Entropy measurement for a hybrid information system with images: an application in attribute reduction

In the real world, there may exist manifold data (e.g., Boolean, categorical, real-valued, set-valued, interval-valued, image, decision and missing data or attributes) in an information system which is referred to as a hybrid information system with images (HISI). Handling an HISI is conducive to generalize applications of rough set theory. This paper studies entropy measurement for a hybrid information system with images and considers an application for attribute reduction. We first give the distance between information values on each attribute in an HISI. Then, we present tolerance relations on the object set of an HISI based on this distance. Next, we define the rough approximations in an HISI by means of the presented tolerance relations. Furthermore, we study entropy measurement for an HISI by using \(\theta \)-information entropy, \(\theta \)-conditional information entropy and \(\theta \)-joint information entropy. Based on Kryszkiewicz’s ideal, we introduce the concepts of \(\theta \)-generalized decision and \(\theta \)-consistent in an HISI. Finally, we apply entropy measurement to perform attribute reduction in a \(\theta \)-consistent HISI. It is worth mentioning that attribute reduction based on generalized decision and common attribute reduction in a \(\theta \)-consistent HISI are the same.