of structures from other classes.

Appendices

Appendix 1Maple Code for Drawing a Modified Compartmental Diagram

We use Listing 1.7 in drawing a modified compartmental diagram of the model structure currently under investigation. When the Explore window associated with Listing 1.8 is launched, the diagram displayed is updated in response to user selections from the drop-down “layout” menu or changes to the input boxes which set parameter values.

Appendix 2Maple Code to Launch an Explore Window

Listing 1.8 presents the Explore command which launches our interactive SGI test dashboard by invoking Listing 1.6. Here we consider the case of three state variables and three outputs; the user can readily change these details. To explain the parameters: $\mathtt{A}$ is the structure’s $\mathbf{A(\boldsymbol{\theta })}$, p1, p2, p3 are the observation gain parameters on the leading diagonal of $\mathbf{C(\boldsymbol{\theta })}$, and p4, p5, p6 are the initial state parameters in $\mathbf{x_{0}}(\boldsymbol{\theta })$. Initially, each of p1,...,p6 are assigned a parameter symbol appropriate for their relationship to $\boldsymbol{\theta }$. Each of these six parameters may be changed through a text-input box. Parameter p7 supplies a graph output style understood by DrawGraph, initially (the widely applicable) “default”. Output from other options (such as “spring”) may be easier to interpret, but return an error when any of p1, p2, or p3 are set to zero, causing the removal of a link between a state variable and its corresponding output. Parameter p8 takes one of the two pre-defined values for theta_mod_type, which dictates the method employed in creating theta_prime from theta (used by theta_prime_creation). The user changes p7 and p8 values by selecting an option from the relevant drop-down menu. If logical-type parameter tracing:=true, Maple will show the output of steps used in constructing the structure’s compartmental diagram.