Towards fast and accurate algorithms for processing fuzzy data: interval computations revisited

In many practical applications, we need to process data – e.g., to predict the future values of different quantities based on their current values. Often, the only information that we have about the current values comes from experts, and is described in informal (“fuzzy”) terms like “small”. To process such data, it is natural to use fuzzy techniques, techniques specifically designed by Lotfi Zadeh to handle such informal information. In this survey, we start by revisiting the motivation behind Zadeh’s formulas for processing fuzzy data, explain how the algorithmic problem of processing fuzzy data can be described in terms of interval computations (α-cuts). Many fuzzy practitioners claim “I tried interval computations, they did not work” – meaning that they got estimates which are much wider than the desired α-cuts. We show that such statements are usually based on a (widely spread) misunderstanding – that interval computations simply means replacing each arithmetic operation with the corresponding operation with intervals. We show that while such straightforward interval techniques indeed often lead to over-wide estimates, the current advanced interval computations techniques result in estimates which are much more accurate. We overview such advanced interval computations techniques, and show that by using them, we can efficiently and accurately process fuzzy data. We wrote this survey with three audiences in mind. First, we want fuzzy researchers and practitioners to understand the current advanced interval computations techniques and to use them to come up with faster and more accurate algorithms for processing fuzzy data. For this “fuzzy” audience, we explain these current techniques in detail. Second, we also want interval researchers to better understand this important application area for their techniques. For this “interval” audience, we want to explain where fuzzy techniques come from, what are possible variants of these techniques, and what are the problems to which interval techniques can be applied. These readers needs to avoid their own frequent misunderstanding – that fuzzy techniques are “magical” heuristic tools that are only justified by intuition and that have no mathematical justification. Readers of both types can skip the parts they already know. Finally, we also want to target people who solve practical data processing problems – and who may not be well familiar neither with the fuzzy nor with the interval techniques. We want these readers to get an understanding of both the problem of processing fuzzy data and of the interval techniques for solving this problem.