The projected dynamic linear model for time series on the sphere

Time series on the unit n-sphere 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.