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Fig. 3.4 Five different hierarchical clustering techniques in the form of dendrograms. a Single
linkage, b Complete method, c Average method, d Centroid method, e Ward
'
s method
3.4.1.1 Single Linkage Clustering
Single linkage is also known as the nearest neighbor method, and is one of the most
straightforward hierarchical clustering techniques. The distance between a pair of
groups is calculated by using the closest pair of elements where one element comes
from each group under consideration as shown in Fig. 3.4 a. In this type of algo-
rithm the linkage function specifying the distance between two clusters is computed
as the minimal object-to-object distance. Assume the clusters are [X, Y] and the
objects (x i ,y j ), then the linkage function can be expressed mathematically as
D min X
ð
;
Y
Þ ¼
x 2 X ; y 2 Y ds x i ;
min
y j
ð 3 : 41 Þ
where ds(x i ,y j ) is the distance between the objects x i and y j , and D min X
ð
;
Y
Þ
is the
minimum object to object linkage distance.
At each step of the hierarchical agglomerative procedure, the two clusters whose
closest members have the smallest distance are merged [ 9 , 25 , 40 , 68 ]. A property
known as chaining is often associated with the single linkage method (although it
can occur with other methods) and is usually considered as a defect, since no
distinct clusters are produced. Chaining occurs when a cluster occurring at low-
level linking increases by the progressive addition of individual elements not
already in a cluster. An example of this can be seen in Fig. 3.4 a.
3.4.1.2 Complete Linkage Clustering
Complete linkage clustering is also known as the farthest neighbor method because
of its mathematical consideration used for calculation of linkage function. It is
based on the distance between the furthest pairs of elements, one from each cluster;
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