Set-Valued Uncertain  Process: Definition and  Some Properties

Abstract Some phenomena are developing over time, while they are uncertain sets at each moment. From an uncertain set, we mean an unsharp concept, such as “illness” and “recovery”, that is not exactly clear, even for an expert. The values of a parameter that are considered “recovery” would guide one to explain the underlying concept quantitatively. For instance, in recovering from some disease, different levels of health might be assumed. Particularly, at each specific time moment, being healthy to some degree would be measured by belonging parameter values to a set of numbers with a specific belief degree. This set might be extracted using imprecisely observed data, while an expert opinion completely expresses the belief degree. Such concepts would direct one to employ uncertainty theory as a strong axiomatic mathematical framework for modeling human reasoning. Another important feature of these sets is their variation over time. For instance, the set defining “recovery” at the beginning stage of recovery in a disease would be completely or partially different from that at other stages. These characteristics result in considering a sequence of evolving sets over time. Analyzing the behavior of such a sequence motivated us to define the set-valued uncertain process. This concept is a combination of uncertain set, uncertain process, and uncertain sequence. Here, we introduce the main concept. Some properties are extracted and clarified, along with some illustrative examples.