Approximate Confidence Intervals for the Number of Clusters

Abstract

We consider clustering for the purpose of data reduction. Similar objects are grouped together in clusters so that one can then work with the few cluster descriptors instead of the many data points. The quality of any given clustering is measured by a loss function that takes into account both the parsimony of the clustering and the loss of information due to clustering. An optimal clustering can be obtained by minimizing the theoretical loss function. It is shown that a sample version of the loss function and optimal clustering converge strongly to their theoretical counterparts as the sample size tends to infinity. We then develop a bootstrap-based procedure for obtaining approximate confidence bounds on the number of clusters in the “best” clustering. The effectiveness of this procedure is evaluated in a simulation study. An application is presented.

Key Words:

• Bootstrap confidence interval
• Cluster analysis
• K-means clusterings
• Simulation study
• Strong consistency