Robust linear regression is one of the most popular problems in the robust statistics community. The parameters of this method are often estimated via least trimmed squares, which minimizes the sum of the k smallest squared residuals. So, the estimation method in contrast to the common least squares estimation method is very computationally expensive. The main idea of this paper is to propose a new estimation method in partial linear models based on minimizing the sum of the k smallest squared residuals which determines the set of outlier point and provides robust estimators. In this regard, first, difference based method in estimation parameters of partial linear models is introduced. Then the method of obtaining robust difference based estimators in partial linear models is introduced which is based on solving an optimization problem minimizing the sum of the k smallest squared residuals. This method can identify outliers. The simulated example and applied numerical example with real data found the proposed robust difference based estimators in the paper produce highly accurate results in compare to the common difference based estimators in partial linear models.

‎In many fields such as econometrics‎, ‎psychology‎, ‎social sciences‎, ‎medical sciences‎, ‎engineering‎, ‎etc.‎, ‎we face with multicollinearity among the explanatory variables and the existence of outliers in data‎. ‎In such situations‎, ‎the ordinary least-squares estimator leads to an inaccurate estimate‎. ‎The robust methods are used to handle the outliers‎. ‎Also‎, ‎to overcome multicollinearity ridge estimators are suggested‎. ‎On the other hand‎, ‎when the error terms are heteroscedastic or correlated‎, ‎the generalized least squares method is used‎. ‎In this paper‎, ‎a fast algorithm for computation of the feasible generalized least trimmed squares ridge estimator in a semiparametric regression model is proposed and then‎, ‎the performance of the proposed estimators is examined through a Monte Carlo simulation study and a real data set.

‎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.

‎With the advancement of biological sciences and modern technologies‎, ‎high-dimensional data‎, ‎in which the number of independent variables exceeds the number of observations‎, ‎have become increasingly prevalent in various fields‎. ‎Gene expression data associated with riboflavin or vitamin $B_2$ production represent an important example of such datasets‎. ‎Riboflavin‎, ‎as an essential micronutrient‎, ‎plays a crucial role in metabolic reactions and vital biological processes through its coenzyme forms‎, ‎$mathit{FAD}$ and $mathit{FMN}$‎. ‎Therefore‎, ‎the biological production of this vitamin using microorganisms as a sustainable and economical approach has gained considerable importance in the food‎, ‎pharmaceutical‎, ‎and animal feed industries‎. ‎The presence of a large number of variables and multicollinearity among them presents major obstacles to classical regression methods‎. ‎Consequently‎, ‎regularisation and dimensionality reduction approaches‎, ‎including Ridge regression‎, ‎Lasso‎, ‎Elastic Net and Partial Least Squares (PLS)‎, ‎have been developed to improve model stability and predictive performance‎. ‎Moreover‎, ‎machine learning methods such as Support Vector Machines (SVM) and Support Vector Regression (SVR) have been widely applied in high-dimensional data analysis due to their ability to capture complex and nonlinear relationships‎. ‎In this study‎, ‎regression and machine learning methods‎, ‎including Ridge‎, ‎Lasso‎, ‎Elastic Net‎, ‎PLS‎, ‎SVM‎, ‎and SVR‎, ‎are implemented and compared using real riboflavin production data and simulated datasets to evaluate their performance in handling multicollinearity‎, ‎dimensionality reduction‎, ‎and prediction of production levels.}