## Using Wavelet Shrinkage in the Cox Proportional Hazards Regression model (simulation study). | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

IRAQI JOURNAL OF STATISTICAL SCIENCES | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Article 2, Volume 20, Issue 1, June 2023, Pages 9-24 PDF (1.85 M)
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DOI: 10.33899/iqjoss.2023.178679 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Authors | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Taha.H Ali^{*} ; Nasradeen Haj Salih Albarwari; Diyar Lazgeen Ramadhan
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^{}Department of Informatics & Statistic, College of Administration and Economics, Salahaddin University, Erbil,, Iraq | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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In this paper, a hybrid method of quantile regression and multivariate wavelet is proposed to deal with the problem of data contamination or the presence of outliers, which uses the median instead of the mean on which the linear regression model and the estimation method for ordinary least squares depend. The paper included a comparison between the proposed (for several wavelets and different threshold) and classical method based on mean absolute error, to get the best fit quantile regression model for the data. The application part dealt with two types of data representing simulation, real data, and analysis using a program designed for this purpose in the MATLAB language, as well as the statistical program SPSS-26 and EasyFit-5.5. The study concluded that the proposed method is more efficient than the classical method in estimating the parameters of a quantile regression model depending on the coefficient of determination and on the mean absolute error and mean squared error criteria. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Through the study of simulation and real data, the following main conclusions and recommendations were summarized:
- Models estimated using the proposed methods (MWQR) are more efficient than the classical method (QR) for all simulation and real data, depending on the MAE.
- The classical method (QR) is better than the classical method (OLS) depending on the criterion MAE, while the proposed methods were the best of them based on the pseudo R
^{2}, MSE, and MAE, for all simulation and real data. - The wavelets (Sym2 and Db2) with the Universal threshold (De-noising Method) were the best compared to the other types of threshold methods.
- The proposed methods (MWQR) treat contamination while not treating outliers in the data, but are robust to it.
- The parameters of the models estimated using the proposed methods (MWQR) are more significant than the classical method (QR), and the F-value was larger.
- For real data, there is a significant effect of the independent variables (food and non-alcoholic beverages, clothes and shoes, and housing, water, gas, electricity and other fuels) on the dependent variable (total household spending), according to the classical and proposed methods.
- For the real data, the proposed methods interpretation ratio (MWQR) was greater than that of the classical method (QR).
- Using the proposed methods (MWQR), because it is more efficient than classical method.
- Using wavelet quantile regression with a generalized regression model.
- Using wavelet quantile regression with gamma regression.
- Use statistical software such as MATLAB and R. To create a package that automatically selects the best wavelet according to the data entered when using wavelet quantile regression.
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Multivariate Wavelet; Quantile regression; De-noise; and threshold | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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**Introduction**
Quantile regression offers the opportunity for a more complete view of the statistical landscape and the relationships among stochastic variables. The simple expedient of replacing the familiar notions of sorting and ranking observations in the most elementary one-sample context by optimization enables us to extend these ideas to a much broader class of statistical models. Just as minimizing sums of squares permits us to estimate a wide variety of models for conditional mean functions, minimizing a simple asymmetric version of absolute errors yields estimates for conditional quantile functions. For linear parametric models, computation is greatly facilitated by the reformulation of the optimization problem as a parametric linear program. Formal duality results for linear programs yield a new approach to rank statistics and rank based inference for linear models (Koenker R. 2005). Wavelet shrinkage is a non-parametric technique used in curve estimation. The idea is to shrink wavelet coefficients towards zero using statistical methods. More specifically, a threshold value is chosen and wavelet coefficients whose absolute values exceed that threshold are kept while others are removed. This has the effect of both reducing the noise contribution and compressing the original data while keeping a good quality of approximation. A key step in the wavelet shrinkage procedure is the choice of the threshold (Raimondo, M. 2002). The aim of our study is estimation of a more appropriate regression model from the classic methods for constructing a regression model by using the proposed hybrid method for multivariate wavelet shrinkage and quantile regression model, and data de-noise using wavelet, and comparison between the classical and the proposed method (multivariate wavelets quantile regression) in estimating regression model. **Theoretical Part (Quantile Regression Model & Wavelet Shrinkage)**
The theoretical part is the most important part of academic research because it will be the viewfinder through which researchers evaluate their study problems. This part provides some information regarding statistical techniques including quantile regression models, and wavelet shrinkage. So, this part describes theories and models for classical and proposed methods that will be applied in the practical part of the paper.
