Set-valued uncertain process: definition and some properties

Abstract

Some phenomena develop over time, while they are uncertain sets at each moment. From an uncertain set, we mean an unsharp concept that is not clearly defined, even for an expert. The potential values of parameters describing this unsharp concept guide one to explain it quantitatively. For instance, in “recovering” from some disease, different levels of health might be defined, meaning that at each specific time, being healthy is measured by belonging some parameter values to a set with a specific belief degree. Clearly speaking, the set defining “recovery” at the beginning stage of a disease would be entirely or partially non-identical with it at other stages. These sets might be extracted using imprecisely observed data, or an expert opinion defines them by some belief degrees. An essential feature of these sets is their variation over time by considering a sequence of evolving sets over time. Such concepts would direct one to employ uncertainty theory as a solid axiomatic mathematical framework for modeling human reasoning. Analyzing the behavior of such a sequence motivated us to define the set-valued uncertain process. This concept combines uncertain set, uncertain process, and uncertain sequence to a new concept. Here, we introduce the main idea; some properties are extracted and clarified, along with some illustrative examples. We also put forward some potential state-of-the-art applications.