Ultrametric inter-object distances found by hierarchical clustering may be used to generate an appropriate constellation of points in a plane. A configuration of points which reproduce the ultrametric distances in a low-dimensional space as closely as possible can be found by Torgerson's method of multidimensional scaling. The resulting display supports the cognition of clusters very well, and for this purpose it seems to be superior to, for instance, the frequently applied principal components display. In certain situations dedrograms are expected to become simpler to interpret when such complementary 'cluster display' is also consulted. An example from analytical chemistry is presented.
KEY WORDS
Cluster analysis Display Interlaboratory test Ultrametrics
Dendrograms are the traditional output of hierarchical clustering algorithms. They allow identification of classes of similar objects and arrangement of these classes in a hierarchical way according to their similarity. Whilst the structure of a dendrogram may sometimes be rather complicated, it is well known that low-dimensional configurations of points (displays) are readily interpretable by human inspection.
A large variety of display methods are available in multivariate analysis: principal components plot, non-linear mapping, display of linear discriminant analysis, etc. Hierarchical clustering may also be regarded as a display method. Using ultrametric distances from a dendrogram rather than the original inter-object distances, a display is obtained which may be expected to reveal sharper structures among the objects. In simple terms, these displays show a constellation of objects as suggested by the clustering algorithm applied.
The way in which distances can be derived from a dendrogram is shown in Figure 1. The ultrametric distance d t corresponds to the lowest level in the dendrogram at which objects i and j belong to the same cluster. The djj are called ultrametric because the inequality djj < max( d y k , dyk) holds for each triplet (i, j , k ) of objects. Given a dendrograrnor a matrix DU of ultrametric distances (for a verification of this equivalence see Reference 1)it is possible to construct a display by means of Torgerson's method' of multidimensional scaling.