Least squares fitting of an affine function and strength of association for interval-valued data

Abstract.

The ultimate goal of this paper is to determine a measure of the degree of dependence between two interval-valued random sets, when the dependence is intended in the sense of an affine function relating these random elements. For this purpose, a general study on the least squares fitting of an affine function for interval-valued data is first carried out, where the least squares method we will present considers that squared residuals are based on a generalized metric on the space of nonempty compact intervals, and output and input random mechanisms are modelled by means of convex compact random sets. For the general case of nondegenerate convex compact random sets, solutions are presented in an algorithmic way, and the few cases leading to nonunique solutions are characterized. On the basis of this regression study we later introduce and analyze a well-defined determination coefficient of two interval-valued random sets, which will allow us to quantify the strength of association between them, and an algorithm for the computation of the coefficient has been also designed. Finally, a real-life example illustrates the study developed in the paper.