Set-Valued Uncertain Process: Definition and Some Properties

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.

A Regression Model Based on Uncertain Set

Traditional regression analysis is a method of statistical data analysis based on probability theory. Regression models play crucial roles in various branches of statistics including design of experiments, econometrics etc. In regression models, the dependent variable is assumed to be of stochastic nature where randomness enters via errors. Further, the independent variables are assumed to be of deterministic nature. The regression coefficients which explain the interdependency between the variables are assumed to be crisp quantities. Whenever, difficulty arises in expressing the values taken by the dependent variable in terms of crisp quantities, traditional regression models become irrelevant. This paper provides a framework for dealing with such situations on using the notion of uncertain sets of various forms. In this paper, a solution for this problem obtained via linear programming technique is introduced along with an illustrative example.