Rapid Bayesian Hamiltonian Monte Carlo Method for Operational Modal Analysis: Theory, Verification, and Application

Abstract

A rapid Bayesian Hamiltonian Monte Carlo (HMC) method for operational modal analysis (OMA) is proposed. This novel approach assumes a conservative system, treating parameter identification as sampling along the gradient direction in phase space. Consequently, it mitigates random walk behavior in probability space by generating samples that adhere to the target probability distribution, achieving rapid sampling of modal parameters in the frequency domain. To accomplish this, the difficulty is twofold. First, formulating a flexible posterior probability density function (PDF) for sampling is essential, because OMA sampling must address the challenges posed by high-dimensional modal spaces and the impact of prediction errors on sampling stability. Second, aligning the Bayesian framework with HMC requires defining structural dynamics-specific hyperparameters, and managing discretization errors due to the evolution of parameters with varying scales in Hamiltonian space. This paper addresses these challenges through mathematical derivation and logical alignment. The optimized posterior PDFs for sampling in the frequency domain naturally disassemble the modal space and function as the potential energy in Hamiltonian. The gradient-defined multivariate mass matrix ensures the generation of kinetic energy, whereas the leapfrog integrator reduces discretization errors. Consequently, the parameters can continue evolving forward under the guidance of the gradient, even through low-probability regions, enabling rapid sampling of unknown parameters. Application cases of synthetic, laboratory, and field test data demonstrated the rapid sampling convergence, even with large measured degrees of freedom.