Describing Rough Approximations by Indiscernibility Relations in Information Tables with Incomplete Information

Rough approximations, a pair of lower and upper approximations, and rule induction are described by directly using indiscernibility relations in information tables containing incomplete information. A set of values is used to express incomplete information. The indiscernibility relations are constructed from viewpoints of both certainty and possibility. First, rough approximations and rule induction are described in information tables with complete information. Second, they are addressed in three cases under incomplete information. One is that a set of objects is approximated by objects with incomplete information. Another is that a set of objects with incomplete information is approximated by objects with complete information. The other is the most general case where a set of objects with incomplete information is approximated by objects with incomplete information. Consequently, we obtain four approximations: certain lower, certain upper, possible lower, and possible upper approximations. Using these approximations, rough approximations are expressed by interval sets. The rough approximations have the complementarity property linked with lower and upper approximations, as is valid under complete information. Last, rule induction are addressed in information tables with incomplete information. Rough approximations under incomplete information do not give sufficient information on rules that an object supports. This is resolved by introducing formulae dealing with pairs of an object and a rule that it supports. The pairs are classified into certain and consistent, possible and consistent, certain and inconsistent, and possible and inconsistent pairs.