Given the sample of size n, {(X
The τ-specific regression parameter (2) where is the quantile loss function and May be reformulated as a linear program by introducing 2n artificial, or “slack,” variables { v_{i}_{ }: 1, . . . , n} to represent the positive and negative parts of the vector of residuals. This yields the new problem(3) where 1n denotes an n-vector of 1. Clearly, in (3) we are minimizing a linear function on a polyhedral constraint set consisting of the intersection of the (2n + 1)-dimensional hyperplane determined by the linear equality constraints (Bickel and Freedman, 1981). Quantile regression problem (2) may be reformulated as a linear program as in (3): (4) Where X now denotes the usual n by p regression design matrix. Again, we have split the residual vector y − Xβ into its positive and negative parts, and so we are minimizing a linear function on a polyhedral constraint set, and most of the important properties of the solutions, which we call “regression quantiles,” again follow immediately from well-known properties of solutions of linear programs. One of the important measures for the efficiency of the estimated regression model is the mean absolute error (MAE) criterion, which is usually used in quantile regression, as in the following formula, (Willmott, C. J., & Matsuura, K. 2005):
And to extract a mean (MAE) for a number of samples or iterations (k), we use the following formula:
Wavelet shrinkage is a well-established technique to data de-noise, it consists of Wavelets and a Shrinkage, as will be explained later.
Wavelets are small waves that can be grouped together to form larger waves or different waves. A few fundamental waves were used, stretched in infinitely many ways, and moved in infinitely many ways to produce a wavelet system that could make an accurate model of any wave. Consider generating an orthogonal wavelet basis for functions (the space of square integrable real functions), starting with two parent wavelets: the scaling function(also called the farther wavelet) and the mother wavelet. Other wavelets are then generated by dilations and translations of and (Percival et al., 2004). The dilation and translated of the functions are defined by formulas (7) and (8).
Where
In general, (dbn) represents the family of small waves of the order n (note that the small wave haar is one of the members of this family and has the following characteristics (Taha H. A., 2009): - The anchor of the small wave (dbn) is on the period [0, 2n-1].
- A wavelet (dbn) has n ephemeral moments, i. e:
From formula (10), the moment - Anomalies increase with rank, (dbn) have (rn) continuous derivatives (r is about 0.2).
Threshold usually involves the conversion of signal values below a certain threshold to zero or equal to the threshold when values are higher than the threshold. In image processing, threshold is commonly used for image segmentation. According to the type of image, different types of threshold mechanisms are used (Ahmadi et al., 2015). Threshold is the simplest method of non-linear wavelet de-noising, in which the wavelet coefficient is divided into two sets, one of which represents the signal while the other represents noise. There are different rules for applying the thresholds of the wavelet coefficients, and several different methods for choosing a threshold value exist, such as: **Universal Threshold Method**
(Donoho and Johnstone, 1994) submitted the universal threshold method, which is given by formula (11).
