Convergence and Consistency of Fuzzy c-means/ISODATA Algorithms

The fuzzy c-means/ISODATA algorithm is usually described in terms of clustering a finite data set. An equivalent point of view is that the algorithm clusters the support points of a finite-support probability distribution. Motivated by recent work on the hard version of the algorithm, this paper extends the definition to arbitrary distributions and considers asymptotic properties. It is shown that fixed points of the algorithm are stationary points of the fuzzy objective functional, and vice versa. When the algorithm is iteratively applied to an initial prototype set, the sequence of prototype sets produced approaches the set of fixed points. If an unknown distribution is approximated by the empirical distribution of stationary, ergodic observations, then as the number of observations grows large, fixed points of the algorithm based on the empirical distribution approach fixed points of the algorithm based on the true distribution. Furthermore, with respect to minimizing the fuzzy objective functional, the algorithm based on the empirical distribution is asymptotically at least as good as the algorithm based on the true distribution.