On Ornstein-Uhlenbeck State Space Models

Abstract

This dissertation develops and applies the Ornstein--Uhlenbeck state space model (OUSSM) for analyzing noisy, irregularly sampled, and multi-dimensional time-series data, with a particular focus on human microbiome dynamics.

We first introduce the OUSSM framework to account for measurement errors in microbiome data, providing more reliable estimation of mean reversion rates and robust profile likelihood confidence intervals. A likelihood-based testing procedure is proposed to compare microbial dynamics between datasets, for example before and after periods of disruption. These methods are validated through simulations and applied to microbiome datasets.

We then investigate the optimal sampling scheme for the OUSSM. Theoretical results and extensive simulations reveal that estimation accuracy exhibits a U-shaped dependence on the sampling interval, with moderate gaps yielding the most efficient inference. Repeated measurements are shown to substantially improve estimation. Allocating 10\%--20\% of the samples to repeated samples at already-sampled time points achieved the most accurate estimation of mean-reversion rate across a range of simulations. The choice of which time points should be repeatedly sampled made very little difference to the estimation.

Finally, we extend the OUSSM framework to the multi-dimensional setting by developing a factor OUSSM, addressing identifiability challenges via parameter constraints and proposing estimation procedures based on the Kalman filter. Simulation studies demonstrate the model's ability to recover latent dynamics under a range of scenarios, while real-data applications illustrate how multidimensional interactions among microbial genera can be captured more effectively than with traditional one-dimensional approaches.

Together, these contributions establish a comprehensive methodological framework for modeling, inference, and sampling design in the OU process with measurement error. The results not only improve the statistical characterization of microbial temporal dynamics but also extend broadly to applications in ecology, finance, and beyond.