CLUSTER_HIERARCHICAL Procedure (PV-WAVE Advantage)

dist—An npt by npt symmetric matrix, where npt is the number of data points to be clustered, containing the distance (or similarity) matrix. dist is a symmetric matrix. On input, only the upper triangular part needs to be present. CLUSTER_HIERARCHICAL saves the upper triangular part of dist in the lower triangle. On return from CLUSTER_HIERARCHICAL, the upper triangular part of dist is restored, and the matrix is made symmetric.

Clevel—Array of length npt – 1 containing the level at which the clusters are joined. Clevel(k – 1) contains the distance (or similarity) level at which cluster npt + k was formed. If the original data in dist was transformed via the Transformation keyword, the inverse transformation is applied to the values in Clevel prior to exit from CLUSTER_HIERARCHICAL.

2. A search is made of the distance matrix to find the two closest clusters. These clusters are merged to form a new cluster, numbered n + k. The cluster numbers of the two clusters joined at this stage are saved in Crson and Clson, and the distance measure between the two clusters is stored in Clevel.

The five methods differ primarily in how the distance matrix is updated after two clusters have been joined. The Method keyword specifies how the distance of the cluster just merged with each of the remaining clusters will be updated. CLUSTER_HIERARCHICAL allows five methods for computing the distances. To understand these measures, suppose in the following discussion that clusters “A” and “B” have just been joined to form cluster “Z”, and interest is in computing the distance of Z with another cluster called “C”.

Table 10-1: Method Values
Method	Method
0	Single linkage method. The distance from Z to C is the minimum of the distances (A to C, B to C).
1	Complete linkage method. The distance from Z to C is the maximum of the distances (A to C, B to C).
2	Average-distance-within-clusters method. The distance from Z to C is the average distance of all objects that would be within the cluster formed by merging clusters Z and C. This average may be computed according to formulas given by Anderberg (1973, page 139).
3	Average-distance-between-clusters method. The distance from Z to C is the average distance of objects within cluster Z to objects within cluster C. This average may be computed according to methods given by Anderberg (1973, page 140).
4	Ward’s method. Clusters are formed so as to minimize the increase in the within-cluster sums of squares. The distance between two clusters is the increase in these sums of squares if the two clusters were merged. A method for computing this distance from a squared Euclidean distance matrix is given by Anderberg (1973, pages 142–145).

In general, single linkage will yield long thin clusters while complete linkage will yield clusters that are more spherical. Average linkage and Ward’s linkage tend to yield clusters that are similar to those obtained with complete linkage.

CLUSTER_HIERARCHICAL produces a unique representation of the binary cluster tree via the following three conventions; the fact that the tree is unique should aid in interpreting the clusters. First, when two clusters are joined and each cluster contains two or more data points, the cluster that was initially formed with the smallest level (in clevel) becomes the left son. Second, when a cluster containing more than one data point is joined with a cluster containing a single data point, the cluster with the single data point becomes the right son. Finally, when two clusters containing only one object are joined, the cluster with the smallest cluster number becomes the right son.

2. Raw correlations, if used as similarities, should be made positive and transformed to a distance measure. One such transformation can be performed by specifying the Transformation keyword, with Transformation = 2 in CLUSTER_HIERARCHICAL.

3. The user may cluster either variables or observations in CLUSTER_HIERARCHICAL since a dissimilarity matrix, not the original data, is used. DISSIMILARITIES may be used to compute the matrix dist for either the variables or observations.

In the following example, the average distance within clusters method is used to perform a hierarchical cluster analysis of the Fisher iris data. The STATDATA function is first used to obtain the Fisher iris data. The example is typical in that after the program obtains the data, function DISSIMILARITIES computes the distance matrix (dist) prior to calling CLUSTER_HIERARCHICAL.