Ridge estimation for uncertain regression model with imprecise observations

In traditional regression analysis, the observed data are all accurate, but the observed data that we can obtain in real life is often not accurate. For that reason, there is the uncertain regression analysis based on the uncertain variable, in the framework of the uncertainty theory. Under the premise of imprecise observations, the data obtained often contains outliers due to human input errors or incorrect measurements. Outliers can affect parameter estimation, resulting in misleading results and making model fitting inaccurate. In parameter estimation, the most commonly used method is least squares estimation, but this method is extremely sensitive to outliers and makes parameter estimation inaccurate. To solve this problem, this paper proposes an uncertain regression model based on ridge estimation, which adds a square penalty term when performing least squares estimation of unknown parameters. The advantage of ridge estimation is that the tolerance of pathological data is much better than other parameter estimation methods, which can reduce the influence of outliers. In this paper, the optimal shrinkage parameter is determined by K-fold cross-validation to estimate the parameters of the regression model, and then we conduct the residual analysis and hypothesis test on the fitted model to obtain the predicted value and the predicted confidence interval. Finally, the validity of the model is demonstrated by two numerical examples.