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FR-RULEINDUCE(C) Input: C, the set of all conditional features; Output: set of fuzzy rules (1) R , A (2) while concepts not covered (3) foreach x (C R) (4) A R x (5) foreach concept X not covered (6) calculate fuzzy lower approximation, AX (7) remove objects covered by AX (8) add generated attribute-value pairs to rulebase (9) a best attribute according to (10) R R a
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Fuzzy-rough rule induction algorithm
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Clearly, one important factor in the process is the choice of search mechanism for the consideration of sets of attribute-value pairs. A simple approach would be a breadth- rst search, though this would be too costly for large datasets. One possible technique can be found in Figure 15.2, which employs a greedy hill-climbing approach to induction. The fuzzy-rough rule induction algorithm considers the addition of individual attributes to the current set of features (initially empty), generating rules from these. Fuzzy lower approximations of concepts are calculated for each attribute, removing objects from a concept if they are covered by the approximation. Once this is complete, the best attribute is selected via the fuzzy-rough measure of dependency, , and added to the set of features R. The process is then repeated for this new subset.
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As mentioned previously, rules can be generated through the use of minimal complexes. Let X be a concept, t an attribute-value pair (a, v), and T a set of attribute-value pairs. The block of t, denoted [t], is the set of objects for which attribute a has value v. A concept X depends on a set of attribute-value pairs T , iff = {[t]|t T } X (15.1)
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T is a minimal complex of X iff X depends on T and no proper subset T of T exists such that X depends on T [121].
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It is often the case that a minimal complex describes a concept only partially, and hence more than one minimal complex is required to cover a concept. A local covering T of a concept X is such a collection of minimal complexes, such that the union of all minimal complexes is exactly X and T is minimal (i.e., contains no spurious attribute-value pairs). The discovery of such local coverings forms the basis of most approaches to rough set rule induction [276]. A partitioning of the universe of discourse by a reduct will always produce equivalence classes that are subsets of the decision concepts and will cover each concept fully. Once a reduct has been found, rules may be extracted from the underlying equivalence classes. In the literature, reducts for the purpose of rule induction are termed global coverings. The most widely used approach to rule induction is the LEM2 algorithm [121], which follows a heuristic strategy for creating an initial rule by choosing sequentially the best elementary conditions according to some heuristic criteria. Learning examples that match this rule are removed from consideration. The process is repeated iteratively while some learning examples remain uncovered. The resulting set of rules covers all learning examples. In [154] additional factors characterizing rules are taken into account: the strength of matched or partly matched rules (the total number of cases correctly classi ed by the rule during training), the number of nonmatched conditions, and the rule speci city (i.e., length of condition parts). All factors are combined and the strongest decision wins. If no rule is matched, the partly matched rules are considered and the most probable decision is chosen.
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15.4.1 Hybrid Approach
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It can be seen that the tasks of feature selection and rule induction (via global coverings) are very similar in rough set theory. Both are concerned with the discovery of reducts. For feature selection, once a reduct has been found, the original set of data features can be reduced to those appearing in the reduct only. For rule induction, the reduct s corresponding equivalence classes can be used to form rules that completely cover the decision concepts appearing in the data. A feature selection step often precedes rule induction in an attempt to speed up the induction process and make the resulting rules more comprehensible. If a dataset contains redundant or misleading features, any extracted knowledge may become opaque or even meaningless. Hence, in rough set theory, it seems natural to combine the two into a hybrid method that possesses the bene ts of feature selection while inducing rulesets. For the purposes of combining the two approaches, rules are constructed from equivalence classes and a corresponding decision concept. An equivalence class corresponds to a set of attribute-value pairs, forming the consequent of the rule, the decision concept forms the consequent. A rule is herein de ned as a tuple E, F, D , where E is an equivalence class, F is the set of features that generated the equivalence class, and D is the set of objects representing decision concept. This formulation is used as it provides a
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