Measures of Inclusion and Closeness of Information Granules: A Rough Set Approach

This article introduces an approach to measures of information 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-valued functions representing analog devices such as proximity (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.