Measures of Inclusion and Closeness of Information Granules: A Rough Set Approach James F. Peters 1 , Andrzej Skowron 2 , Zbigniew Suraj 3 , Maciej Borkowski 1 , and Wojciech Rz¸ asa 4 1 Department of Electrical and Computer Engineering, University of Manitoba Winnipeg, Manitoba R3T 5V6 Canada, jfpeters@ee.umanitoba.ca 2 Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland, skowron@mimuw.edu.pl 3 Univ. of Information Technology and Management H. Sucharskiego 2, 35-225 Rzesz´ow, Poland, zsuraj@wenus.wsiz.rzeszow.pl 4 InstituteofMathematics,Rzesz´owUniversity Rejtana 16A 35-310 Rzesz´ ow, Poland, wrzasa@univ.rzeszow.pl Abstract. This article introduces an approach to measures of infor- mation granules based on rough set theory. The information granules considered in this paper are partially ordered multisets of sample sensor signal values, where it is possible for such granules to contain duplicates of the same values obtained in different moments of time. Such granules are also associated with a feature set in an information system. Information granules considered in this paper are collections of sample values derived from sensors that are modelled as continuous real-valuedfunctionsrepresentinganalogdevicessuchasproximity(e.g., ultrasonic) sensors. The idea of sampling sensor signals is fundamental, since granule approximations and granule measures are defined relative to non-empty temporally ordered multisets of sample signal values. The contribution of this article is the introduction of measures of granule inclusion and closeness based on an indistinguishability relation that partitions real-valued universes into subintervals (equivalence classes). Such partitions are useful in measuring closeness and inclusion of granules containing sample signal values. The measures introduced in this article lead to the discovery of clusters of sample signal values. Keywords: Closeness, inclusion, indistinguishability, information gran- ule, measure, rough sets, sensor. 1 Introduction This article introduces an approach to measures of a particular class of infor- mation granules based on rough set theory. Informally, a granule is a multiset (or bag) [19]-[20] of real-world objects that are somehow indistinguishable (e.g., J.J. Alpigini et al. (Eds.): RSCTC 2002, LNAI 2475, pp. 300–307, 2002. c  Springer-Verlag Berlin Heidelberg 2002