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Many clustering algorithms require globular (gaussian) clusters. Main example is k-means. If clusters are not globular, these methods can bring wrong results.

In social sciences, data is often clustered by these algorithms (market segmentation is very common example). My question then is whether there are some reasons why we should regard clusters from social surveys as globular (and therefore appropriate for methods like k-means).

An example of social data that could be clustered is mentioned in my previous question. I asked my students to rate selected interest (1-5 scale) and the task then could be to cluster students according to similar interests.

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  • $\begingroup$ You may be disappointed, but one of the main reasons for k-means' popularity is its simplicity, not necessarily its theoretical justifications. k-means is known to have several important flaws. $\endgroup$ – Marc Claesen Aug 8 '13 at 16:14
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    $\begingroup$ Many clustering methods indeed prefer to create spherical clusters, but far not all. Clusters in social sciences are usually assumed to be ellipsoid, not necessarily spherical. (In other domains, such as image recognition for instance, clusters often are curved.) The issue of shape, however, implies that the data exist in euclidean or similar metrical space. In social sciences we often deal with categorical data for which such space is inappropriate, so we won't speak of shape. $\endgroup$ – ttnphns Aug 8 '13 at 16:15
  • $\begingroup$ @ttnphns, categorical data are also used in social sciences, but in this question, I speak only about numerical data. For example, when factors are created from likert items and then data is clustered. $\endgroup$ – Miroslav Sabo Aug 8 '13 at 16:19
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    $\begingroup$ It is not necessarily the case that Likert scaled items are Euclidean; change the coding from 1-2-3-4-5 to 1-4-5-6-7 and you change the distance, and that's OK if they are only ordinal. $\endgroup$ – Peter Flom Aug 8 '13 at 17:07
  • $\begingroup$ With likert-type items much depends on how subjects perceive the task. If they feel the survey is "normative" or "formal" they are likely to narrow the range they use on the scale, so if they think in terms of prototypes on some items and not other items there will be spherical clusters in the data. But if they feel the survey is "personal" and asks their subjective expertise they will use all the range of the scale for those items where they don't think in terms of prototypes, and the clusters will be oblong. $\endgroup$ – ttnphns Aug 8 '13 at 17:19
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Technically, the clusters in k-means are not globular, but Voronoi cells. It's repeated over and over in books; and indeed there are good reasons why k-means works best when the clusters are spherical and of the same extend.

But: does the clustering actually need to be perfect?

In many use cases, there is no such thing as a "correct" clustering. Most often anything better than random will do.

k-means is then very convenient, because it not only partitions your data into a guaranteed low number of clusters (i.e. k) much better than random. But also these clusters are convex (i.e. of relatively simple shape), exhaustive (all objects are clustered, not "outliers" remain), don't overlap (more advanced techniques may produce overlapping clusters or hierarchies of clusters) and - which probably is most important to many users - are represented by a simple, data-like representative: the cluster mean.

So if you just want to have a heuristic to reduce your data set to k representative objects, k-means is a really good heuristic.

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