Fuzzy c-means in high dimensional spaces

Abstract

High dimensions seem to have a devastating effect on the FCM algorithm and alike. One of these effects is that the prototypes tend to run into the centre of gravity of the entire data set. Obviously, the objective function must have a local minimum in the centre of gravity that causes FCM to its ill behaviour. In this paper, we concentrate on this problem rather than on new algorithms. We answer the following questions: How many dimensions are at least necessary to cause an ill behaviour of FCM? How does the number of prototypes influence the ill behaviour? Why has the objective function a local minimum in the centre of gravity? How must FCM be initialised to avoid the local minima in the centre of gravity? To understand the ill behaviour of the FCM algorithm and to answer the above questions, we take a closer look at the values of the objective function. We develop three test environments that consist of artificially generated data sets to provide a controlled environment. Finally, we conclude that FCM can only be applied successfully in high dimensions, if the prototypes are initialized very close to the cluster centres.