Spatial and Spatio-temporal Circular Processes with Application to Wave Direction

In application we often find directional data that is associated with locations in space and time. Examples include wind directions, animal movement directions, and our motivating application here, wave directions. To analyze such data we need models for observations at locations $mathbf{s}$ and times $t$, so-called geostatistical models providing structured dependence which is assumed to decay in distance and time. For wave directions in a body of water, we conceptualize a conceptual wave direction at every location and every time. Thus, the challenge is to introduce structured dependence into angular data. The approach we take begins with models for linear variables over space and time using Gaussian processes. Then, we use either wrapping or projection to obtain Gaussian processes for circular data. Altogether, this chapter reviews work developed by the authors in a set of papers published over the last few years focusing on spatial and spatio-temporal modeling for wave directions. All are cast as hierarchical models, with fitting and inference within a Bayesian inference framework. The hierarchical specification is vital; latent variables are introduced to facilitate passage from linear variables to circular variables. The Bayesian framework is arguably most attractive for this setting. We obtain full posterior inference including uncertainties, we avoid potentially inappropriate asymptotics, and we enable routine prediction (kriging) over space and time. We use a wave direction dataset as a running example through the chapter.