In this work we develop new state space models and Bayesian inference techniques for generating accurate probabilistic predictions of time series that possess various non-Gaussian properties. In the first chapter, we consider multivariate time series data that include a heterogeneous mix of non-Gaussian distributional features (asymmetry, multimodality, heavy tails, etc) and data types (continuous and discrete variables). Traditional multivariate time series methods based on convenient parametric families of probability distributions are typically ill-equipped to model this heterogeneity. Copula models provide an appealing alternative, but it is challenging to estimate them in a fully Bayesian way that incorporates uncertainty from all model unobservables, which is crucial for probabilistic time series forecasting. To meet this challenge, we propose a novel method for posterior approximation in copula time series models, and we apply it to a Gaussian copula built from a dynamic factor model. This framework provides flexible, scalable, and computationally tractable Bayesian inference for both the dependence structure and the heterogeneous marginal behavior of a multivariate time series. We validate our posterior approximation by providing model-trusting posterior consistency theory, and we provide simulation evidence that consistency is still achieved under model misspecification. In a diverse array of forecast comparisons on real and simulated data, we show that our proposed approach provides accurate point, interval, and density forecasts compared to a basket of popular alternatives. Taken together, these results demonstrate that our proposed method is a versatile, general-purpose utility for multivariate time series forecasting that works well across of range of applications with minimal user-intensive tuning.

In the second chapter, we consider time series on the unit n-sphere, which arise in directional statistics, compositional data analysis, and many scientific fields. There are few models for such data, and the ones that exist suffer from several limitations: they are often computationally challenging to fit, many of them apply only to the circular case of n=2, and they are usually based on families of distributions that are not flexible enough to capture the complexities observed in real data. Furthermore, there is little work on Bayesian methods for spherical time series. To address these shortcomings, we propose a state space model based on the projected normal distribution that can be applied to spherical time series of arbitrary dimension. We describe how to perform fully Bayesian offline inference for this model using a simple and efficient Gibbs sampling algorithm, and we develop a Rao-Blackwellized particle filter to perform online inference for streaming data. In analyses of wind direction and energy market time series, we show that the proposed model outperforms competitors in terms of point, set, and density forecasting.

In the last chapter, we develop new methods for data on the Stiefel manifold. Such data arise again in directional statistics, where we measure the orientation of an object in space and record this information in the form of an orthonormal matrix. There is a rich literature on distributional theory and inference for these data from a classical point of view, but there has been very little Bayesian work until recently, and almost no discussion of time series. To fill these gaps, we describe how to use data augmentation to perform Bayesian inference for a class of models on the Stiefel manifold that are based on the matrix normal distribution. The chapter culminates in the first fully Bayesian method for time series on the Stiefel manifold. We propose a new state space model for this purpose, and show how to access its full posterior using a Markov chain Monte Carlo sampler.