Digital Signal Processing Reference
In-Depth Information
(a) (b)
Figure 7.2
Different cluster shapes: (a) compact, and (b) spherical.
7.3
Fuzzy Clustering Concepts
Clustering partitions a data set in groups of similar pattern, each group
having a representant that is characteristic of the considered feature
class. Within each group or cluster, patterns have the largest similarity
to each other. In pattern recognition, we distinguish between crisp
and fuzzy clustering. Fuzzy clustering has a major advantage in real-
world application where the belonging of a pattern to a certain class
is ambiguous. To obtain such a fuzzy partitioning, the membership
function is allowed to have elements with values between 0 and 1, as
shown in the previous section, In other words, in fuzzy clustering a
pattern belongs
simultaneously
to more than one cluster, with the degree
of belonging specified by membership grades between 0 and 1, whereas
in traditional statistical approaches it belongs
exclusively
to only one
cluster.
Clustering is based on minimizing a cost or objective function
J
of dissimilarity (or distance) measure. This predefined measure
J
is a
function of the input data and of an unknown parameter vector set
L
.
The number of clusters
n
is assumed in the following to be predefined
and fixed. Algorithms with growing or pruning cluster numbers and
geometries are more sophisticated and are described in [264].
An optimal clustering is achieved by determining the parameter
L
such that the cluster structure of the input data is as captured as well
as possible. It is plausible that this parameter depends on the type of
geometry of the cluster: compact or spherical as visualized in figure 7.2.
While compact clusters can be accurately described by a set of
n
points
L
i
∈
L
representing these clusters, spherical clusters are described
by the centers of the cluster
V
and by the radii
R
of the clusters.
In the following, we will review the most important fuzzy clustering
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