Rough Sets, Similarity, and Optimal Approximations

Abstract

Rough sets have been studied for over 30 years, and the basic concepts of lower and upper approximations have been analysed in detail, yet nowhere has the idea of an `optimal' rough approximation been proposed or investigated. In this thesis, several concepts are used in proposing a generalized definition: measures, rough sets, similarity, and approximation are each surveyed. Measure Theory allows us to generalize the definition of the `size' for a set. Rough set theory is the foundation that we use to define the term `optimal' and what constitutes an `optimal rough set'. Similarity indexes are used to compare two sets, and determine how alike or different they are. These sets can be rough or exact. We use similarity indexes to compare sets to intermediate approximations, and isolate the optimal rough sets. The historical roots of these concepts are explored, and the foundations are formally defined. A definition of an optimal rough set is proposed, as well as a simple algorithm to find it. Properties of optimal approximations such as minimum, maximum, and symmetry, are explored, and examples are provided to demonstrate algebraic properties and illustrate the mechanics of the algorithm.