Clustering in Low-Dimensional Space

It is often asserted that clustering techniques and multidimensional scaling (MDS) have mutually exclusive roles in the analysis of a given set of data, the former being especially indicated when the configuration of points is high-dimensional and has local regions of high density, while the latter would be more useful when the points are low-dimensional and more evenly distributed. This view has been challenged by, amongst others, Critchley and Heiser (1988). They showed how a set of objects with ultrametric distances can be mapped as a series of points on a line. Taking one step further, it may be argued that clustering is a way to stabilize and robustify the multidimensional scaling task, the aim being to fit a low-dimensional distance model to groups of points, rather than to single points. Thus finding clusters is not an end in itself, but subordinate to some other task. This idea is discussed for the unfolding situation, in which we have to deal with single-peaked variables defined over a common set of points, and for the general MDS situation. A convergent least squares algorithm is described and illustrated. The approach leads to a useful decomposition of the badness-of-fit function into between and within components. For a fixed partitioning of the points a least squares version of Gower’s (1989) canonical distance analysis is obtained.