Bayesian Analysis of Circular Data Using Wrapped Distributions

Abstract

Circular data arise in a number of different areas such as geological, meteorological, biological and industrial sciences. We cannot use standard statistical techniques to model circular data, due to the circular geometry of the sample space. One of the common methods used to analyze such data is the wrapping approach. Using the wrapping approach, we assume that, by wrapping a probability distribution from the real line onto the circle, we obtain the probability distribution for circular data. This approach creates a vast class of probability distributions that are flexible to account for different features of circular data. However, the likelihood-based inference for such distributions can be very complicated and computationally intensive. The EM algorithm used to compute the MLE is feasible, but is computationally unsatisfactory. Instead, we use Markov Chain Monte Carlo (MCMC) methods with a data augmentation step, to overcome such computational difficulties. Given a probability distribution on the circle, we assume that the original distribution was distributed on the real line, and then wrapped onto the circle. If we can "unwrap" the distribution off the circle and obtain a distribution on the real line, then the standard statistical techniques for data on the real line can be used. Our proposed methods are flexible and computationally efficient to fit a wide class of wrapped distributions. Furthermore, we can easily compute the usual summary statistics. We present extensive simulation studies to validate the performance of our method. We apply our method to several real data sets and compare our results to parameter estimates available in the literature. We find that the Wrapped Double Exponential family produces robust parameter estimates with good frequentist coverage probability. We extend our method to the regression model. As an example, we analyze the association between ozone data and wind direction. A major contribution of this dissertation is to illustrate a technique to interpret the circular regression coefficients in terms of the linear regression model setup. Regression diagnostics can be developed after augmenting wrapping numbers to the circular data (refer Section 3.5). We extend our method to fit time-correlated data. We can compute other statistics such as circular autocorrelation functions and their standard errors very easily. We use the Wrapped Normal model to analyze the hourly wind directions, which is an example of the time series circular data.