Extracting cover sets from free fuzzy sorting data

Assignment of items to multiple categories requires suitable statistical methods. The present paper provides a new approach to solve this task. The concept of fuzzy sets is extended to cover sets (sets of overlapping clusters) in a simple manner introducing a vector of item membership sums. The application of the new concept is exemplified by modifying the fuzzy cluster analysis algorithm of Kaufman and Rousseeuw (Finding groups in data: an introduction to cluster analysis, 1990) to cover set cluster analysis appropriately. Wide equivalence of the numerical problems is demonstrated from Lagrange multipliers and Karush-Kuhn-Tucker conditions. Additionally, some extensions are introduced to the algorithm to improve its behavior for suboptimal large or small numbers of clusters. The adapted algorithm in most cases reproduces single sortings for correct numbers of clusters. Two applications to empirical free fuzzy sorting data sets are provided. Limitations of the algorithm are discussed.