Where is the standard deviation estimator of details coefficients, and equal to . Where MAD is the median absolute deviation of the wavelet coefficients at the finest scale. **Minimax Threshold Method**
The optimal minimax threshold method, submitted by (Donoho and Johnstone, 1994) as an improvement to the universal threshold method. Minimax is based on an estimator
Where
Where and , denote the vectors of true and estimated sample values. The threshold minimax estimator is different from universal counter parts, in which the minimax threshold method is concentrate on reducing the overall mean square error (MSE) but the estimates are not over-smoothing. **SURE Threshold Method**
The sure threshold proposed by (Donoho and Johnstone, 1994), which based upon the minimization of stein's risk estimator. In sure threshold method specifies a threshold estimate of at each level Where
On the other hand, there are many rules for threshold (Threshold Rules). The two types used in this research will be discussed. **Soft Threshold**
The other standard technique for wavelet de-noising is Soft threshold of the wavelet coefficient, also proposed by Donoho and Johnostone, which is defined as follows (Joy et al., 2013).
Where
Coefficients smaller than the threshold are set to zero, and additionally, all coefficients greater than the threshold are reduced by the amount of the threshold. Thus, soft threshold is a continuous mapping. **Hard Threshold**
Donoho and Johnstone proposed Hard threshold, it is a simplest scheme threshold interpreting the statement of (keep or kill). The Hard threshold used straightforward technique for implementing wavelet de-noising (Hagiwara, 2021). The wavelet coefficient is set to the vector
exceeding are left untouched, while smaller than or equal to are eliminated or set to 0. Thus, the operation of hard threshold is not continuous mapping.
To display noise detection in observations () when i=0,1,..,n, researcher (Greenblatt, 1996) suggested a way to do this work by using discrete wavelet transform (DWT), specifically, which represent the discrete wavelet transformation coefficients at the first level for observation . Express the test statistics using the following formula:
Where () represents the mean of the sample to () of the wavelet coefficients at the first level, and () represents the standard deviation of the wavelet coefficients of the first level. The above test statistic is used to test for the presence of noise in observations. The noise represents a high oscillation event or phenomenon , which is captured and well detected by () because its coefficients are related to frequencies (
The value of () determines the upper limit that is acceptable and that values greater than them are considered undesirable noise values, while (- ) represents the minimum and that values less than them are also considered undesirable noise values, and on this basis, will be determined by determining the values that fall outside the limits of the specified period, i.e. [- , ] and trying to remove their influence and obtaining the candidate observations that fall within the period (2011عبدالقادر، ).
we will treat the data from contamination (de-noising of data) before creating Quantile Regression Models using multivariate wavelet shrinking. In particular, Daubechies and Symlets wavelets, and Universal, Minimax, and SURE as Threshold Method, and Soft and Hard as threshold Rules. The de-noise data (for the dependent and independent variables) is used to estimate parameters of the quantile regression model and then compare its efficiency with the classical method depending on the (MAE). **Practical part (Simulation and Real Data)**
For the purpose of comparing the proposed and classical method, two types of data were used, namely simulation and real data.
Simulation of the first experiment using a program designed in MATLAB language (Appendix) to generate a multiple regression model with three independent variables and a sample size equal to (100), and assuming that the parameter vector is [1.5, 3, 2, 2.5] with a random error that has an identical independent
Figure 1. Scatter plot for Regression Standardized Residual (OLS) Figure (1) shows that there are three outliers (points: 29, 36, and 74) because they are greater than (2.5). The information of the linear model estimated by (OLS) is summarized in the table (1):
Table (1) shows that the F-Statistic (14.89) supports the fit of the linear model to the data because it is greater than its tabulated value under the significant level (0.05) and degrees of freedom (3, 96) which is equal to (2.7114). All parameters are significant and the coefficient of determination is 31.8%, MSE and MAE are (3.834 and 1.3684) respectively. Quantile regression results for the same data and when τ = (0.5) were as follow:
Figure (2) shows that there are four outliers (points: 29, 36, 74, and 94) because they are greater than (2.5), because Quantile regression does not treat outliers, but is robust to them when estimating the parameters of the linear model. The information of the linear model estimated by (QR) is summarized in the table (2):
Table (2) shows that all parameters are significant and the pseudo R Proposed method (Multivariate Wavelet Quantile regression), Beginning with the use of the Multivariate Wavelet (Sym2), Minimax method, and Soft rule (threshold). We get the following results (using a program prepared for this purpose in MATLAB language in the appendix).
