A circular hidden Markov model for directional time series data

Abstract

Modeling directional time series data such as wind or ocean current direction presents several interesting challenges. Standard linear time series techniques do not account for the circularity of the observations, while existing circular modeling approaches typically work best when the entirety of the data only spans a small arc. Motivated by a series of fire burn experiments collecting wind direction in south eastern Australia, we propose a new method for directional time series data by combining hidden Markov models with a conditional circular distribution given the latent state. The resulting circular hidden Markov model, or cHMM, can allow for multimodality and/or varying amounts of circular dispersion over time. Furthermore, by utilizing a von Mises distribution whose mean direction depends on previous observations, we can accommodate strong serial correlations within a specific hidden state. We employ direct maximum likelihood estimation to fit the cHMM, making use of recursive probability formulas to efficiently compute the marginal log-likelihood function, and examine three approaches to perform forecasting based on extrapolating the latent state sequence and then direction observations conditional on this sequence. Simulation studies demonstrate the validity of our estimation procedure, while an application to three motivating wind direction datasets from fire burn experiments reveals that cHMMs produces similar or better point/probabilistic forecasting performance compared with several established time series methods.