Models for Environmental Cylindrical Time Series

Environmental studies often produce observations characterized by a bivariate response, where one observation is linear and the other is circular, i.e., an angle or direction. These data are frequently referred to as cylindrical, as they can be represented by points on a cylinder of radius one. Typical examples include wind direction and speed, sea wave direction and height, and changes in direction and speed in animal movement. When such observations are collected at discrete time points, the resulting data are usually referred to as cylindrical time series. In this chapter, we review various models for bivariate linear-circular time series that are flexible enough to account for possible regime-switching behavior in the underlying process, as well as the within-regime serial dependence through an autoregressive (AR) structure. Regime switching may be random (as in a hidden Markov model, HMM) or driven by past observations (as in a threshold autoregressive model, TAR). We provide empirical examples to compare these models using information criteria such as AIC and mean squared prediction errors, with applications to forecasting problems.