Figure (3) shows the original data (red color) and the data processed using multivariate wavelet (blue color), which confirms the presence of noise in the data that has been processed or de-noise. Multivariate Wavelet Quantile regression results for the same data and when τ = (0.5) were as follow:
Figure (4) shows that there are three outliers (points: 9, 10, and 66) Because they were outside the limits of the interval (-2.5 and 2.5). The information of the linear model estimated by (MWQR) is summarized in the table (3):
Table (3) shows that all parameters are significant (p-value = 0.000), and It is more significant than the parameters estimated by the classical method. The pseudo R
Table (4) shows that all the proposed methods were better than the classical method because the values of MAE were less than their value in the classical method (MAE = 0.8523, 0.7412, and 0.6711 respectively) for all different sample sizes. When the sample size is equal to (30), the proposed method (Db2-universal, MAE = 0.2810) was better than the other proposed methods, while for the sample sizes (50 and 100), the proposed method (Sym2-universal, MAE = 0.2608 and 0.1178, respectively) was the best compared to the other proposed methods. For all the methods used, the values of MAE decrease when the sample size increases, so the estimated models were better.
In this part, we will analyze real data with multiple linear regression and quantile regression and then proposed method by using some multivariate wavelets, and compare them and choose the method that gives us better results. The data in (Appendix) have been retrieved from a book located in the Directorate of Statistics in Dohuk under the name (IHSEES II Household Social and Economic Survey 2012), where the first edition was in 2014. The available copy can be obtained on the website (www.cost.gov.iq) where the census process was comprehensive in all governorates of Iraq and the sample size in each governorate was 1512 samples (family). The dependent variable represents (total household spending x, and housing, water, gas, electricity and other fuels _{2}x). Figure (1) shows the scatter plot of the regression standardized predicted Values versus regression standardized residual (using OLS method)._{3}
Figure (5) shows that there is one outlier (point 7) because it was less than (-2.5). The hypotheses of the linear regression model were verified and they are summarized in the following table:
Table (5) shows that to test the hypothesis of a normal distribution for random error (H The estimated linear model information (OLS) is summarized in Table (6):
Table (6) shows that the F-Statistic (157.7) supports the fit of the linear model to the data because it is greater than its tabulated value under the significant level (0.05) and degrees of freedom (3, 14) which is equal to (3.3439). All parameters (slops) are significant and the coefficient of determination is 97.1%, MSE and MAE are (8267.4 and 54.118) respectively. Quantile regression results for real data, when τ = (0.5) were as follow:
Figure (6) shows that also there is one outlier (point 7) because it was less than (-2.5). The hypotheses of the linear regression model were verified and they are summarized in the following table:
Table (7) shows that all parameters are significant and the pseudo R Proposed method (Multivariate Wavelet Quantile regression), Beginning with the use of the Multivariate Wavelet (Sym2), Minimax method, and Soft rule (threshold). We get the following results (using a program prepared for this purpose in MATLAB language in the appendix).
Figure (7) shows the original data (red color) and the data processed using multivariate wavelet (blue color), which confirms the presence of noise in the data that has been processed or de-noise. Multivariate Wavelet Quantile regression results for the real data and when τ = (0.5) were as follow:
Figure (8) shows that also there is one outlier (point 2) because it was less than (-2.5)., The information of the linear model estimated by (MWQR) is summarized in the table (8):
Table (8) shows that all parameters are significant (p-value = 0.000), and It is more significant than the parameters estimated by the classical method. The pseudo R All proposed and classical methods were applied, then the mean absolute error was calculated, and the results were summarized in the following table:
Table (9) shows that all the proposed methods were better than the classical method because the values of (MAE = 3.3344, and 3.2841, respectively) were less than their value in the classical method (MAE = 49.7139). The proposed methods (Sym2 and Db2-universal) performed better than the other proposed methods. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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