Bayesian Methods for Circular Regression Using Wrapped Distributions

Circular data arise from a number of activities in our daily life where the sample space is considered to be a circle. Some common examples are the migration paths of birds and animals, wind directions and ocean current directions. Many examples of circular data can be found in the various scientific fields like earth sciences, meteorology, biology, physics, psychology and medicine. See Mardia and Jupp (1999) for examples. In this article, we explore the association of a circular random variable with other variables. Fisher (1993) provides several examples of regression involving circular data. One such example includes a study performed at a weather station in Milwaukee in 1975. The data was collected on wind direction and ozone concentration. There seemed to be some evidence of association between these variables. Figure 1 shows the joint data plot, where each dot represents the direction of the wind and the length of the dot from the center is proportional to the ozone concentration. It appears that ozone concentrations are higher for wind coming from the East direction. However, this plot is not very informative about the association between wind direction and ozone. Standard definition of correlation does not work for this data, as one of the variables (wind direction) is not linear. Therefore, we need to use circular association to analyse this data. Note that, the sample mean for circular data is not the usual sample mean. The circular data are considered as vectors, and a vector sum is performed to obtain the circular sample mean. Let μ be the direction of the mean vector and ρ be the length of the mean vector. We are interested in modeling the variation of a mean direction, μ of a circular response y in terms of one or more covariates or explanatory variables. In the literature several approaches to model cir90