Bayesian Estimation for the Extended t-process Regression Models with Independent Errors

Abstract

Gaussian process regression (GPR) model is well-known to be susceptible to outliers. Robust process regression models based on t-process or other heavy-tailed processes have been developed to address the problem. However, due to the nature of the current definition for heavy-tailed processes, the unknown process regression function and the random errors are always defined jointly and thus dependently. This definition, mainly owing to the dependence assumption involved, is not justified in many practical problems and thus limits the application of those robust approaches. It also results in a limitation of the theory of robust analysis. In this paper, we propose a new robust process regression model enabling independent random errors. An efficient estimation procedure is developed. Statistical properties, such as unbiasness and information consistency, are provided. Numerical studies show that the proposed method is robust against outliers and has a better performance in prediction compared with the existing models. We illustrate that the estimated random-effects are useful in detecting outlying curves.Remark 2 Under a special case of r i0 = r i1 = ... = r iJ = r which is actually a joint error model, µ ri in (10) is independent of r, thus it becomes the conditional mean of f i (X i )|D n , the same mean as the one from a GPR model. Equation (11) shows that

A robust estimation for the extended t-process regression model

“…This paper assumes that prior of the unknown function and the error term have a joint ETP, which is an unnatural way to define a process model (see discussions in Wang et al, 2018). A better way is to use an independent processes model, i.e.…”