We know that the partition coefficient index has the monotonic tendency with cluster number c. Moreover, they always select the smallest cluster number c=2 as a optimal cluster number estimate when data contains some noise points. In this paper, we will discuss this problem by defining the validity measure of each single data point. We then define the singular point that has equal memberships to each cluster. By analyzing the influence of the singular point on the validity index, we can then give some guidelines for designing the fuzzy c-partitions based validity indexes that can avoid the influence of the noise.
Fuzzy Systems, 2008. FUZZ-IEEE 2008. (IEEE World Congress on Computational Intelligence). IEEE International Conference on, pp.372-376