New data induced metric for density based clustering

Cluster Analysis is a fundamental data exploration technique in which there exist a variety of algorithms arising in many different areas of the literature. One of the most popular clustering algorithms is fuzzy c-means which minimises the weighted within-group sum of squares functional. However, this algorithm has a major drawback in that it will only find clusters of the same simple convex shape. In reality datasets can contain clusters of a variety shapes, including complex non-convex structures the geometry of which cannot be analytically described. This paper proposes a new data induced metric to be used in the fuzzy c-means functional that results in an algorithm able to find clusters of unknown arbitrary shapes. The proposed metric is based on density information obtained using the Delaunay triangulation. The resulting optimisation problem is now non-smooth, and the alternating optimisation technique used in fuzzy c-means can no longer be applied. Instead we have made use of a recent method of non-smooth optimisation to solve the problem.