From fuzzy regression to gradual regression: Interval-based analysis and extensions

This paper proposes an analysis of parametric interval-based regression methodologies according to ontic and epistemic visions of intervals. When assuming an epistemic point of view, a new interpretation of fuzzy regression through the notion of gradual intervals is developed, which leads to gradual regression. Gradual regression is viewed as an extension of the imprecise interval-based regression, which is obtained by integrating an uncertain dimension. Gradual intervals can yield improved specificity compared to conventional intervals and jointly consider the concepts of imprecision and uncertainty through a single and coherent formalism. The formulation of the gradual regression problem, its resolution and the propagation of the information through the obtained regressive models are carried out via gradual interval arithmetic. The proposed method allows not only the extension of the interval vision to the gradual case but also interesting interpretations according to non-additive confidence measure theories (possibility and belief functions).