Cluster Analysis

Development of Cluster Analysis

Prior to 1960, many clustering problems were solved separately in different disciplines. Progress was fragmented. The early 1960s saw attempts to provide general treatments of cluster analysis, given these many developments. Sokal and Sneath (1963) provided an extensive discussion and helped set the framework for the development of cluster analysis as a data-analytic field. Specifying clustering problems is not difficult. Nor are the mathematical foundations for expressing and creating most solutions to the clustering problem. The difficulty of cluster analysis comes [Page 129]from the computational complexities in establishing solutions to the clustering problem. As a result, the field has been driven primarily by the evolution of computing technology. Generally, this has been beneficial, with substantive interpretations being enriched by useful clusterings. In addition, many technical developments have stemmed from exploring substantive applications in new domains. There are now many national societies of cluster analysts that are linked through the International Federation of Classification Societies.

Solving Clustering Problems

In general, the clustering problem can be stated as establishing one (or more) clustering(s) with r clusters that have the minimized value of a well-defined criterion function over all feasible clusterings. The criterion function provides a measure of fit for all clusterings. In practice, however, the criterion function often is left implicit or ignored. In most applications, the clustering is a partition, but “fuzzy clustering” with overlapping clusters is possible. Once the units of analysis have been selected, there are five broad steps in conducting cluster analyses:

1.
measuring the relevant variables (both QUANTITATIVE VARIABLES and CATEGORICAL VARIABLES can be included, and some form of standardization may be necessary),
2.
creating a (dis)similarity MATRIX for an appropriate measure of (dis)similarity,
3.
creating one or more clusterings via a clustering algorithm,
4.
providing some assessment of the obtained clustering(s), and
5.
interpreting the clustering(s) in substantive terms.

Although all steps are fraught with hazard, Steps 2 and 3 are the most hazardous, and Step 4 is ignored often. In Step 2, dissimilarity measures (e.g., Euclidean, Manhattan, and Minkowsky distances) or similarity measures (e.g., CORRELATION and matching COEFFICIENTS) can be used. The choice of a measure is critical: Different measures can lead to different clusterings. In Step 3, there are many ALGORITHMS for establishing clusterings. Each pair of choices (of measures and algorithms), in principle, can lead to different clusterings of the units.

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