On the foundations of granular computing paradigm

This paper investigates the foundations of granular computing, which is a paradigm oriented towards capturing and processing meaningful pieces of information, the so-called information granules. Each level of granulation induces a higher complexity in the semantic structure of the universe of discourse we have to deal with. Equipping the space of such granules with suitable metric concepts is the key to disclose the cognitive structures underlying granular information systems and to abstract or highlight their hidden meaning. We first interrogate on the epistemological foundations of granular computing and address its hard core: the representation formalism. As far as we are concerning with representing fuzzy granules, the accent is shifted from norms and metrics that equip numeric spaces to those equipping function spaces. The Hilbert space of square-integrable functions is the most suitable choice when specific goals are assumed, such as deriving second order statistics (variance or covariance) for random fuzzy-valued variables, or using the projection theorem for solving minimum norm problems. We interrogate on the foundations of univariate and multivariate granular statistics, fuzzy extension of principal component analysis, granular clustering, granular graphical models, granular least squares estimates and discuss some related approaches.