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MULTIVARIATE CLUSTER ANALYSIS AND DISCUSSION 249

First object in pair (SIC code group)

1 2 3 … n
Second object in pair (SIC code group) 2 d 21 d – – – –
1
–
–
–
–
–
d
3
–
–
–
31
32
–
–
n d n1 d n2 d n3 … – Figure 15.7 Format of cluster
analysis resemblance matrix.
row contains the value of the resemblance coefficient for the given pair of objects
(Romesburg, 1984). For these reasons, only the lower half of the resemblance matrix
contains values. Figure 15.7 provides an example.

15.4.4 STEP 4: EXECUTE THE CLUSTER METHOD

The k-means method, a nonhierarchical clustering procedure was applied to group the
SIC code groups. The k-means method applies an iterative process to find the optimal
clustering for a specified number of groups (k). The k-means clustering method splits
a set of objects into a selected number of groups by maximizing between-cluster vari-
ation (SSA) relative to within-cluster variation (SSE). The following calculations are
used to calculate the sums of squares; the nomenclature of the cluster matrix is shown
below as well:

WASTE GROUPS (CLUSTERS)

Attribute 1 1 2 .... k

y 11 y 21 y k1
y 12 y 22 y k2

y 1n y 2n y kn

n
k
SST = ∑ ∑ ( y − y ) 2 = total sum of squares
..
ij
i=1 1 j=1
k
SSA = n ∑ ( y − y ) 2 = within cluster sum of squares
s
.
.
.
i
i=1
n
k
SSE = ∑ ∑ (y − y . ) = between cluster sum of squares
2
r
ij
i
i=1 j=1
+
SST = SSA SSE