Performs a K‑means (centroid) cluster analysis.

Routine KMEAN is an implementation of Algorithm AS 136 by Hartigan and Wong (1979). It computes K‑means (centroid) Euclidean metric clusters for an input matrix starting with initial estimates of the K cluster means. Routine KMEAN allows for missing values (coded as NaN, “not a number”) and for weights and frequencies.

Let p = NVAR denote the number of variables to be used in computing the Euclidean distance between observations. The idea in K‑means cluster analysis is to find a clustering (or grouping) of the observations so as to minimize the total within‑cluster sums of squares. In this case, the total sums of squares within each cluster is computed as the sum of the centered sum of squares over all nonmissing values of each variable. That is,

where νim denotes the row index of the m‑th observation in the i‑th cluster in the matrix X; ni is the number of rows of X assigned to group i; f denotes the frequency of the observation; w denotes its weight; δ is zero if the j‑th variable on observation νim is missing, otherwise δ is one; and

is the average of the nonmissing observations for variable j in group i. This method sequentially processes each observation and reassigns it to another cluster if doing so results in a decrease in the total within‑cluster sums of squares. The user in referred to Hartigan and Wong (1979) or Hartigan (1975) for the details.

This example performs K‑means cluster analysis on Fisher’s iris data, which is first obtained via routine GDATA (see Chapter 19, “Utilities”). The initial cluster seed for each iris type is an observation known to be in the iris type.