Structural identifiability of partially-observed stochastic processes: from single-particle trajectories to total particle density data

Technical Summary: Structural Identifiability of Partially-Observed Stochastic Processes

Problem Statement
The increasing availability of experimental data has intensified the need to calibrate stochastic models to gain quantitative insights. A fundamental prerequisite for parameter estimation is structural identifiability: determining whether model parameters can be uniquely recovered from idealized, noise-free data. While extensive methods exist for Ordinary Differential Equation (ODE) models, there is a lack of general methodologies to assess the structural identifiability of stochastic processes. This challenge is exacerbated by the randomness inherent in these processes and the nature of the available data. Specifically, the paper addresses how identifiability depends on the observation regime, distinguishing between single-particle trajectory data (individual-based) and total particle density data (population-level). For partially-observed Markov processes, the challenge is compounded by the fact that data may hide the internal states of particles, complicating parameter inference.

Methodology
The authors propose a novel methodology to analyze structural identifiability for a class of spatio-temporal stochastic processes, specifically velocity-jump models where a particle's internal state evolves as a continuous-time Markov chain within a network of states, each characterized by a fixed velocity. The study employs two distinct mathematical representations corresponding to the two data types:

1. Individual-Based Model (Single-Particle Trajectories):

   • The authors utilize the explicit individual-based description of the continuous-time Markov process.
   • They analyze the trajectory $x(t)$ and its derivative $x'(t)$, which is piecewise constant and takes values from the set of state velocities.
   • Identifiability is assessed by deriving the cumulative distribution of time spent in each state (to identify switching rates) and the frequency of transitions between states (to identify transition probabilities).
   • Initial conditions (location and state distribution) are identified by analyzing the distribution of starting points and initial velocities across a countably infinite number of trajectories.

2. Population-Level Model (Total Particle Density):

   • The authors derive a Partial Differential Equation (PDE) system (a Fokker-Planck formulation) describing the evolution of particle density $n_s(x,t)$ in each state $s$.
   • They apply the differential algebra approach, a technique common in ODE analysis, to the PDE system. This involves symbolically eliminating unobserved state variables ($n_1, n_2, n_3$) to derive input-output equations dependent only on the observable total density $N(x,t)$ and model parameters.
   • Novel Initial Condition Analysis: Recognizing that standard differential algebra at $t=0$ may not capture all identifiability information (as spatial and temporal derivatives can be proportional), the authors introduce a new method. They utilize characteristic equations of the PDE system to construct a Taylor expansion of the density evolution about $t=0$. This allows for the identification of additional parameter combinations related to the initial condition that are not apparent through standard elimination.

Key Results
The methodology is applied to a three-state velocity-jump model with nine parameters (velocities, switching rates, and transition probabilities).

• Single-Particle Trajectory Data:

  • The model is structurally globally identifiable (up to state relabelling) when a countably infinite number of infinitely long single-particle trajectories are available.
  • All parameters, including velocities, switching rates, transition probabilities, and the initial condition distribution, can be uniquely recovered.

• Total Particle Density Data:

  • Using the differential algebra approach on the PDE system, the authors derive input-output equations with coefficients ($c_1$ through $c_8$) that are functions of the model parameters.
  • Velocities and switching rates are found to be structurally identifiable (up to state relabelling).
  • However, the transition probability parameters are not globally identifiable using only the input-output coefficients derived from the PDE alone; they satisfy quadratic equations yielding multiple solutions.
  • Impact of Initial Conditions: The standard differential algebra approach at $t=0$ yields only one identifiable coefficient ($c_9$) related to the initial condition, which is insufficient to identify the initial state probabilities ($a_1, a_2, a_3$).
  • Taylor Expansion Contribution: By applying the novel Taylor expansion method based on characteristic equations, the authors identify additional coefficients ($c_{10}$ through $c_{15}$). These coefficients allow for the identification of the initial state probabilities.
  • Local Identifiability: Even with the additional coefficients from the Taylor expansion, the model is only locally structurally identifiable with respect to the probability parameters. This means that while parameters are uniquely identifiable in a small neighborhood of the true values, there exist distinct parameter sets (equivalent models) that produce identical total particle density evolutions under specific initial conditions.

• Special Cases (Shared Velocities):

  • If two or three states share the same velocity, identifiability degrades. If all three states share the same velocity, only the velocity is identifiable; all rate and probability parameters become non-identifiable.
  • If two states share a velocity, the sum of their switching rates is identifiable, but individual rates and probabilities generally remain non-identifiable, even with the Taylor expansion method.

Significance and Claims
The paper claims to provide the first general methodology for analyzing the structural identifiability of spatio-temporal stochastic processes, bridging the gap between individual-based descriptions and population-level PDE formulations.

• Data Type Dependence: The study highlights a critical distinction: single-particle trajectory data leads to global structural identifiability, whereas total particle density data leads only to local identifiability. This underscores that the choice of experimental observation regime fundamentally dictates the feasibility of unique parameter inference.
• Role of Initial Conditions: The work demonstrates that initial conditions significantly impact structural identifiability in PDE models. The authors claim that standard differential algebra approaches are often insufficient for capturing initial condition information in these stochastic contexts and that their proposed Taylor expansion method based on characteristic equations is necessary to uncover additional identifiable parameter combinations.
• Experimental Design Implications: The existence of structurally equivalent models (different parameters producing identical density evolutions) suggests that varying initial conditions could serve as a practical strategy to improve parameter identifiability in experimental design.
• Methodological Extension: The authors assert that their approach, particularly the use of characteristic equations for Taylor expansions, is applicable to other stochastic models where such equations can be written in terms of model parameters, potentially extending to chemical reaction networks with spatial components.

The authors remain modest regarding practical applications, noting that their results characterize identifiability in idealized, noise-free settings. They explicitly state that future work is required to address practical identifiability with finite, noisy data and to extend these methods to models with more complex initial or boundary conditions.