A Thesis Submitted to The Council of the College of Administration & Economics – University of Kerbala, Partial Fulfillment of the Requirements for The Degree of Master of Science in Statistics 

By

Naeem Malik Jasim

Supervised by
Prof. Dr. Mushtaq Kareem Abd Al-Rahem

Summary of the thesis

       Fuzzy Regression Model (FRM) is one of the important regression models used to represent data that is characterized by ambiguity and imprecision, especially the response variable, for the purpose of predicting the phenomenon under study according to the different membership functions of its data and parameters or some of them, the most famous of which is the triangular membership function, which is relied upon to represent each observation with a fuzzy number and according to the type of membership function.

      Many researches and studies have adopted two methods in estimation, the first of which is the formulation of the Linear Programming (LP) problem for the fuzzy model proposed by Tanaka, and the second is the use of the Fuzzy Least Squares (FLS) method proposed by Diamond. FLS is the most widely used method in research because it is characterized by its efficiency and ease of implementation. Despite this, researchers’ attempts to improve its efficiency in estimation have not stopped. In this thesis, the fixed point theorem was used by employing its properties to estimate the parameters of the fuzzy regression model, which requires the stability of the estimation point (outputs) despite the change in the inputs to the function. To achieve this improvement, it is necessary to rely on iterative methods according to the fixed point principle in estimation to reach convergence required is that there should be no significant difference or equality between two estimates of each of the FRM parameters. Therefore, the Newton-Raphson (Fixed Point) (N-R(FP)) method and the Expectation-Maximization (Fixed Point) (E-M(FP)) algorithm were used. The simulation was conducted for the purpose of comparing three FRM parameter estimation methods to determine the best one, which are N-R(FP), E-M(FP) and FLS, for sample sizes (10, 20, 35, 50, 75, 100) and with different fuzzy factor values (cut-off level) which are (0.1, 0.5, 0.8). The simulation results demonstrated that the N-R(FP) method outperformed the other methods, as it had the lowest values for the trade-off metrics. Furthermore, it was thhttps://drive.google.com/file/d/1mf4Uc8MmKDlGabcnBI4L0SYuymv-ql7m/view?usp=sharinge most representative of the response variable values, demonstrating the estimation efficiency of this method. Based on this, real data on patients with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in a hospital in Karbala Governorate were analyzed. One of the most important conclusions reached by the thesis is that the N-R(FP) method is the best in the estimation process and representation of the data of the fuzzy regression model, as the characteristics and fulfillment of the conditions of the fixed point theory led to improving the FLS method. Therefore, the thesis recommended adopting it in the estimation process.