Decision Rules
On the basis of the approximations obtained by means of the dominance relations, it is possible to induce a generalized description of the preferential information contained in the decision table, in terms of decision rules. The decision rules are expressions of the form if then, that represent a form of dependency between condition criteria and decision criteria. Procedures for generating decision rules from a decision table use an inducive learning principle. We can distinguish three types of rules: certain, possible and approximate. Certain rules are generated from lower approximations of unions of classes; possible rules are generated from upper approximations of unions of classes and approximate rules are generated from boundary regions.
Certain rules has the following form:
if and and then
if and and then
Possible rules has a similar syntax, however the consequent part of the rule has the form: could belong to or the form: could belong to .
Finally, approximate rules has the syntax:
if and and and and and then
The certain, possible and approximate rules represent certain, possible and ambiguous knowledge extracted from the decision table.
Each decision rule should be minimal. Since a decision rule is an implication, by a minimal decision rule we understand such an implication that there is no other implication with an antecedent of at least the same weakness (in other words, rule using a subset of elementary conditions or/and weaker elementary conditions) and a consequent of at least the same strength (in other words, rule assigning objects to the same union or sub-union of classes).
A set of decision rules is complete if it is able to cover all objects from the decision table in such a way that consistent objects are re-classified to their original classes and inconsistent objects are classified to clusters of classes referring to this inconsistency. We call minimal each set of decision rules that is complete and non-redundant, i.e. exclusion of any rule from this set makes it non-complete. One of three induction strategies can be adopted to obtain a set of decision rules:
- generation of a minimal description, i.e. a minimal set of rules,
- generation of an exhaustive description, i.e. all rules for a given data matrix,
- generation of a characteristic description, i.e. a set of rules covering relatively many objects each, however, all together not necessarily all objects from the decision table
The most popular rule induction algorithm for dominance-based rough set approach is DOMLEM, which generates minimal set of rules.
Read more about this topic: Dominance-based Rough Set Approach
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