Mathematics and Statistics 10(4): 759-772, 2022 DOI: 10.13189/ms.2022.100407 http://www.hrpub.org Uncertainty Optimization-Based Rough Set for Incomplete Information Systems Arvind Kumar Sinha, Pradeep Shende * , Department of Mathematics, National Institute of Technology Raipur, Chhattisgarh, India Received February 22, 2022; Revised May 18, 2022; Accepted June 21, 2022 Cite This Paper in the following Citation Styles (a): [1] Arvind Kumar Sinha, Pradeep Shende, ”Uncertainty Optimization-Based Rough Set for Incomplete Information Systems,” Mathematics and Statistics, Vol.10, No.4, pp. 759-772, 2022. DOI: 10.13189/ms.2022.100407 (b): Arvind Kumar Sinha, Pradeep Shende, (2022). Uncertainty Optimization-Based Rough Set for Incomplete Information Systems. Mathematics and Statistics, 10(4), 759-772 DOI: 10.13189/ms.2022.100407 Copyright ©2022 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Often the information in the surrounding world is incomplete, and such incomplete information gives rise to un- certainties. Pawlak’s rough set model is an approach to approximation under uncertainty. It uses a tolerance relation to obtain single granulation of the incomplete information system for approximation. In this work, we extend the single granulation rough set for the incomplete information system to an uncertainty optimization-based rough set (UOBRS). The proposed approach is used to minimize the uncertainty using multiple tolerance relations. We list properties of the UOBRS for incomplete information systems. We compare UOBRS with the classical single granulation rough set (SGRS) and multi-granular rough set (MGRS). We list the basic properties of the UOBMGRS. We introduce the application of the UOBRS for attribute subset selection in case of incomplete information system. We use the measure of approximation quality to assess the uncertainties of the attributes. We compare the approximation quality of the attributes using UOBRS with the approximation quality using SGRS and MGRS. We get higher approximation quality with the less number of attributes using UOBRS as compared to SGRS and MGRS. The proposed method is a novel approach to dealing with incomplete information systems for more effective dataset analysis. Keywords Incomplete Information System, Rough Set, Uncertainty Optimization, Approximation Quality, Feature Subset Selection 1 Introduction Decision making under uncertainty is a problem of concern in many applications related to artificial intelligence. Missing data lead to uncertainties in the dataset and may lead to wrong decisions; therefore, optimization under uncertainty is imperative for expert systems. The attribute set is considered as the granular space [1] and is the main reason of the granular structure of the knowledge. Partitioning the universe of discourse, knowledge granulation and set approximating are the extensively used methods of human’s reasoning under uncertainty [2, 3]. Pawlak’s rough set theory [4, 5, 6] is a well established tool to handle uncertainty involved in many problems of science and technology. The rough set framework is a very useful tool to handle uncertainty and is used in features extraction [7], noise reduction [8] and pattern recognition [9]. It is the general fact that every attribute preserves a different amount of objects information and hence every attribute set is considered a granular space that brings about different granular structure of the knowledge. The rough set methodology uses objects-attribute relation in the form of table known as information system (IS). Mathematically, the IS is an ordered touple (U, At ∪{D},f ), where U is the non-empty finite set of objects, At is the attribute set (conditional features), D is the decision attribute (decision feature) and f A : U → V A for any A ∈ At, where V A is the attribute value set of A. There are many extentions of the rough set model available in the literature including variable precision rough set model [10], multi-granular rough set model [1], neighborhood-based multigranulation rough set model [11], covering based rough set