HYBRID RULE INDUCTION in .NET

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fast way of determining rule coverage (the cardinality of E) and rule speci city (the cardinality of F ). To generate a human-readable form of the rules, the values of the features in F for any object appearing in E forms the antecedent, while the value of any object appearing in D for the decision feature(s) will form the consequent. A rule can be induced from any given equivalence class E if E is a subset of a decision concept D appearing in the dataset. From a rough set perspective, equivalence class E will appear in the lower approximation of D, and hence the features that generated E are predictive of this concept. The strength of a rule, constructed from equivalence class E and decision concept D, can be de ned as simply |E| |D| (15.2)
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For example, if E = {1} and D = {1, 2, 3} the strength is 1 . If E = {1, 2, 3}, the 3 strength is 1, meaning all objects are covered for this decision concept.
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15.4.2 Rule Search
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The most basic approach to integrating rule induction and feature selection would be to generate rules from the global covering produced as a result of a typical rough set feature selection process. However, at this point, the generated rules would be too speci c as each antecedent would include every feature appearing in the nal reduct. A greedy hill-climbing approach based on the QuickReduct algorithm is proposed for the purpose (although any search method may be used) and can be seen in Figure 15.3. The progress through the search space is identical to that of QuickReduct, but at each stage of subset evaluation the resulting equivalence classes are considered for rule construction and inclusion in the
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QUICKRULES(C,D) Input: C, the set of all conditional attributes; D, the set of decision attributes Output: R, the feature subset; a set of rules (1) R {} (2) while R (D) = C (D) (3) T R (4) foreach x (C R) (5) if GAMMARULESx, R > T (D) (6) T R {x} (7) R T PRUNE() (8) (9) return R
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Figure 15.3 QuickRules algorithm
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GAMMARULES(a,R) Input: a, the feature under consideration for inclusion in R; R, the current reduct candidate Output: R (D), the dependency (1) Pos (2) foreach eq U/{R a} //each equivalence class //each decision concept (3) foreach dec U/D (4) if eq dec //lower approximation (5) DETERMINEADD(eq, R {a}, dec) (6) Pos Pos eq (7) return
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|Pos| |U|
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//dependency degree
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GammaRules algorithm
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current ruleset. The search continues until the maximum degree of dependency is reached for a subset. At this point, a global covering has been discovered and all concepts can be adequately described by the ruleset. (All equivalence classes will be a subset of a decision concept, and hence all concepts are covered.) Once the search has terminated, a simple pruning procedure is employed to remove super uous features from rules. This is the same procedure as that employed by LEM2, except that it is employed at the end of the search rather than repeatedly during the search. The de nition of GammaRules can be found in Figure 15.4. Given an attribute a and the current reduct candidate R, this function constructs the positive region for R {a} and returns the resulting dependency degree. During this process, if an equivalence class is evaluated that is a subset of a decision concept (and hence included in the lower approximation of the concept), the function DetermineAdd (Figure 15.5) is executed to determine whether a rule representing this is constructed. A rule will be constructed and added to the ruleset if the equivalence class has not been encountered before or if the equivalence class is a superset of an existing rule s equivalence class (i.e., has a larger rule strength). In the latter case the existing rule is removed as this covers fewer objects. For this type of search (i.e., hill-climbing), equality of equivalence classes is not considered in the function. This is because the newer equivalence class will have been produced by the same or greater number of features, and so will have an equal or greater speci city. Alternative heuristics can also be employed, for example, by replacing the GammaRules function with that of Figure 15.6 and minimizing entropy instead of maximizing dependency degree. Reducts (or super-reducts) are found when the entropy for a feature subset reaches zero, and hence all decision concepts are covered fully. For any hill-climbing approach, the nal subsets are not guaranteed to be minimal, but pruning during search would ensure this. It should be noted that in
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