A robust regression method based on exponential-type kernel functions

Abstract

Robust regression methods appear commonly in practical situations due the presence of outliers. In this paper we propose a robust regression method that penalize bad fitted observations (outliers) through the use of exponential-type kernel functions in the parameter estimator iterative process. Thus, the weights given to each observation are updated at each iteration in order to optimize a suitable objective function. The convergence of the parameter estimation algorithm is guaranteed with a low computational cost. Its performance is sensitive to the choice of the initial values for the vector of parameters of the regression model as well as to the width hyper-parameter estimator of the kernel functions. A simulation study with synthetic data sets revealed that some width hyper-parameter estimators can improve the performance of the proposed approach and that the ordinary least squares (OLS) method is a suitable choice for the initial values for the vector of parameters of the proposed regression method. A comparative study between the proposed method against some classical robust approaches (WLS, M-Estimator, MM-Estimator and L1 regression) and the OLS method is also considered. The performance of these methods are evaluated based on the bias and mean squared error (MSE) of the parameter estimates, considering synthetic data sets with X-space outliers, Y-space outliers and leverage points, different sample sizes and percentage of outliers in a Monte Carlo framework. The results suggest that the proposed approach presents a competitive performance (or best) in outliers scenarios that are comparable to those found in real problems. The proposed method also exhibits a similar performance to the OLS method when no outliers are considered and about half of the computational time if compared with MM-Estimator method. Applications to real data sets corroborates the usefulness of the proposed method.