Factor State Space Modelling of the Ornstein-Uhlenbeck Process with Measurement Error and its Application

2026-05-02

AI Summary

This study addresses the limitations of the standard Ornstein–Uhlenbeck (OU) model, which suffers from biased parameter estimates when observation noise is ignored and encounters non-identifiability issues in multivariate settings. To overcome these challenges, the authors introduce a factor structure into the OU state-space framework for the first time, imposing identifiability constraints that effectively correct for measurement error and enable reliable estimation of parameters in multivariate mean-reverting systems. The proposed method demonstrates robust performance in simulations and is successfully applied to real-world datasets—namely, human gut microbiome time series and North Atlantic sea surface temperature records—revealing underlying low-dimensional dynamical structures. This advancement significantly extends the applicability of OU models to high-dimensional biological and environmental time series analysis.

Abstract

Standard Ornstein-Uhlenbeck (OU) models often yield biased parameter estimates when measurement error is ignored. While the Ornstein-Uhlenbeck State Space Model (OUSSM) addresses this in univariate settings, multidimensional extensions remain limited. This paper introduces the factor OUSSM to model multi-dimensional, mean-reverting systems with observational noise. We resolve critical identifiability challenges in parameter estimation by establishing necessary constraints and validating the method through extensive simulations. We demonstrate the model's versatility by analyzing human gut microbiome dynamics and North Atlantic Sea Surface Temperature (SST) data. The results reveal distinct latent temporal structures in both biological and environmental systems, establishing the factor OUSSM as a robust framework for multivariate time series analysis.

Problem

• Ornstein-Uhlenbeck process
• measurement error
• state space model
• multivariate time series
• identifiability

Innovation

• Factor OUSSM
• Measurement Error
• Multivariate Time Series
• Identifiability
• Mean-Reverting Process