  Cluster analysis is used to explore structure in unlabeled data sets in a
wide range of applications. An important part of cluster analysis is validating
the quality of computationally obtained clusters. A large number of different
internal indices have been developed for validation in the offline setting.
However, this concept has not been extended to the online setting. A key
challenge is to find an efficient incremental formulation of an index that can
capture both cohesion and separation of the clusters over potentially infinite
data streams. In this paper, we develop two online versions (with and without
forgetting factors) of the Xie-Beni and Davies-Bouldin internal validity
indices, and analyze their characteristics, using two streaming clustering
algorithms (sk-means and online ellipsoidal clustering), and illustrate their
use in monitoring evolving clusters in streaming data. We also show that
incremental cluster validity indices are capable of sending a distress signal
to online monitors when evolving clusters go awry. Our numerical examples
indicate that the incremental Xie-Beni index with forgetting factor is superior
to the other three indices tested.
