Visualization

Visualization of high-dimensional data clusters can be largely divided into four popular approaches:

1. Dimensionality reduction by selection of 2 or 3 dimensions, or, more generally, projecting the data down to 2 or 3 dimensions. Often these dimensions correspond to principal components or a scalable approximation thereof (e.g., FASTMAP [FL95]). Chen, for example, creates a browsable 2-dimensional space of authors through co-citations [Che99]. Another noteworthy method is CViz [DMS98], which projects onto the plane that passes through three selected cluster centroids to yield a `discrimination optimal' 2-dimensional projection. These projections are useful for a medium number of dimensions, i.e., if $d$ is not too large ($<$ 100).^2.3Nonlinear projections have also been studied [CG01]. Recreating a 2- or 3-dimensional space from a similarity graph can also be done through multi-dimensional scaling [Tor52].
2. Parallel axis plots show each object as a line along $d$ parallel axis. However, this technique is rendered ineffective if the number of dimensions $d$ or the number of objects gets too high.
3. Kohonen's Self Organizing Map (SOM) [Koh90] provides an innovative and powerful way of clustering while enforcing constraints on a logical topology imposed on the cluster centers. If this topology is 2-dimensional, one can readily "visualize" the clustering of data. Essentially a 2-dimensional manifold is mapped onto the (typically higher dimensional) feature space, trying to approximate data density while maintaining topological constraints. Since the mapping is not bijective, the quality can degrade very rapidly with increasing dimensionality of feature space, unless the data is largely confined to a much lower order manifold within this space [CG01]. Multi-Dimensional Scaling (MDS) and associated methods also face similar issues.
4. Visualization can also be done by showing the data matrix as an image by converting entries to brightness values. The ordering of data points for visualization has previously been used in conjunction with clustering in different contexts. For example, in OPTICS [ABKS99] instead of producing an explicit clustering, an augmented ordering of the database is produced. Subsequently, this ordering is used to display various metrics such as reachability values. In cluster analysis of genome data [ESBB98] re-ordering the primary data matrix and representing it graphically has been explored. This visualization takes place in the primary data space rather than in the relationship-space. Sparse primary data matrix reorderings have also been considered for browsing hypertext [BHR96].

A useful survey of visualization methods for data mining in general can be found in [KK96]. The popular books by E. Tufte [Tuf83] on visualizing information are also recommended.