Optimization in Discovery of Compound Granules

The problem considered in this paper is the evaluation of perception as a means of optimizing various tasks. The solution to this problem hearkens back to early research on rough set theory and approximation. For example, in 1982, Ewa Orlowska observed that approximation spaces serve as a formal counterpart of perception. In this paper, the evaluation of perception is at the level of approximation spaces. The quality of an approximation space relative to a given approximated set of objects is a function of the description length of an approximation of the set of objects and the approximation quality of this set. In granular computing (GC), the focus is on discovering granules satisfying selected criteria. These criteria take inspiration from the minimal description length (MDL) principle proposed by Jorma Rissanen in 1983. In this paper, the role of approximation spaces in modeling compound granules satisfying such criteria is discussed. For example, in terms of approximation itself, this paper introduces an approach to function approximation in the context of a reinterpretation of the rough integral originally proposed by Zdzislaw Pawlak in 1993. We also discuss some other examples of compound granule discovery problems that are related to compound granules representing process models and models of interaction between processes or approximation of trajectories of processes. All such granules should be discovered from data and domain knowledge. The contribution of this article is a proposed solution approach to evaluating perception that provides a basis for optimizing various tasks related to discovery of compound granules representing rough integrals, process models, their interaction, or approximation of trajectories of discovered models of processes.