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.