Interval regression by tolerance analysis approach

In interval linear regression analysis, we are given crisp or interval data and we are to determine appropriate interval regression parameters. There are various methods for interval regression: many of them possess the property that while some of the resulting interval regression parameters are very wide, the other parameters are crisp. This drawback is the main limiting factor for such methods and much effort has been devoted to overcoming it. We propose a method motivated by tolerance analysis in linear systems. Our method yields intervals for regression parameters the widths of which are proportional to an in-advance given vector of parameters. Moreover, the method is computationally very cheap, and provides a natural measure of quality of a model. First we formulate the method for the basic model of crisp input-crisp output data and then extend it to crisp input-interval output and interval input-interval output models. For the interval-valued cases we study several formulations of the solution concept: possibility, strong possibility, weak possibility, necessity. Here, strong possibility is a new concept proposed as a natural counterpart to the remaining ones. We prove that the method provides optimal interval parameters meeting centrality and proportionality requirements. We also show that the method provides interval regression parameters satisfying various versions of Tanaka-Lee's inclusion property. We also derive a form of a complementarity theorem for the weak possibility and necessity solution concepts. Since practical problems may be affected by outlying observations we show that our approach is easily adapted to deal with them. We illustrate the theory by examples. (C) 2011 Elsevier B.V. All rights reserved.