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Statistical theory in clustering. (English) Zbl 0575.62058
A number of statistical models for forming and evaluating clusters are reviewed. Hierarchical algorithms are evaluated by their ability to discover high density regions in a population, and complete linkage hopelessly fails; the others don’t do too well either. Single linkage is at least of mathematical interest because it is related to the minimum spanning tree and percolation.
Mixture methods are examined, related to k-means, and the failure of likelihood tests for the number of components is noted. The DIP test for estimating the number of modes in a univariate population measures the distance between the empirical distribution function and the closest unimodal distribution function (or k-modal distribution function when testing for k modes). Its properties are examined and multivariate extensions are proposed. Ultrametric and evolutionary distances on trees are considered briefly.

MSC:
62H30 Classification and discrimination; cluster analysis (statistical aspects)
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