S.V. Dronov, N.N. Strizhov
On the Cluster Variable Quantification Consistent with the Results of Post-Hoc Analysis
Let us consider a task of studying some objects defined by a certain collection of their numerical (nominal) characteristics. These characteristics are said to be forming. We assume that the objects in view were already divided into clusters and the forming characteristics were ordered by the degree of their influence on the existing cluster structure of the objects. It means that so-called post-hoc problem for the cluster analysis is solved. We deal with a quantification problem for the cluster variable. Generally speaking, this variable is nonnumeric and represents some symbol for the cluster to which the object belongs. Our purpose is the assigning of some numerical label to each cluster in such a way that the labels assigned would be coordinated with the post-hoc ordering of the forming characteristics in the best possible way. In contrast to the existing methods of solving such a problem, the assignment of the labels is organized in a direct way, no iterative procedures of any kind are used. Exact formulae for the best possible labels are obtained and theoretically confirmed. We discuss the differences between the proposed and preexisting methods. Some recommendations for the transfer of the resulting labels to integer ones are provided. Suggestions have been made on the possible use of the results produced by the quantification, including problems in evidence-based medicine. An example of such processing of the real data of medical examination is included and discussed as well.
Key words: quantification, cluster partition, posthoc analysis of variables
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