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Permeability index (PI) and magnesium absorption ratio (MAR) are both primary irrigation water quality indicators (IWQI) used to evaluate the efficacy of agricultural water supplies. This is considered a complex environmental issue to reliably forecast IWQI parameters without its appropriate time series and limited input sequences. Hence, this research develops an innovative hybrid intelligence framework for the first time to forecast the PI and MAR indices at the Karun River, Iran. The proposed framework includes a new hybrid machine learning (ML) model based on generalized ridge regression and kernel ridge regression with a regularized locally weighted (GRKR) method. This research developed an optimized multivariate variational mode decomposition (OMVMD) technique, optimized by the Runge-Kutta algorithm (RUN), to decompose the input variables. The light gradient boosting machine model (LGBM) is also implemented to select the influential input variables. The main contribution of the intelligence framework lies in developing a new hybrid ML model based on GRKR coupled with OMVMD. Five water quality parameters from the Karun River at two stations (Ahvaz and Molasani) over 40 years are used to forecast the PI and MAR indices monthly. Statistical metrics confirmed that the proposed OMVMD-GRKR model, concerning the best efficiency in the Ahvaz ( R  = 0.987, RMSE = 0.761, and U95% = 2.108) and Molasani ( R  = 0.963, RMSE = 1.379, and U95% = 3.828) stations, outperformed the OMVMD and simple-based methods such as ridge regression (Ridge), least squares support vector machine (LSSVM), deep random vector functional link (DRVFL), and deep extreme learning machine (DELM). For this reason, the suggested OMVMD-GRKR model serves as a valuable framework for predicting IWQI parameters.

The Poisson-Inverse Gaussian regression model is a widely used method for analyzing count data, particularly in over-dispersion. However, the reliability of parameter estimates obtained through maximum likelihood estimation in this model can be compromised when multicollinearity exists among the explanatory variables. Multicollinearity means that high correlations between explanatory variables inflate the variance of the maximum likelihood estimates and increase the mean squared error. To handle this problem, the Poisson-Inverse Gaussian ridge regression estimator has been proposed as a viable alternative. This paper introduces a generalized ridge estimator to estimate regression coefficients in the Poisson-Inverse Gaussian regression model under multicollinearity. The performance of the proposed estimator is evaluated through a comprehensive simulation study, covering various scenarios and employing the mean squared error as the evaluation criterion. Furthermore, the practical applicability of the estimator is demonstrated using two real-life datasets, with its performance again assessed based on mean squared error. Theoretical analyses, supported by simulation and empirical findings, suggest that the proposed estimator outperforms existing methods, offering a more reliable solution in multicollinearity.

This paper introduces two new estimators based on the philosophy of unbiased ridge regression estimation, where the parameters are part of a partial linear model suffering from multicollinearity. These proposed estimators are called the Difference-Based Unbiased Ridge Estimator$$ {\\hat{\\beta}}_{DB- URR} $$and the Difference-Based Modified Unbiased Ridge Estimator$$ {\\hat{\\beta}}_{DB- MUR} $$for the regression parameters β . The Mean Squared Error Matrix (MSEM) criterion is employed to compare the proposed estimators against the Difference-Based Ordinary Least Squares estimator$$ {\\hat{\\beta}}_{DB- OLS} $$and the Difference-Based Ordinary Ridge Estimator$$ {\\hat{\\beta}}_{DB- ORR} $$ . Finally, the performance of the new estimators is evaluated through a comprehensive simulation study and a numerical example.

This study focuses on the use of machine learning (ML) models to predict the biodistribution of nanoparticles in various organs, using a dataset derived from research on nanoparticle behavior for cancer treatment. The dataset includes both categorical and numerical variables related to nanoparticle properties, with a focus on their distribution across organs such as the tumor, heart, liver, spleen, lung, and kidney tissues. In order to address the complex and non-linear nature of the data, three machine learning models were utilized: Bayesian Ridge Regression (BRR), Kernel Ridge Regression (KRR), and K-Nearest Neighbors (KNN). The selection of these models was based on their wide range of capabilities in dealing with non-linear relationships and data complexity. To further model performance and strength, the study also applied cutting-edge methods including the Firefly Algorithm for hyperparameter tuning and Recursive Feature Elimination (RFE) for feature selection. Based on higher R² and lower RMSE values for most output parameters, the study concluded that Kernel Ridge Regression (KRR) did better compared to other models in predicting biodistribution outcomes. The study revealed that machine learning models, particularly KRR, exhibit a high level of efficiency in accurately representing the non-linear characteristics of nanoparticle biodistribution. The results obtained provide valuable insights into the optimization of predictive models for the behavior of nanoparticles. These models can be further enhanced by the use of advanced feature selection and hyperparameter tuning techniques.

