Funções de covariância e robustez a outliers em processos Gaussianos

Abstract

In Regression Theory, Gaussian Processes emerge as an efficient alternative to simplify the estimation process in Bayesian Models. A Gaussian Process is a generalization of the Gaussian distribution, fully defined by its mean and covariance function (or kernel). This non-parametric model has gained prominence due to its computational ease and effectiveness in predictions. In Gaussian Process predictions, the choice of the covariance function plays a crucial role, as it directly depends on the input values of the observations. However, the presence of outliers in the data can compromise model inferences, leading to inadequate predictions. This study analyzes, through simulations with data containing outliers, how different covariance functions affect the performance of Predictive Gaussian Process Models. The objective is to identify which kernels are more sensitive to extreme data points and which are more robust, as well as to categorize these functions within their respective families of variation. Additionally, a new kernel, called DC-LogExp, is proposed as an alternative to improve the robustness of Gaussian Processes. The results show that model performance is significantly impacted by the choice of kernel and the proportion of outliers in the samples. Covariance functions classified as belonging to the regular variation family, specifically Rational Quadratic and DC-LogExp, stood out for their low sensitivity to the presence of outliers. In particular, the DC-LogExp function proved to be a promising alternative, providing good fits in scenarios with and without outliers, as well as satisfactory results in terms of parameter convergence and predictive errors.