Characteristics of Methods for Clustering Observations

Many simulation studies comparing various methods of cluster analysis have been performed. In these studies, artificial data sets containing known clusters are produced using pseudo-random-number generators. The data sets are analyzed by a variety of clustering methods, and the degree to which each clustering method recovers the known cluster structure is evaluated. See Milligan (1981) for a review of such studies. In most of these studies, the clustering method with the best overall performance has been either average linkage or Ward’s minimum variance method. The method with the poorest overall performance has almost invariably been single linkage. However, in many respects, the results of simulation studies are inconsistent and confusing.

When you attempt to evaluate clustering methods, it is essential to realize that most methods are biased toward finding clusters possessing certain characteristics related to size (number of members), shape, or dispersion. Methods based on the least squares criterion (Sarle; 1982), such as k-means and Ward’s minimum variance method, tend to find clusters with roughly the same number of observations in each cluster. Average linkage is somewhat biased toward finding clusters of equal variance. Many clustering methods tend to produce compact, roughly hyperspherical clusters and are incapable of detecting clusters with highly elongated or irregular shapes. The methods with the least bias are those based on nonparametric density estimation such as single linkage and density linkage.

Most simulation studies have generated compact (often multivariate normal) clusters of roughly equal size or dispersion. Such studies naturally favor average linkage and Ward’s method over most other hierarchical methods, especially single linkage. It would be easy, however, to design a study that uses elongated or irregular clusters in which single linkage would perform much better than average linkage or Ward’s method (see some of the following examples). Even studies that compare clustering methods that use "realistic" data might unfairly favor particular methods. For example, in all the data sets used by Mezzich and Solomon (1980), the clusters established by field experts are of equal size. When interpreting simulation or other comparative studies, you must, therefore, decide whether the artificially generated clusters in the study resemble the clusters you suspect might exist in your data in terms of size, shape, and dispersion. If, like many people doing exploratory cluster analysis, you have no idea what kinds of clusters to expect, you should include at least one of the relatively unbiased methods, such as density linkage, in your analysis.

The rest of this section consists of a series of examples that illustrate the performance of various clustering methods under various conditions. The first, and simplest, example shows a case of well-separated clusters. The other examples show cases of poorly separated clusters, clusters of unequal size, parallel elongated clusters, and nonconvex clusters.