A Hierarchical Ornstein-Uhlenbeck Model for Stochastic Time Series Analysis

Abstract

Longitudinal data is ubiquitous in research, and often complemented by broad collections of static background information. There is, however, a shortage of general-purpose statistical tools for studying the temporal dynamics of complex and stochastic dynamical systems especially when data is scarce, and the underlying mechanisms that generate the observation are poorly understood. Contemporary microbiome research provides a topical example, where vast cross-sectional and longitudinal collections of taxonomic profiling data from the human body and other environments are now being collected in various research laboratories world-wide. Many classical algorithms rely on long and densely sampled time series, whereas human microbiome studies typically have more limited sample sizes, short time spans, sparse sampling intervals, lack of replicates and high levels of unaccounted technical and biological variation. We demonstrate how non-parametric models can help to quantify key properties of a dynamical system when the actual data-generating mechanisms are largely unknown. Such properties include the locations of stable states, resilience of the system, and the levels of stochastic fluctuations. Moreover, we show how limited data availability can be compensated by pooling statistical evidence across multiple individuals or studies, and by incorporating prior information in the models. In particular, we derive and implement a hierarchical Bayesian variant of Ornstein-Uhlenbeck driven t-processes. This can be used to characterize universal dynamics in univariate, unimodal, and mean reversible systems based on multiple short time series. We validate the model with simulated data and investigate its applicability in characterizing temporal dynamics of human gut microbiome.