Fuzzy quantifiers for data summarization and their role in granular computing

Data summarization is an enabling technique of granular computing, as it is able to abstract from individual observations and to view a phenomenon as a whole. The linguistic summaries are built around a fuzzy quantifier which functions as the 'summarizer'. Linguistic data summarization therefore presupposes an underlying model of fuzzy quantifiers, which is of crucial importance to the adequacy of the generated summaries. In the paper, we present an axiomatic theory of fuzzy quantification. It attempts to formalize the notion of 'linguistic adequacy', in order to eliminate the implausible results observed with existing approaches. We provide evidence that the models of the theory are plausible from linguistic considerations. Finally, we present three practical models and discuss some of their properties. These models are computational, and systems for data summarization can directly profit from our improvements by plugging in the new algorithms.