An information system as a database that represents relationships between objects and attributes is an important mathematical model. An interval-valued information system is a generalized model of single-valued information systems. As important evaluation tools in the field of machine learning, measures of uncertainty can quantify the dependence and similarity between two targets. However, the existing measures of uncertainty for intervalvalued information systems have not been thoroughly researched. This paper is devoted to the study of new measures of uncertainty for an interval-valued information system. Information structures are first introduced in a given interval-valued information system. Then, the dependence between two information structures is depicted. Next, new measures of uncertainty for an interval-valued information system are investigated by using the information structures. As an application of the proposed measures, the rough entropy of a rough set is proposed by means of information granulation. Finally, a numerical experiment on the Face recognition dataset is presented to demonstrate the feasibility of the proposed measures, and a statistical effectiveness analysis is conducted. The results are helpful for understanding the essence of uncertainty in interval-valued information systems.

Introduction

Rough set theory, a mathematical tool to address imprecision and uncertainty in data analysis, has been successfully applied to intelligent systems, machine learning, knowledge discovery, expert systems, decision analysis, inductive reasoning, pattern recognition, split theory and signal analysis [17,18,22–۲۴]. An information system based on rough sets was introduced by Pawlak [18]. An interval-valued information system is a generalization of the classic information system. To address interval data, scholars have used the methods for managing classic information systems to manage interval-valued information systems. For example, Yao and Li [34] proposed an interval set model for interval-valued information systems with upper and lower approximations and introduced generalized decision logic. Dai et al. [3] studied the algebraic structures of interval-set-valued rough sets generated from an approximation space. Leung et al. [10] investigated a rough set approach on the basis of a knowledge induction process for selecting decision rules with minimum feature sets in interval-valued information systems. Qian et al. [25] presented a dominance relation for interval-valued information systems. Yang et al. [35] proposed a dominance relation and generated the optimal decision rules in incomplete interval-valued information systems. Wu et al. [30] considered the real formal concept analysis of grey-rough set theory by using grey numbers and proposed a grey-rough set approach to Galois lattice reductions. Sakai et al. [29] developed a rule generation prototype system for incomplete information databases in Lipski that can process interval-valued information systems. To measure the uncertainty of a system, Shannon [28] introduced the concept of entropy, which is a very useful mechanism for characterizing information content in various modes and has been used in diverse fields. Some scholars have applied the extension of entropy and its variants to information systems or rough sets. For example, Liang and Qian [12] studied the theory of information particles and entropy in information systems. Liang et al. [13] researched information entropy, rough entropy and knowledge granularity in incomplete information systems. Dai and Tian [4] considered the entropy measure and granularity measure of set-valued information systems. Wang and Yue [32] discussed the entropy measure and granularity measure of interval-valued and set-valued information systems. Qian et al. [27] researched fuzzy information entropy and the granularity of fuzzy information in fuzzy granular structures. Qian et al. [26] proposed fuzzy granularity structure distance. Xu et al. [33] presented rough entropy of rough sets in ordered information systems. Dai et al. [7] studied uncertainty measurement based on the α-weak similarity for incomplete interval-valued information systems. Dai et al. [5] introduced θ-similarity entropy and proposed the θ-rough degree to measure the uncertainty of concepts or rough sets in an interval-valued information system. Dai et al. [6] constructed an extended conditional entropy for interval-valued decision information systems. Zhang et al. [41] investigated uncertainty measures in a fully fuzzy information system.