بناء توزيع احتمالي في نظام الاحداثي القطبي بزوايا منتظمة مع تطبيق عملي

RP-“Building a Probability Distribution in Polar Coordinate System with Uniform Angles”.pdf

Abstract
There are numerous data sets that are expressed with variables having angles and distance, particularly in the fields of mathematics, physics, marine engineering, robotics, unmanned aerial vehicles, receivers, telecommunication, and others. Researchers face difficulties in selecting a probability distribution that accurately represents a given phenomenon in the best possible way. In such cases, it is not possible to effectively express the data within Cartesian coordinates since it requires expressing them in a coordinate system that incorporates angles and distance, namely the polar coordinate system. In this thesis, the joint probability function of the Weibull distribution in the polar coordinate system is constructed, followed by finding the marginal distribution function of the polar Weibull distribution and deriving some important properties of the new distribution.. Four estimation methods (MLE,MOM,OLS,WLS ) are then discussed to estimate the parameters of the distribution. To achieve the objective of the thesis, the results are analyzed and discussed using both simulated data and real data representing wind speed and direction in the holy city of Karbala.
Simulation was employed by using various sample sizes( 30,50,75,100,150) and different hypothetical parameter values, and the experiment was repeated 1000 times. After obtaining the results, a comparison was made using the mean square error criterion, revealing that the maximum likelihood method is the most suitable for parameter estimation among the other methods. In the applied aspect, real data representing wind direction and speed in the city of Karbala period (1/10/2022 to 30/11/2022 ) were recorded. The suitability of the real data for the new distribution was examined using some tests(Kolmogorov-Smirnov & Anderson –Darling Test). Subsequently, the estimated parameters for the real data were obtained using the maximum likelihood method, and a comparison was made between the estimation results of the new polar Weibull distribution and the Weibull distribution in Cartesian coordinates using some comparison and preference criteria (AIC,BIC) , which showed the superiority of the new distribution, aligning with the thesis’s objective.