This paper considers an extension of the linear mixed model, called semiparametric mixed effects model, for longitudinal data, when multicollinearity is present. To overcome this problem, a new mixed Ridge estimator is proposed while the nonparametric function in the semiparametric model is approximated by the kernel method. The proposed approache integrates Ridge method into the semiparametric mixed effects modeling framework in order to account for both the correlation induced by repeatedly measuring an outcome on each individual over time, as well as the potentially high degree of correlation among possible predictor variables. The asymptotic normality of the exhibited estimator is established. To improve efficiency, the estimation of the covariance function is accomplished using an iterative algorithm. Performance of the proposed estimator is compared through a simulation study and analysis of CD4 data.

Process capability indices show the ability of a process to produce products according to the pre-specified requirements. Since final quality characteristics of a product are usually interrelated to its previous amounts in earlier workstations, one need to model and consider the relationship among them to assess the process capability properly. Hence, conducting process capability analysis in multivariate environment is inevitable; unfortunately, the analysis in multivariate environment is usually complex and requires extensive calculations. Sometimes it is preferable to simplify the analysis by assuming independency among quality characteristics and evaluating process performance with respect to each individual quality characteristic using univariate process capability indices such as CP, CPK, CPM, and CPMK. However, this simplification introduces some error in the analysis leading to under or overestimation of the process capability index. This paper models the interrelationship among quality characteristics that are produced in different workstations to provide an overall process capability index. Ridge residual Regression is used as a vehicle to evaluate process capability and helps quality engineers to provide a reasonable quality policy for controlling and reducing variation in quality characteristics.

‎The high-dimensional data analysis using classical Regression approaches is not applicable, and the consequences may need to be more accurate. This study tried to analyze such data by introducing new and powerful approaches such as support vector Regression, functional Regression, LASSO and Ridge Regression. On this subject, by investigating two high-dimensional data sets (riboflavin and simulated data sets) using the suggested approaches, it is progressed to derive the most efficient model based on three criteria (correlation squared, mean squared error and mean absolute error percentage deviation) according to the type of data.

This paper deals with Ridge estimation of fuzzy nonparametric Regression models using triangular fuzzy numbers. This estimation method is obtained by implementing Ridge Regression learning algorithm in the La- grangian dual space. The distance measure for fuzzy numbers that suggested by Diamond is used and the local linear smoothing technique with the cross- validation procedure for selecting the optimal value of the smoothing param- eter is fuzzified to fit the presented model. Some simulation experiments are then presented which indicate the performance of the proposed method.

Radial Basis Functions (RBFs) have gained significant attention in various machine learning applications, including Regression modeling, due to their ability to approximate complex, nonlinear relationships. RBFs offer a flexible approach to capturing intricate dependencies between input features and the target variable, making them particularly useful in high-dimensional and nonparametric settings. This paper investigates the use of a specific class of compactly supported RBFs, known as Wendland functions, within the framework of kernel Ridge Regression (KRR). We discuss their theoretical advantages—such as sparsity enforcement and computational efficiency as well as practical challenges, including parameter selection and scalability. A comprehensive overview of RBFs is provided, along with their mathematical formulation and a comparison of different RBF kernels in terms of smoothness and locality. We detail the integration of Wendland functions into KRR models, emphasizing their suitability for problems requiring robustness and interpretability. Through extensive simulation studies, the performance of the proposed approach is evaluated against conventional RBF kernels and other widely used Regression techniques. Our results demonstrate that Wendland-based KRR achieves competitive accuracy while offering improved stability in the presence of noise and outliers. Furthermore, real-world case studies illustrate the effectiveness of Wendland functions in handling datasets with high collinearity, where traditional kernels often struggle. The practical implications of our findings are discussed, along with guidelines for implementation and potential extensions to large-scale or sparse data scenarios. This work contributes to the growing body of research on interpretable and efficient kernel methods, providing insights for both theoretical and applied machine learning practitioners.

The study of Regression diagnostic, including identification of the influential observations and outliers, is of particular importance. The sensitivity of least squares estimators to the outliers and influential observations lead to extending the Regression diagnostic in order to provide criteria to assess the anomalous observations. Detecting influential observations and outliers in the presence of collinearity is a complicated task, in the sense that collinearity may cover some of the unusual data. One of the considerable methods to identify outliers is the mean shift outliers method. In this article, we extend the mean shift outliers method to the Ridge estimates under linear stochastic restrictions, which is used to reduce the effect of collinearity, and to provide the test statistic to identify the outliers in these estimators. Finally, we show the ability of our proposed method using a practical example of real data.

