A Generalization of Gustafson-Kessel Algorithm Using a New Constraint Parameter

In this paper one presents a new fuzzy clustering algorithm based on a dissimilarity function determined by three parameters. This algorithm can be considered a generalization of the Gustafson-Kessel algorithm for fuzzy clustering.

💡 Research Summary

The paper addresses a well‑known limitation of the Gustafson‑Kessel (GK) fuzzy clustering algorithm: its inability to handle clusters with markedly different volumes because the algorithm forces a constant volume through fixed λ parameters. To overcome this, the author proposes a generalized version of GK that introduces three additional parameters into the dissimilarity measure, thereby allowing each cluster to adapt its own scale (volume) and density.

The new algorithm models each cluster j by four quantities: a centroid (m_j), a fuzzy covariance matrix (C_j), a volume‑related scalar (V_j), and a density‑related scalar (\rho_j). The scalar (V_j) is defined to be close to the cardinality of the cluster (i.e., the effective number of points belonging to it) and thus serves as a direct measure of cluster volume. The density measure (\rho_j) is defined as the ratio (n_j/V_j), where (n_j) is the fuzzy cardinality of the cluster.

The core of the method is a new dissimilarity function
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