Combining Interval, Probabilistic, and Fuzzy Uncertainty: Foundations, Algorithms, Challenges – An Overview

Abstract

Summary. Since the 1960s, many algorithms have been designed to deal with interval uncertainty. In the last decade, there has been a lot of progress in extending these algorithms to the case when we have a combination of interval, probabilistic, and fuzzy uncertainty. We provide an overview of related algorithms, results, and remaining open problems. 1 Main Problem Why indirect measurements? In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. Examples of such quantities are the distance to a star and the amount of oil in a given well. Since we cannot measure y directly, a natural idea is to measure y indirectly. Specifically, we find some easier-to-measure quantities x1,..., xn which are related to y by a known relation y = f(x1,..., xn); this relation may be a simple functional transformation, or complex algorithm (e.g., for the amount of oil, numerical solution to an inverse problem). Then, to estimate y, we first measure the values of the quantities x1,..., xn, an