Modified Objective Function Algorithms

Abstract

In this chapter we discuss three modifications of previous algorithms which have been proposed in an attempt to compensate for the difficulties caused by variations in cluster shape. The basic dilemma is that “clusters” defined by criterion functions usually take mathematical substance via metrical distances in data space. Each metric induces its own unseen but quite pervasive topological structure on ℝ^p due to the geometric shape of the open balls it defines. This often forces the criterion function employing d to unwittingly favor clusters in X having this basic shape—even when none are present! In S21, we discuss a novel approach due to Backer which “inverts” several previous strategies. S22 considers an interesting modification of the FCM functional J _m due to Gustafson and Kessel,⁽⁵⁴⁾ which uses a different norm for each cluster! S23 and S24 discuss generalization of the fuzzy c-means algorithms (A11.1) in a different way—the prototypes v _i for J _m (U, v) become r-dimensional linear varieties in ℝ^p, 0 ≤ r ≤ p − 1.