CONVEX DECOMPOSITIONS OF FUZZY PARTITIONS

In this paper we investigate some algebraic and geometric properties of fuzzy partition spaces (convex hulls of hard or conventional partition spaces). In particular, we obtain their dimensions, and describe a number of algorithms for effecting convex decompositions. Two of these are easily programmable, and each affords a different insight about data structures suggested by the fuzzy partition decomposed. We also show how the sequence of partitions in any convex decomposition leads to a matrix for which the norm of the corresponding coefficient vector equals a scalar measure of partition fuzziness used with certain fuzzy clustering algorithms. © 1979.