On the Validity of Minimin and Minimax Methods for Support Vector Regression with Interval Data

In the recent years, generalizations of support vector methods for analyzing interval-valued data have been suggested in both the regression and classification contexts. Standard Support Vector methods for precise data formalize these statistical problems as optimization problems that can be based on various loss functions. In the case of Support Vector Regression (SVR), on which we focus here, the function that best describes the relationship between a response and some explanatory variables is derived as the solution of the minimization problem associated with the expectation of some function of the residual, which is called the risk functional. The key idea of SVR is that even when considering an infinite-dimensional space of arbitrary regression functions, given a finitedimensional data set, the function minimizing the risk can be represented as the finite weighted sum of kernel functions. This allows to practically determine the SVR estimate by solving a much simpler optimization problem, even in the case of nonlinear regression. In case that only interval-valued observations of the variables of interest are available, it has been suggested to minimize the minimal or maximal risk values that are compatible with the imprecise data, yielding precise SVR estimates on the basis of interval data. In this paper, we show that also in the case of an interval-valued response the optimal function can be represented as the finite weighted sum of kernel functions. Thus, the minimin and minimax SVR estimates can be obtained by minimizing the corresponding simplified expressions of the empirical lower and upper risks, respectively.