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In cluster analysis, selecting the number of clusters is an "ill-posed" problem of crucial importance. In this paper we propose a re-sampling method for assessing cluster stability. Our model suggests that samples' occurrences in clusters can be considered as realizations of the same random variable in the case of the "true" number of clusters. Thus, similarity between different cluster solutions is measured by means of compound and simple probability metrics. Compound criteria result in validation rules employing the stability content of clusters. Simple probability metrics, in particular those based on kernels, provide more flexible geometrical criteria. We analyze several applications of probability metrics Editors: Süreyya Özögür-Akyüz, Devrim Ünay, and Alex Smola. Learn (2011) 85:209-248 combined with methods intended to simulate cluster occurrences. Numerical experiments are provided to demonstrate and compare the different metrics and simulation approaches.

K-Nearest Neighbors is a widely used technique for classifying and clustering data. In the current article, we address the cluster stability problem based upon probabilistic characteristics of this approach. We estimate the stability of partitions obtained from clustering pairs of samples. Partitions are presumed to be consistent if their clusters are stable. Clusters validity is quantified through the amount of K-Nearest Neighbors belonging to the point's sample. The null-hypothesis, of the well-mixed samples within the clusters, suggests Binomial Distribution of this quantity with K trials and the success probability 0 5. A cluster is represented by a summarizing index, of the p-values calculated over all cluster objects, under the null hypothesis for the alternative, and the partition quality is evaluated via the worst partition cluster. The true number of clusters is attained by the empirical index distribution having maximal suitable asymmetry. The proposed methodology offers to produce the index distributions sequentially and to assess their asymmetry. Numerical experiments exhibit a good capability of the methodology to expose the true number of clusters.

The paper is, for the most part, devoted to a survey of the analytical properties of generalized convolution algebras and their realizations. This issue appears to be the state of the art until now because intensive research on the generalized convolution and the related models still persists.

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