From fuzzy sets to shadowed sets: Interpretation and computing

In this study, we discuss a concept of shadowed sets and present their applications. To establish some sound compromise between the qualitative Boolean (two-valued) description of data and quantitative membership grades, we introduce an interpretation framework of shadowed sets. Shadowed sets are discussed as three-valued constructs induced by fuzzy sets assuming three values (that could be interpreted as full membership, full exclusion, and uncertain membership). The algorithm of converting membership functions into this quantification is a result of a certain optimization problem guided by the principle of uncertainty localization. We revisit fundamental ideas of relational calculus in the setting of shadowed sets. We demonstrate how shadowed sets help in problems in data interpretation in fuzzy clustering by leading to the three-valued quantification of data structure that consists of core, shadowed, and uncertain structure.