Employing observability rank conditions for taking into account experimental information a priori

TL;DR

The paper investigates how observability-based rank tests can inform practical identifiability (PI) in dynamical models by treating structural identifiability as a special case of observability. It extends the standard observability rank condition (ORC) with a differential-geometric framework that augments the state to include parameters, forming the observability-identifiability matrix $\mathcal{O}_I$ and using its rank to assess SIO. The authors further adapt the analysis to practical settings by considering multiple experiments, input characteristics, initial conditions, and the availability of output derivatives, and they propose numerical tools such as SVD of $\mathcal{O}_I$ and bootstrap validations to quantify PI. The work shows that experimental design choices (e.g., using ramps, multiple experiments, and informative inputs) can improve identifiability and provides guidance on when ORC-based analyses may be sufficient or require augmentation, while also highlighting limitations related to unknown inputs and initial-condition sensitivity.

Abstract

The concept of identifiability describes the possibility of inferring the parameters of a dynamic model by observing its output. It is common and useful to distinguish between structural and practical identifiability. The former property is fully determined by the model equations, while the latter is also influenced by the characteristics of the available experimental data. Structural identifiability can be determined by means of symbolic computations, which may be performed before collecting experimental data, and are hence sometimes called a priori analyses. Practical identifiability is typically assessed numerically, with methods that require simulations - and often also optimization - and are applied a posteriori. An approach to study structural local identifiability is to consider it as a particular case of observability, which is the possibility of inferring the internal state of a system from its output. Thus, both properties can be analysed jointly, by building a generalized observability matrix and computing its rank. The aim of this paper is to investigate to which extent such observability-based methods can also inform about practical aspects related with the experimental setup, which are usually not approached in this way. To this end, we explore a number of possible extensions of the rank tests, and discuss the purposes for which they can be informative as well as others for which they cannot.

Figures (5)

• Figure 1: Model $\mathcal{M}_{3.2}$. The panel on the left shows simulations of the model input, $u(t),$ and the output $y(t)=x_1(t);$ the blue circles represent the artificial noiseless data used for parameter estimation. The panel on the right shows the resulting bootstrap of parameter $k_{12}$.
• Figure 2: Model $\mathcal{M}_{3.4}$. The panel on the left shows the noisy data (blue circles) of one of the five experiments used for parameter estimation, as well as the noiseless simulation of the output (red line). The panel on the right shows the resulting bootstrap of parameter $\theta_1$. It can be seen that the parameter is practically identifiable.
• Figure 3: Singular values of models $\mathcal{M}_{3.2}$, $\mathcal{M}_{3.4}$, and $\mathcal{M}_{3.5}$, in logarithmic scale.
• Figure 4: Model $\mathcal{M}_{3.5}$. Noisy data (circles) of the experiment used for parameter estimation, as well as the noiseless simulation of the output, glucose.
• Figure 5: Model $\mathcal{M}_{3.5}$. Bootstrap results: histograms of all estimated variables.

Theorems & Definitions (1)

• Theorem 1: Observability Rank Condition, ORC hermann1977nonlinear