A New Method for Measuring the Uncertainty in Incomplete Information Systems

Based on an intuitionistic knowledge content nature of information gain, the concept of combination entropy CE(A) in incomplete information systems is first introduced, and some of its important properties are given. Then, the conditional combination entropy CE(Q | P) and the mutual information CE(P;Q) are defined. Unlike all existing measures for the uncertainty in incomplete information systems, the relationships among these three concepts can be established, which are formally expressed as CE(Q | P) = CE(P ⋃ Q)-CE(P) and CE(P;Q) = CE(P)-CE(P | Q). Furthermore, a variant CE(CA) of the combination entropy with maximal consistent block nature is introduced to measure the uncertainty of an incomplete information system in the view of maximal consistent block technique. Its monotonicity is the same as that of the combination entropy. Finally, the combination granulation CG(A) and its variant CG(CA) with maximal consistent block nature are defined to measure discernibility ability of an incomplete information system, and the relationship between the combination entropy and the combination granulation is established as well. These results will be very helpful for understanding the essence of knowledge content and uncertainty measurement in incomplete information systems. Note that the combination entropy also can be further extended to measure the uncertainty in non-equivalence-based information systems.