Employing The Discrete Wavelet Transformation In Estimating The Harmonic Regression Model Parameters

A Thesis

Submitted To The Council Of The College Of Administration And Economics – University Of Kerbala As A Partial Fulfillment Of  The  Requirements For  The Degree Of  Ph.D. Of The Science In Statistics

By

Hamzah Imad Abbas Al-Dibis

Supervised By

Prof. Dr. Shrook A. S. Al-Sabbah

Abstract _:

    Discrete wavelet transformation, or what is called in this research, the total discontinuous wavelet transformation in the Harmonic Regression Model, through which the response variable observations (Y) and the explanatory variables observations (X’s) are transformed The harmonic in the wavelet field (frequency and time) of the harmonic regression model when estimating its parameters, as well as making the transformation on the matrix of observations of explanatory variables (X’s) only in the harmonic regression model as a proposal by the researcher to employ the wavelet transformation or what is called in this research the partial wavelet transformation ( Partial Wavelet Transform).

 The researcher used three methods to estimate the parameters of the harmonic regression model, which are (Partial Least Squares Method, Bootstrap Method, General Maximum Entropy), as the harmonic regression model was obtained through the use of harmonics Between the sine and cosine functions of the sine equation or the cosine equation that is used to represent data that is characterized by periodicity and is called periodic data.

The harmonic regression model is characterized as a nonlinear regression and by making some transformations on the model that can be dealt with when analyzed statistically as a multiple linear regression through which all available methods can be used to analyze the model of ordinary multiple linear regression, and the research also dealt with the proof of the analysis of the nonlinear harmonic regression model As an ordinary linear regression model after making some transformations, and this is what was discussed in the theoretical side of the research, as well as a reference to how to build a harmonic regression model for dual data (Panel Data) to achieve a secondary goal in the research. The main objective of the research is to identify the usefulness of employing the intermittent wavelet transformation (total and partial) in the process of estimating the parameters of the harmonic regression model to improve the estimates of the estimation methods used in the research and referred to previously and to rely on them in prediction, so a simulation experiment was conducted to determine the most efficient method in the estimation process before and after employing the wavelet transform (total and partial) for the harmonic regression model and for sample sizes They are different (16,32,64,128) and the results of the simulation experiment were obtained with the help of the program (Matlab language R14), The results showed the preference of the bootstrap method when employing the partial discontinuous wavelet transform to estimate the parameters of the harmonic regression model compared with the rest of the methods before and after employing the wavelet transform (total and partial) for the harmonic regression model at a sample size (16), as for the sample sizes (32,64,128) The results of the simulation experiment showed the preference of the general maximum entropy method when employing the total discrete wavelet transformation of the harmonic regression model compared to the rest of the other estimation methods before and after the discrete wavelet transformation (total and partial), noting that the comparison process between the aforementioned estimation methods relied on the values ​​of Comparison Standard (Mean Absolute Error Rate) (MAPE) The optimal and most efficient method of estimating is the one that has the lowest value (average absolute error rate) (MAPE).

And based on the simulation results and according to the size of the real data sample, which amounted to (128) views on the phenomenon of the number of daily infections with Corona virus (Covid-19) in the governorates (Karbala, Najaf Al-Ashraf, Babylon, Al-Qadisiyah), the practical application was carried out by employing the total wavelet transformation and using The general maximum entropy method in the estimation process, as it has proven its efficiency in simulation experiment. Among the most important conclusions that have been reached is the superiority of the estimation of the general great entropy method (FWGME) when employing the total wavelet transform, as well as the superiority of the bootstrap method (PWBS) when employing the partial wavelet transform at a sample size (16), and the deterioration of the estimate of the partial least squares method (PWPLS) When employing the partial wavelet transform while its values ​​are equal with the values ​​of the total wavelet transform of the same method, and the results of the practical application showed a confirmation of the quality of the general maximum entropy method (FWGME) in the estimation process of the harmonic regression model when employing the total wavelet transform, and this was indicated by the values ​​of the comparison standard ( MAPE), and the number of  Forecasting the injuries for a later time period based on the estimated harmonic regression model by the general maximum entropy method, and the results showed the extent of their conformity and closeness to the real data, which indicates the efficiency of the estimations obtained in the harmonic regression model .