Evapotranspiration is one of the most important parameters in the Planning and operation of reservoirs, designing of irrigation systems. The practical importance of accurate estimates of evaporation and the complexity of effect phenomenon, shows the use of new methods of data mining. In this study, the simulation of pan evaporation in Tabriz station using multiple Regression models were investigated. Meteorological data, including maximum and minimum air temperature, dew point, maximum and minimum air relative humidity, number of sunshine hours and Daily wind speed during (1992-2012) were used in synoptic Tabriz stations. Various models of multiple linear Regression and nonlinear one were derived for Tabriz station. The selected multiple linear Regression model were tested by Ridge Regression method to be considered multi-collinearity among inputs in the model. Variance inflation factor, values for each variable were calculated. The results showed that all Variance inflation factor, s had the value less than 10. In addition, the ratio lmax/lmax for two- variable selected model was 3.34. Therefore, there was no multi-collinearity in the selected multiple linear Regression model f (Tmin, n). Durbin-Watson statistic for the selected model was 1.45 that shows the reliability of the selected multiple linear Regression model. RMSE and R2 values of the selected models (multiple linear Regression and Non- Linear Regression) was calculated as 2.45 and 0.67 and 2.58 and 0.65, respectively. This results demonstrate the ability of Regression techniques to estimate Pan evaporation in Tabriz station.

This paper focuses on estimating Ridge in a type-2 fuzzy non-parametric Regression model that utilizes non-fuzzy inputs, type-2 fuzzy output data, and type-2 fuzzy coefficients within a dual Lagrange space. It begins with definitions of type-2 fuzzy sets (T2FSs) and presents a closed parametric form for complete triangular T2FSs. The proposed framework underpins a local linear smoothing method that incorporates a cross-validation procedure for optimizing Ridge parameters and smoothing values. The research advances statistical modeling with type-2 fuzzy systems, offering innovative techniques for Regression analysis in complex data situations. The combination of Ridge estimation, local linear smoothing, and cross-validation is highlighted for its potential to yield precise results. Our work is able to model complex and nonlinear relationships between variables, which more effectively deals with uncertainties and ambiguities in the data, prevents overfitting, and ultimately improves the accuracy and reliability of predictions. Numerical simulations are included to validate the theoretical findings and demonstrate the method's effectiveness.

It is well known that bias in parameter estimates arises when there are measurement errors in the covariates of Regression mod-els. One solution for decreasing such biases is the use of prior informa-tion concerning the measurement error, which is often called replication data. In this paper, we present a Ridge estimator in replicated measure-ment error (RMER) to overcome the multicollinearity problem in such models. The performance of RMER against some other estimators is investigated. Large sample properties of our estimator are derived and compared with other estimators using a simulation study as well as a real data set.

Introduction: Estimation of age has an important role in legal medicine, endocrine diseases and clinical dentistry. Correspondingly, evaluation of dental development stages is more valuable than tooth erosion. In this research, the modeling of calendar age has been done using new and rich statistical methods. Considerably, it can be considering as a practicable method in medical science that is a combination of some new statistical methods. Materials and Methods: Among the methods used to determine age, the most commonly used method in the world is the modern modified Demirjian’ s method based on the calcification of the permanent tooth in panoramic radiography. The study population is consisted of 87 patients who referred to Khatam-ol-Anbia Clinic of Yazd in a simple randomized method during the 12 months of the 2014-2015 year. Using the estimated age of third molar tooth and gender variables, we evaluated the calendar age. In the analysis of Regression issues and especially the statistical modeling of many data such as economic data, psychology, social sciences, medical sciences, engineering, etc., we faced with the problem of collinearity among the predictor variables and the presence of remote areas in the data set. The least squares error method in estimation of the parameters of Regression model was very sensitive to the outliers. Most of the existing methods for estimating the parameters of these models based on the least squares error approach, affected by the outliers, were yielded to inappropriate estimates, unexpected and high error rates. Robust methods were used to overcome the problem of the outlier observations. It is also recommended the Ridge Regression to fix the multicollinearity problem. Therefore, in this study, the robust Ridge Regression estimators will be introduced in the modeling of dependent variables that are less sensitive to the outliers. Results: The mean age of the subjects was 17. 21 years old, with a gender difference of 67% female and 33% male. Additionally, in the relationship between the estimated age of 4 teeth lower right (LR), lower left (LL), upper right (UR), upper left (UL) with a correlation coefficient were above 70%. Correlation between upper and lower jaw teeth was about 30% and between the left and right teeth was 60%. The reason of using robust Ridge Regression model in this study is the existence of outlier data and collinearity between independent variables. Conclusion: The necessity of using advanced statistical methods in medical sciences in the recent research is very important. In order to choose the best model, we need to study the data carefully. In this research, the fitted model for prediction of age based on the robust Ridge Regression method was more efficient with respect to the other methods.