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The topic of this book is the domain of cluster analysis. Two fundamental problems in this domain are studied. First, because cluster analysis is an unsupervised process, it is impossible to define a criterion, that is to be optimized in order to find the best clustering, in terms of deviations between given inputs and given desired outputs. Rather, one has to define such criterions, called cluster validation measures, in terms of desired, intuitive characteristics of a clustering. However, a general framework for such measures does not exist. Thus our first problem: can we define axioms for cluster validation measures? Another fundamental problem is related to the robustness of clustering. The cluster analysis task is typically performed by a clustering algorithm that has some parameters to be determined by the user, e.g. the number of clusters. As cluster analysis is an unsupervised process, we should not expect the user to know the most suitable values for these parameters. However, in practice the resulting clustering is strongly influenced by the choice of these parameter values. Thus our second problem: can we define a suitable measure for the robustness of a clustering?