NUMERICAL CONVERGENCE AND INTERPRETATION OF THE FUZZY-C-SHELLS CLUSTERING-ALGORITHM

Dave's version of fuzzy c-shells is an iterative clustering algorithm which requires the application of Newton's method or a similar general optimization technique at each half step in any sequence of iterates for minimizing the associated objective function. An important computational question concerns the accuracy of the solution required at each half step within the overall iteration. This note applies the general convergence theory for grouped coordinate minimization to this question to show that numerically exact solution of the half-step subproblems in Dave's algorithm is not necessary. We show that one iteration of Newton's method in each coordinate minimization half step yields a sequence of iterates with the same local convergence properties as sequences obtained using the fuzzy c-shells algorithm with numerically exact coordinate minimization at each half step. Finally, we show that fuzzy c-shells generates hyperspherical prototypes to the clusters it finds for certain special cases of the measure of dissimilarity used.