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Abstract: In this study, we develop a concept of dynamic data granulation realized in presence of incoming data organized in the form of so-called data snapshots. For each of these snapshots we reveal a structure by running fuzzy clustering. The proposed algorithm of adjustable fuzzy C-means (FCM) exhibits a number of useful features which directly associate with the dynamic nature of the underlying data: (a) the number of clusters is adjusted from one data snapshot to another in order to capture the varying structure of patterns and its complexity, (b) continuity between the consecutively discovered structures is retained, viz the clusters formed for a certain data snapshot are constructed as a result of evolving the clusters discovered in the predeceasing snapshot. We present a detailed clustering algorithm in which the mechanisms of adjustment of information granularity (the number of clusters) become the result of solutions to well-defined optimization tasks. The cluster splitting is guided by conditional fuzzy C-means (FCM) while cluster merging involves two neighboring prototypes. The criterion used to control the level of information granularity throughout the process is guided by a reconstruction criterion which quantifies an error resulting from pattern granulation and de-granulation. Numeric experiments provide a suitable illustration of the approach. [Copyright &y& Elsevier]

Abstract: Given the rapidly growing diversity of techniques and applications of fuzzy clustering, an interpretation of grouping results becomes of paramount relevance. Fuzzy clusters offer a lot of detailed information about the structure in data by allocating patterns to clusters with numeric degrees of membership. While this information could be highly beneficial, its level of detail could be too overwhelming and in some sense somewhat detrimental to the formation of the global view of the structure. 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). The algorithm of converting membership functions into this quantification is a result of a certain optimization problem guided by the principle of uncertainty localization. With the shadowed sets of clusters in place, discussed is a taxonomy of patterns leading to the three-valued quantification of data structure that consists of core, shadowed, and uncertain structure. [Copyright &y& Elsevier]