With the in-depth implementation of the rural revitalization strategy, it is of great significance to study the supervision and governance mechanism of national audit in the implementation of the rural revitalization strategy. As the main components with smaller contribution rates will be discarded when evaluating and analyzing the influencing factors of rural revitalization, it may affect the analytical judgment of the actual problem. Therefore, based on stepwise regression, this paper adopts ridge regression to correct multicollinearity, with the help of the ridge trace diagram to judge the correlation between independent variables intuitively and quickly, and through the ridge calculation as much as possible to retain the independent variables that have a greater impact on rural revitalization. The principal component-ridge regression model is proposed because it takes into account the advantages and disadvantages of various regression methods. To solve the problem of unstable regression coefficients caused by multiple covariances, to analyze the important influencing factors of state auditing on rural revitalization according to the regression coefficients, and to establish the governance optimization path of state auditing to promote rural revitalization. According to the regression results, the government audit variable has a significance level of 1%. The comprehensive governance function of government audit has a regression coefficient of -0.038 that shows significant and negative results at a 1% confidence level. The performance scores of rural sustainability as well as financial dimensions are in the range of 0.8 to 1, which is a good level.

Traditional regression estimators like Ordinary Least Squares (OLS) and classical ridge regression often fail under multicollinearity and outlier contamination respectively. Although recently developed two-parameter ridge regression (TPRR) estimators improve efficiency by introducing dual shrinkage parameters, they remain sensitive to extreme observations. This study develops a new class of Two-Parameter Robust Ridge M-Estimators (TPRRM) that integrate dual shrinkage with robust M-estimation to simultaneously address multicollinearity and outliers. A Monte Carlo simulation study, conducted under varying sample sizes, predictor dimensions, correlation levels, and contamination structures, compares the proposed estimators with OLS, ridge, and the most recent TPRR estimators. The results demonstrate that TPRRM consistently achieves the lowest Mean Squared Error (MSE), particularly in heavy-tailed and outlier-prone scenarios. Application to the Tobacco and Gasoline Consumption datasets further validates the superiority of the proposed methods in real-world conditions. The findings confirm that the proposed TPRRM fills a critical methodological gap by offering estimators that are not only efficient under multicollinearity, but also robust against departures from normality.

We propose a fast algorithm for computing the entire ridge regression regularization path in nearly linear time. Our method constructs a basis on which the solution of ridge regression can be computed instantly for any value of the regularization parameter. Consequently, linear models can be tuned via cross-validation or other risk estimation strategies with substantially better efficiency. The algorithm is based on iteratively sketching the Krylov subspace with a binomial decomposition over the regularization path. We provide a convergence analysis with various sketching matrices and show that it improves the state-of-the-art computational complexity. We also provide a technique to adaptively estimate the sketching dimension. This algorithm works for both the over-determined and under-determined problems. We also provide an extension for matrix-valued ridge regression. The numerical results on real medium and large-scale ridge regression tasks illustrate the effectiveness of the proposed method compared to standard baselines which require super-linear computational time.

The Gamma ridge regression estimator (GRRE) is commonly used to solve the problem of multicollinearity, when the response variable follows the gamma distribution. Estimation of the ridge parameter estimator is an important issue in the GRRE as well as for other models. Numerous ridge parameter estimators are proposed for the linear and other regression models. So, in this study, we generalized these estimators for the Gamma ridge regression model. A Monte Carlo simulation study and two real-life applications are carried out to evaluate the performance of the proposed ridge regression estimators and then compared with the maximum likelihood method and some existing ridge regression estimators. Based on the simulation study and real-life applications results, we suggest some better choices of the ridge regression estimators for practitioners by applying the Gamma regression model with correlated explanatory variables.

For the linear model Y=Xb+error, where the number of regressors (p) exceeds the number of observations (n), the Elastic Net (EN) was proposed, in 2005, to estimate b. The EN uses both the Lasso, proposed in 1996, and ordinary Ridge Regression (RR), proposed in 1970, to estimate b. However, when p>n, using only RR to estimate b has not been considered in the literature thus far. Because RR is based on the least-squares framework, only using RR to estimate b is computationally much simpler than using the EN. We propose a generalized ridge regression (GRR) algorithm, a superior alternative to the EN, for estimating b as follows: partition X from left to right so that every partition, but the last one, has 3 observations per regressor; for each partition, we estimate Y with the regressors in that partition using ordinary RR; retain the regressors with statistically significant t-ratios and the corresponding RR tuning parameter k, by partition; use the retained regressors and k values to re-estimate Y by GRR across all partitions, which yields b. Algorithmic efficacy is compared using 4 metrics by simulation, because the algorithm is mathematically intractable. Three metrics, with their probabilities of RR’s superiority over EN in parentheses, are: the proportion of true regressors discovered (99%); the squared distance, from the true coefficients, of the significant coefficients (86%); and the squared distance, from the true coefficients, of estimated coefficients that are both significant and true (74%). The fourth metric is the probability that none of the regressors discovered are true, which for RR and EN is 4% and 25%, respectively. This indicates the additional advantage RR has over the EN in terms of discovering causal regressors.

Ridge regression model is a widely used model with many successful applications, especially in managing correlated covariates in a multiple regression model. Multicollinearity represents a serious threat in fuzzy regression models as well. We address this issue by combining ridge regression with the fuzzy regression model. Our proposed algorithm uses the α -level estimation method to evaluate the parameters of the ridge fuzzy regression model. Two examples are given to illustrate the ridge fuzzy regression model with crisp input/fuzzy output and fuzzy coefficients.