Rule Induction Based on Indiscernible Classes from Rough Sets in Information Tables with Continuous Values

Rule induction based on indiscernible classes from neighborhood rough sets is described in information tables with continuous values. An indiscernible range that a value has in an attribute is determined by a threshold on that attribute. The indiscernible class of every object is derived from using the indiscernible range. First, lower and upper approximations are described in complete information tables by using indiscernible classes. Rules are obtained from the approximations. A rule that an object supports, which is called a single rule, is short of applicability. To improve the applicability of rules, a series of single rules is put into one rule expressed in an interval value, which is called a combined rule. Second, these are addressed in incomplete information tables. Incomplete information is expressed in a set of values or an interval value. Two types of indiscernible classes; namely, certainly and possibly indiscernible ones, are obtained from in an information table. The actual indiscernibility class is between the certainly and possibly indiscernible classes. The family of indiscernible classes of an object has a lattice structure. The minimal element is the certainly indiscernible class while the maximal one is the possibly indiscernible class. By using certainly and possibly indiscernible classes, we obtain four types of approximations: certain lower, certain upper, possible lower, and possible upper approximations. From these approximations we obtain four types of combined rules: certain and consistent, certain and inconsistent, possible and consistent, and possible and inconsistent ones. These combined rules have greater applicability than single rules that individual objects support.