SET THEORY

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extensions with sample applications. Further details can be found in the literature (e.g., LNCS Transactions on Rough Sets) and on the Internet (e.g., www. roughsets.org). Rough set theory is an extension of conventional set theory that supports approximations in decision making. It possesses many features in common (to a certain extent) with the Dempster-Shafer theory of evidence [334] and fuzzy set theory [262,389]. A rough set is itself the approximation of a vague concept (set) by a pair of precise concepts, called lower and upper approximations, that are a classi cation of the domain of interest into disjoint categories. The lower approximation is a description of the domain objects that are known with certainty to belong to the subset of interest, whereas the upper approximation is a description of the objects that possibly belong to the subset. It works by exploring and exploiting the granularity structure of the data only. This is a major difference when compared with Dempster Shafer theory [76,315] and fuzzy set theory [408], 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 are a true and accurate re ection of the real world (which may not be the case). The numerical and other contextual aspects of the data are ignored, which may seem to be a signi cant omission but keeps model assumptions to a minimum.

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2.3.1 Information and Decision Systems

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An information system can be viewed as a table of data, consisting of objects (rows in the table) and attributes (columns). In medical datasets, for example, patients might be represented as objects and measurements such as blood pressure, form attributes. The attribute values for a particular patient is their speci c reading for that measurement. Throughout this book, the terms attribute, feature, and variable are used interchangeably. An information system may be extended by the inclusion of decision attributes. Such a system is termed a decision system. For example, the medical information system mentioned previously could be extended to include patient classi cation information, such as whether a patient is ill or healthy. A more abstract example of a decision system can be found in Table 2.1. The table consists of four conditional features (a, b, c, d), a decision feature (e), and eight objects. A decision system is consistent if for every set of objects whose attribute values are the same, the corresponding decision attributes are identical. More formally, I = (U, A) is an information system, where U is a nonempty set of nite objects (the universe of discourse) and A is a nonempty nite set of attributes such that a : U Va for every a A. Va is the set of values that attribute a may take. For decision systems, A = {C D}, where C is the set of input features and D is the set of class indexes. Here, a class index d D is itself a variable d : U {0, 1} such that for a U, d(a) = 1 if a has class d and d(a) = 0 otherwise.

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ROUGH SET THEORY

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TABLE 2.1 An example dataset x U a b c 0 S R T 1 R S S T R R 2 S S R 3 4 S R T T T R 5 6 T S S 7 R S S

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d T S S T R S S R

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e R T S T S S T S

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Indiscernibility

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With any P A there is an associated equivalence relation IND(P ): IND(P ) = {(x, y) U2 | a P , a(x) = a(y)} (2.8)

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Note that this relation corresponds to the equivalence relation for which two objects are equivalent if and only if they have the same vectors of attribute values for the attributes in P . The partition of U, determined by IND(P), is denoted U/IND(P ) or U/P , which is simply the set of equivalence classes generated by IND(P ): U/IND(P ) = {U/IND({a})|a P } where A B = {X Y |X A, Y B, X Y = } (2.10) (2.9)

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If (x, y) IND(P ), then x and y are indiscernible by attributes from P . The equivalence classes of the indiscernibility relation with respect to P are denoted [x]P , x U. For the illustrative example, if P = {b, c}, then objects 1, 6, and 7 are indiscernible, as are objects 0 and 4. IND(P) creates the following partition of U: U/IND(P ) = U/IND({b}) U/IND({c}) = {{0, 2, 4}, {1, 3, 6, 7}, {5}} {{2, 3, 5}, {1, 6, 7}, {0, 4}} = {{2}, {0, 4}, {3}, {1, 6, 7}, {5}}

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