Building and estimating an angular regression model for circular data

A Thesis
Submitted to the council of the College of Administration and Economic- University of Karbala in partial fulfillment of the Requirements for the Degree  of  Philosophy of Statistics Science

By
Ali Mohammed Jawad
Under supervision of
Prof . Dr . Jassim Nasser Hussein

 

The study of variables with circular data is one of the important processes in our present time due to the presence of some phenomena that are described by this type of data, and the angular regression (circular) is one of the most important statistical methods to represent this type of phenomena, and the method of maximum likelihood estimation (MLE) is described as one of the most important methods that are used to estimate its parameters. The random error distribution of the angular regression model is also important because the method of MLE depends mainly on maximizing the probability density function of the random error distribution.

Therefore, the study included the use of three (circular) angular regression models, namely, the angular regression model with von Mises error, the simple angular regression model with an error that follows the wrapped Cauchy distribution, and the proposed model, which is the simple angular regression model with an error that follows the wrapped normal distribution. A goodness of fit test for the three models was proposed based on the Fisher distribution. A comparison was made between the three models using three criteria, which are the Akaki information criterion (AIC), the Bayesian information criterion (BIC), and the Adjusted Akaki information criterion (AIC_c) .

The results showed on the experimental side that each of the three models has a preference if the dependent variable data were generated in accordance with the random error distribution, after the goodness-of-fit test for these models showed that most of these models are significant, that is, when a dependent variable was generated that follows the Von Mises distribution. The best model is the simple angular regression model with error that follows Von Mises. If the dependent variable is generated that follows the wrapped Cauchy distribution, then the best model is the simple angular regression model with error that follows the wrapped Cauchy distribution. When the dependent variable data was generated with the wrapped normal distribution, the best model for this type of data is the proposed model, which is a simple angular regression model with an error that follows the wrapped normal distribution, according to what was shown by the values ​​of the three criteria (AIC,AIC_c,BIC) and the goodness of fit test for the models showed that most of the models are significant models.

 The practical application was carried out on two circular variables: the wind direction in the city of Nasiriyah as a dependent variable and the direction of high atmospheric pressure as an independent variable. The results showed that all models represented the data correctly by passing the estimated models to test the goodness of fit, and that the relationship between the direction of high atmospheric pressure and the direction of the winds in the city of Nasiriyah is a positive relationship. The proposed model, which is the simple angular regression model with an error that follows the wrapped normal distribution, is considered the best model for this data, as shown by the values ​​of the three criteria.