Necessary conditions for the confidence level of the randomized algorithm of finding the true number of clusters

One of the most difficult problems in cluster analysis is the identification of the number of groups in a given data set. In this paper we offer the approach in the framework of the common “elbow” methodology such that the true number of clusters is recognized as the slope discontinuity of the index function. A randomized algorithm has been suggested to allocate this position. The scenario approach is used to significantly reduce the computational complexity. We present weaker necessary conditions to provide a priori chosen level of confidence. In addition, we present a number of simulation examples of unknown huge number of groups clustering to demonstrate theoretical results. Finally, we note that necessary conditions can be relaxed more and ideas considered potentially can be extended to a wide range of real-time decision-making problems in control systems.