Rough Set Theory — Fundamental Concepts, Principals, Data Extraction, and Applications
Silvia das Dores Rissino, Germano Lambert‐Torres · 2009
Data Mining and Knowledge Discovery in Real Life Applications 36outset, rough set theory has been a methodology of database mining or knowledge discovery in relational databases.This section presents the concepts of Rough Set Theory; which coincide partly with the concepts of other theories that treat uncertain and vagueness information.Among the existent, most traditional approaches for the modeling and treatment of uncertainties, they are the Theories of the Uncertainty of Dempster-Shafer and Fuzzy Set (Pawlak et al., 1995).The main concepts related to Rough Set Theory are presented as the following: SetA set of objects that possesses similar characteristics it is a fundamental part of mathematics.All the mathematical objects, such as relations, functions and numbers can be considered as a set.However, the concept of the classical set within mathematics is contradictory; since a set is considered to be "grouping" without all elements are absent and is know as an empty set (Stoll, 1979).The various components of a set are known as elements, and relationship between an element and a set is called of a pertinence relation.Cardinality is the way of measuring the number of elements of a set.Examples of specific sets that treat vague and imprecise date are described below: a. Fuzzy Set Proposed by mathematician Loft Zadeh in the second half of the sixties, it has as its objective the treatment of the mathematical concept of vague and approximate, for subsequent programming and storage on computers.In order for Zadeh to obtain the mathematical formalism for fuzzy set, it was necessary to use the classic set theory, where any set can be characterized by a function.In the case of the fuzzy set, the characteristic function can be generalized so that the values are designated as elements of the Universe Set U belong to the interval of real numbers [0,1].The characteristic Function Fuzzy is µA: U å [0,1], where the values indicate the degree of pertinence of the elements of set U in relation to the set A, which indicated as it is possible for an element of x of U to belong to A, this function is known as Function of Pertinence and the set A is the Fuzzy Set (Zadeh, 1965).b.Rough Set An approach first forwarded by mathematician Zdzislaw Pawlak at the beginning of the eighties; it is used as a mathematical tool to treat the vague and the imprecise.Rough Set Theory is similar to Fuzzy Set Theory, however the uncertain and imprecision in this approach is expressed by a boundary region of a set, and not by a partial membership as in Fuzzy Set Theory.Rough Set concept can be defined quite generally by means of interior and closure topological operations know approximations (Pawlak, 1982).Observation: It is interesting to compare definitions of classical sets, fuzzy sets and rough sets.Classical set is a primitive notion and is defined intuitively or axiomatically.Fuzzy set is defined by employing the fuzzy membership function, which involves advanced mathematical structures, numbers and functions.Rough set is defined by topological operations called approximations, thus this definition also requires advanced mathematical concepts. Information system or information tableAn information system or information table can be viewed as a table, consisting of objects (rows) and attributes (columns).It is used in the representation of data that will be utilized by Rough Set, where each object has a given amount of attributes (Lin, 1997). www.intechopen.comRough Set Theory -Fundamental Concepts, Principals, Data Extraction, and Applications 37 These objects are described in accordance with the format of the data table, in which rows are considered objects for analysis and columns as attributes (Wu et al., 2004).Below is shown an example of an information Table 1.Attributes Patient Headache Vomiting Temperature Viral illness #1 No Yes High Yes #2 Yes No High Yes #3 Yes Yes Very high Yes #4 No Yes Normal No #5 Yes No High No #6 No Yes Very high Yes Conditional Attributes Decision Attribute Patient blotched_red_skin muscular_pain_articu lations