TopBackground
Lately there has been great interest in developing methodologies which are capable of dealing with imprecision and uncertainty, and the resounding amount of research currently being done in the areas related to fuzzy (Zadeh, 1965) and rough sets (Pawlak, 1991) is representative of this. The success of rough set theory is due in part to three aspects of the theory. Firstly, only the facts hidden in data are analyzed. Secondly, no additional information about the data is required for data analysis such as thresholds or expert knowledge on a particular domain. Thirdly, it finds a minimal knowledge representation for data. As rough set theory handles only one type of imperfection found in data, it is complementary to other concepts for the purpose, such as fuzzy set theory. The two fields may be considered analogous in the sense that both can tolerate inconsistency and uncertainty – the difference being the type of uncertainty and their approach to it.
Fuzzy sets are concerned with modeling the vagueness that is present in real world data through the extension of classical set theory (with binary set membership) to the case of gradual memberships, i.e. elements belong to (fuzzy) sets to a certain degree. This allows greater flexibility in modeling concepts that are inherently vague. For example, the concept “Tall,” where elements are people, could be defined differently by different people based on their subjective views of what constitutes tallness. Indeed, it is a valid observation that there are degrees of tallness: some people are definitely tall, however other people might be slightly tall or moderately tall. This vagueness is difficult to model with classical set theory, but straightforward to do so with fuzzy set theory. With this basis, tools and techniques can be developed for representation and reasoning.
Rough set theory has been used as a tool to discover data dependencies, induce rules, and to reduce the number of features contained in a dataset using the data alone, requiring no additional information (Pawlak, 1991; Polkowski, 2002; Skowron et al., 2002). The rough set itself is the approximation of a vague concept (set) by a pair of precise concepts, called lower and upper approximations, which are a classification of the domain of interest into disjoint categories. The lower approximation is a description of the domain objects which are known with certainty to belong to the subset of interest, whereas the upper approximation is a description of the objects which possibly belong to the subset. The approximations are constructed with regard to a particular subset of features, and works by making use of the granularity structure of the data only. This is a major difference when compared with Dempster-Shafer theory and fuzzy set theory which require probability assignments and membership values respectively. However, this does not mean that no model assumptions are made. In fact by using only the given information, the theory assumes that the data is a true and accurate reflection of the real world (which may not be the case).