Bayesian Estimator with Cubic splines by Using the Binary Algorithms to Estimate the Fuzzy Regression Function with Practical Implementation

A Thesis

submitted to the counsel of the college of administration and economics / University of Kerbela as partial fulfillment of the requirements for the degree of Philosophy of Statistics Sciences

Presented by

Zainb Hassan Radhy

 Under supervision

Prof. Dr. Mahdi Wahab Nea’ama

Abstract

The statisticians are interested in finding an efficient model to represent the phenomena. The model should be capable to demonstrate the phenomena under study, specially, if the phenomena are uncertain, fuzzy, or the data has a determined functional form. So, it becomes compulsory to model the different phenomena test the efficiency of the model sufficiency under different conditions. This must lead to find modern estimation approaches which should be able to deal with such data or phenomena.

In this thesis, two different approaches are proposed which use the smoothing spline methods by Bayesian estimator. The first approach uses the Bayesian method in variable selections and estimating the fuzzy nonparametric regression function. In this approach, the algorithm of focused sampling is used to update the independent variables matrix and also updating the column vector of the parameters in each iteration. In the second approach, the Bayesian method is applied by using the linear Bayesian regression form under the same fuzzy thoughts and theories. This approach depends on using the spline modelling to generate the random numbers rather than using one of the conventional algorithms like Gibbs or the focused sampling.

Both methods base on using the smoothing methods of cubic spline and determining the number of node locations. Then, the model is examined under different conditions. The methods are applied in two cases. The first case is the normal condition of the model when the random errors are of normal distribution. The second case happens when the model assumptions are not valid. This occurs if there is a contamination in the random error, where normal random errors with zero mean  and constant variance are generated and added to another type of random errors which follow T(m) distribution with m degree of freedom. Then, on of the robust methods, M-estimation, is employed. The findings are compared with the findings of two proposed methods for nodes shaking.        

Bayesian Estimator with Cubic splines by Using the Binary Algorithms to Estimate the Fuzzy Regression Function with Practical Implementation

A Thesis

submitted to the counsel of the college of administration and economics / University of Kerbela as partial fulfillment of the requirements for the degree of Philosophy of Statistics Sciences

Presented by

Zainb Hassan Radhy

 Under supervision

Prof. Dr. Mahdi Wahab Nea’ama

Abstract

The statisticians are interested in finding an efficient model to represent the phenomena. The model should be capable to demonstrate the phenomena under study, specially, if the phenomena are uncertain, fuzzy, or the data has a determined functional form. So, it becomes compulsory to model the different phenomena test the efficiency of the model sufficiency under different conditions. This must lead to find modern estimation approaches which should be able to deal with such data or phenomena.

In this thesis, two different approaches are proposed which use the smoothing spline methods by Bayesian estimator. The first approach uses the Bayesian method in variable selections and estimating the fuzzy nonparametric regression function. In this approach, the algorithm of focused sampling is used to update the independent variables matrix and also updating the column vector of the parameters in each iteration. In the second approach, the Bayesian method is applied by using the linear Bayesian regression form under the same fuzzy thoughts and theories. This approach depends on using the spline modelling to generate the random numbers rather than using one of the conventional algorithms like Gibbs or the focused sampling.

Both methods base on using the smoothing methods of cubic spline and determining the number of node locations. Then, the model is examined under different conditions. The methods are applied in two cases. The first case is the normal condition of the model when the random errors are of normal distribution. The second case happens when the model assumptions are not valid. This occurs if there is a contamination in the random error, where normal random errors with zero mean  and constant variance are generated and added to another type of random errors which follow T(m) distribution with m degree of freedom. Then, on of the robust methods, M-estimation, is employed. The findings are compared with the findings of two proposed methods for nodes shaking.