Parameter Estimation of a Partially Observed Hypoelliptic Stochastic Linear System

• Published in: Mathematics (Basel) Vol. 13; no. 3; p. 529
• Main Authors: Ávido, Nilton O. B., Milheiro-Oliveira, Paula
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.02.2025
• Subjects: Algorithms; Approximation; Computing time; Continuous time systems; Damping; Diffusion; EM algorithm; Error analysis; Error reduction; harmonic oscillator; Harmonic oscillators; hypoelliptic models; Linear systems; Maximum likelihood estimators; Parameter estimation; partially observed systems; Random variables; stochastic differential equations; Stochastic systems; Structural engineering; Time series
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math13030529

In this article, we address the problem of the parameter estimation of a partially observed linear hypoelliptic stochastic system in continuous time, a relevant problem in various fields, including mechanical and structural engineering. We propose an online approach which is an approximation to the expectation–maximization (EM) algorithm. This approach combines the Kalman–Bucy filter, to deal with partial observations, with the maximum likelihood estimator for a degenerate n-dimensional system under complete observation. The performance of the proposed approach is illustrated by means of a simulation study undertaken on a harmonic oscillator that describes the dynamic behavior of an elementary engineering structure subject to random vibrations. The unknown parameters represent the oscillator’s stiffness and damping coefficients. The simulation results indicate that, as the variance of the observation error vanishes, the proposed approach remains reasonably close to the output of the EM algorithm, with the advantage of a significant reduction in computing time.