Abstract
In this study, a paradigm of fuzzy clustering is augmented by available domain knowledge expressed in the form of relational constraints built with the aid of a collection of fuzzy sets. These constraints are described as a collection of Cartesian products of fuzzy sets or their logic expressions are used to form an augmented data space and transform nonlinearly original data. Depending upon the nature of the constraints, discussed are two categories of resulting representations (clustering spaces), namely homogeneous spaces (in case when the transformations are fully expressed by means of the constraints) and heterogeneous spaces (when the resulting space is composed of some original variables present in the initially available data space and those being transformed and expressed by means of satisfaction levels of the constraints). The role of information granules of order-2 is revealed with regard to results of clustering produced in the transformed space. A generalization of the proposed approach is also discussed in case the clustered data are not numeric but are provided in the form of information granules; in this case a special attention is paid to a way in which a representation (description) of information granules is realized through relational constraints. We elaborate on the formation of the space (induced by constraints) and original data as well as discuss the detailed algorithmic developments. (C) 2015 Elsevier B.V. All rights reserved.