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Table 2.5
The results derived by Algorithm-FSC with different
λ
levels on the modified data sets
Modified data set I
λ
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
SI
0.466
0.466
0.466
0.466
0.466
0.466
0.466
0.466
0.999
K
2
2
2
2
2
2
2
2
2
Modified data set II
λ
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
SI
0.474
0.474
0.474
0.474
0.474
0.474
0.474
0.474
0.999
K
2
2
2
2
2
2
2
2
2
levels, Algorithm 2.2 produces
three clusters, and the smallest SI values are exactly the same as 0.437. In fact, if
we take a closer look at the assigned cluster label of each IFS, then we can find
that Algorithm 2.2 recognizes the cluster structure perfectly under these
As can be seen from Table
2.4
, for most of the
λ
levels.
Clearly, by incorporating the uncertainty degree into the correlation computation
of IFSs, Algorithm 2.2 has the ability to identify all the three classes. However,
this is not the case for traditional clustering algorithms for fuzzy sets. To illustrate
this, we also exploit Algorithm-FSC on the simulated data set. As mentioned above,
Algorithm-FSC does not take into account the uncertain information. Therefore, to
make sure
λ
μ(
x
)
+
v
(
x
)
=
1 for any
x
in the simulated data set, we should modify
the data set by adding
. We produce the two modified data
sets and then exploit Algorithm-FSC on them. The results can be found in Table
2.5
(Xu et al. 2008).
As can be seen in Table
2.5
, the clustering results of Algorithm-FSC on the two
modified data sets are poor, since it cannot identify all the three classes precisely.
This further justifies the importance of the uncertain information in IFSs.
In summary, by comparing the performance of Algorithm-IFSC with that of
Algorithm-FSC on the simulated data set, we know that (1) Algorithm-IFSC is capa-
ble to cluster large scale IFSs; and (2) the uncertain information captured by IFSs is
crucial for the success of some clustering tasks.
π(
x
)
to either
v
(
x
)
or
μ(
x
)
2.3 Intuitionistic Fuzzy Hierarchical Clustering Algorithms
Xu (2009) introduced an intuitionistic fuzzy hierarchical algorithm for clustering
IFSs, which is based on the traditional hierarchical clustering procedure, the intuition-
istic fuzzy aggregation operator, and the basic distancemeasures between IFSs. Then,
the algorithm was extended for clustering IVIFSs. The algorithm and its extended
form were applied to the classifications of building materials and enterprises respec-
tively. In this section, we shall give a detailed introduction to the intuitionistic fuzzy
hierarchical algorithms.
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