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Approximate Bayesian computation for stochastic hybrid systems with ergodic behaviour

Sascha Desmettre, Agnes Mallinger, Amira Meddah, Irene Tubikanec

TL;DR

This work develops an ABC-based framework for parameter inference in piecewise diffusion Markov processes (PDifMPs), a class of stochastic hybrid systems combining diffusion dynamics with random regime switching. It provides detailed PDifMP path simulation algorithms and extends ABC summary statistics to account for hybrid dynamics, emphasizing ergodic systems where single long trajectories can reveal invariant properties. The method demonstrates robust parameter recovery across representative ergodic PDifMPs, including scenarios with partial observations and state-dependent jump rates, and it accommodates extensions to additional parameters such as the drift-rate $\eta$. The approach offers a practical, simulation-based tool for inference in complex stochastic hybrids and shows promise for broader applications in science and engineering. The accompanying code enables practitioners to implement PDifMP simulation and ABC-based inference for ergodic stochastic hybrids.

Abstract

Piecewise diffusion Markov processes (PDifMPs) form a versatile class of stochastic hybrid systems that combine continuous diffusion processes with discrete event-driven dynamics, enabling flexible modelling of complex real-world hybrid phenomena. The practical utility of PDifMP models, however, depends critically on accurate estimation of their underlying parameters. In this work, we present a novel framework for parameter inference in PDifMPs based on approximate Bayesian computation (ABC). Our contributions are threefold. First, we provide detailed simulation algorithms for PDifMP sample paths. Second, we extend existing ABC summary statistics for diffusion processes to account for the hybrid nature of PDifMPs, showing particular effectiveness for ergodic systems. Third, we demonstrate our approach on several representative example PDifMPs that empirically exhibit ergodic behaviour. Our results show that the proposed ABC method reliably recovers model parameters across all examples, even in challenging scenarios where only partial information on jumps and diffusion is available or when parameters appear in state-dependent jump rate functions. These findings highlight the potential of ABC as a practical tool for inference in various complex stochastic hybrid systems.

Approximate Bayesian computation for stochastic hybrid systems with ergodic behaviour

TL;DR

This work develops an ABC-based framework for parameter inference in piecewise diffusion Markov processes (PDifMPs), a class of stochastic hybrid systems combining diffusion dynamics with random regime switching. It provides detailed PDifMP path simulation algorithms and extends ABC summary statistics to account for hybrid dynamics, emphasizing ergodic systems where single long trajectories can reveal invariant properties. The method demonstrates robust parameter recovery across representative ergodic PDifMPs, including scenarios with partial observations and state-dependent jump rates, and it accommodates extensions to additional parameters such as the drift-rate . The approach offers a practical, simulation-based tool for inference in complex stochastic hybrids and shows promise for broader applications in science and engineering. The accompanying code enables practitioners to implement PDifMP simulation and ABC-based inference for ergodic stochastic hybrids.

Abstract

Piecewise diffusion Markov processes (PDifMPs) form a versatile class of stochastic hybrid systems that combine continuous diffusion processes with discrete event-driven dynamics, enabling flexible modelling of complex real-world hybrid phenomena. The practical utility of PDifMP models, however, depends critically on accurate estimation of their underlying parameters. In this work, we present a novel framework for parameter inference in PDifMPs based on approximate Bayesian computation (ABC). Our contributions are threefold. First, we provide detailed simulation algorithms for PDifMP sample paths. Second, we extend existing ABC summary statistics for diffusion processes to account for the hybrid nature of PDifMPs, showing particular effectiveness for ergodic systems. Third, we demonstrate our approach on several representative example PDifMPs that empirically exhibit ergodic behaviour. Our results show that the proposed ABC method reliably recovers model parameters across all examples, even in challenging scenarios where only partial information on jumps and diffusion is available or when parameters appear in state-dependent jump rate functions. These findings highlight the potential of ABC as a practical tool for inference in various complex stochastic hybrid systems.

Paper Structure

This paper contains 44 sections, 37 equations, 19 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Piecewise diffusion Markov processes
  3. General framework of PDifMPs
  4. Description of PDifMP components
  5. Simulation of PDifMPs
  6. Simulation of SDE paths over interjump intervals
  7. Simulation of PDifMP paths with constant jump rate
  8. Simulation of PDifMP paths with process-dependent jump rate function
  9. Representative example PDifMPs
  10. Underlying SDEs
  11. Wiener process with drift (WPWD)
  12. Ornstein-Uhlenbeck process (OU)
  13. Weakly damped stochastic harmonic oscillator (WDSHO)
  14. Simple stochastic harmonic oscillator (SimpleSHO)
  15. PDifMP test problems
  16. ...and 29 more sections

Figures (19)

  • Figure 1: Example time grid for $N_k=9$.
  • Figure 2: Example time grid for $N^j=3$ and $N^x=12$.
  • Figure 3: Paths of the four test problems.
  • Figure 4: Space and time averages of the four test problems.
  • Figure 5: Marginal ABC posterior densities of the parameter vector $\theta$\ref{['eq:theta']} for TP 1 (OU-PDifMP), derived under three different observation settings.
  • ...and 14 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5