A model of granular data: a design problem with the Tchebyschev FCM

Abstract

In this study, we propose a model of granular data emerging through a summarization and processing of numeric data. This model supports data analysis and contributes to further interpretation activities. The structure of data is revealed through the FCM equipped with the Tchebyschev (l_∞) metric. The paper offers a novel contribution of a gradient-based learning of the prototypes developed in the l_∞-based FCM. The l_∞ metric promotes development of easily interpretable information granules, namely hyperboxes. A detailed discussion of their geometry is provided. In particular, we discuss a deformation effect of the hyperbox-shape of granules due to an interaction between the granules. It is shown how the deformation effect can be quantified. Subsequently, we show how the clustering gives rise to a two-level topology of information granules: the core part of the topology comes in the form of hyperbox information granules. A residual structure is expressed through detailed, yet difficult to interpret, membership grades. Illustrative examples including synthetic data are studied.