![](bg3.png)
u⊲x,
Q
P
i
⊳ D 1−
d
mean
⊲x,
Q
P
i
⊳
D
max
(5)
Next, the objects are affected to one or more
poles. In a single-affectation approach, each object
x is assigned to the pole maximising the membership
u⊲x,
Q
P
j
⊳. In a multi-affectation approach, the object is
affected to the poles whose memberships are greater
than some reference values given by a linear approx-
imation on the set of object memberships, sorted in
decreasing order.
3 HIERARCHICAL CLUSTERING
Classical approaches for hierarchical clustering ob-
tain the cluster solution by iterative mergings or di-
visions of clusters (Everitt, 1974; Kaufmann and
Rousseeuw, 1990). Two major hierarchical ap-
proaches can be distinguished: agglomerative and di-
visive.
Hierarchical Agglomerative Approaches. Ag-
glomerative algorithms are the so-called bottom-up
approaches, starting with all points as individual
clusters and successively merging the closest pair
of clusters until all patterns are enclosed in a single
cluster. The algorithms can be visualised using a
graphical tree structure called dendogram where
the pair of clusters that are merged at each iteration
can be observed. The final cluster solution is se-
lected by the user, by specifying a level to cut the
dendogram or, equivalently, a desired number k of
clusters. Different agglomerative approaches can be
distinguished, depending on the proximity criterion
to merge the next pair of clusters. For example,
while the single linkage algorithm selects the pair of
clusters with the minimum distance between their
closest elements, the complete linkage algorithm
selects the clusters with minimum distance between
the farthest objects. In a similar way, the average
linkage and centroid algorithms choose the clusters
with the minimum average inter cluster distance and
the minimum distance between their centroid objects,
respectively.
Hierarchical Divisive Approaches. As opposed to
agglomerative algorithms, a divisive approach, such
as the divisive analysis (DiANA) algorithm, starts at
the top dendogram level where all objects compound
a unique cluster and iteratively splits the biggest clus-
ter until each object is in its own cluster. The reader is
referred to (Everitt, 1974) for more details about the
divisive analysis algorithm.
4 NEW HIERARCHICAL
POLE-BASED APPROACH
The new clustering method is combination of the
PoBOC algorithm and hierarchical divisive clustering
strategies. In a divisive manner, the proposed hier-
archical approach is initialised with the set of poles
identified by the PoBOC algorithm, and recursively
applied to each obtained pole, searching for possible
subclusters.
4.1 Pole-based Clustering Basis Module
In order to detect the set of poles in the new hierarchi-
cal approach, the graph construction, pole construc-
tion and pole restriction stages of POBOC have been
preserved, but the affectation step has been replaced
by a new procedure called pole regrowth:
Algorithm 2. Pole regrowth (
Q
P, R, D).
Input: sets of poles and residual from the pole-reduction
step:
Q
P, R; dissimilarity matrix D
Output: set of regrown poles
O
P
Initialise:
O
P D
Q
P
while R 6D
/
0 do
Find the pair (x
i
∈ R,
O
P
j
∈
O
P) with minimum distance:
⊲x
i
,
O
P
j
⊳ D argmin
x∈R,
O
P∈
O
P
D
min
⊲x,
O
P⊳,
with D
min
⊲x
i
,
O
P
j
⊳ D min
x
k
∈
O
P
j
D
ik
Attach the point x
i
to its closest pole and remove it
from the residual set:
O
P
j
D
O
P
j
∪ x
i
R D R − x
i
end while
Return
O
P
The pole-regrowth procedure is an alternative to
the PoBOC single affectation for reallocating overlap-
ping objects into one of the restricted poles. As it can
be observed in Figure 1(a), not only a pole but also an
overlapping region may contain potential subclusters.
If each overlapping object x
i
is individually assigned
to the pole maximising the membership u⊲x
i
I
Q
P⊳, the
objects inside a single cluster might be assigned to
different poles
1
. The pole regrowth procedure is in-
tended to avoid any undesired partitioning of clusters
existing in overlapping areas while reallocating resid-
ual objects.
An example of the pole regrowth method is shown
in Figure 1. Figure 1(a) shows two restricted poles
1
Note that the hierarchical approach is independently
applied to the grown poles.
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