A novel fuzzy clustering algorithm based on a fuzzy scatter matrix ...

644 K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652
will move away from x with a corresponding
weighted term g i . Since g i is a monotone increasing
function of b, the cluster centers obtained by FCS
with a large b value will be more far away from x
than the centers obtained with a small b value. The
proposed FCS <strong>clustering</strong> <strong>algorithm</strong> is summarized
as follows:
FCS Algorithm 1. (see also Yang et al.
(2003)) Set the iteration counter ‘ = 0 and choose
the initial values a ð0Þ
i , i =1,...,c. Given b, e >0
Step 1. Find g ð‘þ1Þ
i using (15)
Step 2. Find l ð‘þ1Þ
ij using (13)
Step 3. Find a ð‘þ1Þ
i using (14)
Increment ‘; until max i ka ð‘þ1Þ
i
a ð‘Þ
i k < e.
We now use a simple example to illustrate the
FCS properties. This is a sampling data set drawn
from a one-dimensional normal mixture (1/
3)N(0, 1) + (1/3)N(4, 1) + (1/3)N(10, 1) with three
populations of means 0, 4 and 10. The histogram
of this data set is shown in Fig. 2(a). Fig. 2(b)
shows the FCMmembership function that belongs
to the mean 0 cluster. Fig. 2(c)–(f) are the FCS
membership functions that belong to the mean 0
cluster with the values 0.9, 0.5, 0.1 and 0.05 of
the parameter b. The cluster kernel volumes always
decrease when b is decreasing. The volume
of the cluster kernel can also be presented using
the range with the value of one of the cluster membership
functions. In our 3-cluster example, Fig.
2(b)–(f) draw only the membership function of
the mean 0 cluster. Thus, the range with the membership
value one presents the volume of the mean
0 cluster kernel, and the range with the membership
value zero presents the volumes of the means
4 and 10 cluster kernels. The crisp membership
values and <strong>fuzzy</strong> membership values co-exist in
FCS. When b = 0, the FCS membership values will
be equivalent to FCMwith no existing cluster kernel.
The three cluster centers obtained using FCM
and FCS are shown in Table 1. When b decreases,
the cluster centers obtained by FCS will then be
closer to the grand mean x and also closer to the
cluster centers obtained by the FCM. These results
coincide with the property shown in Eq. (15).
Histogram
FCM
FCS
beta=0.9
15
1.0
1.0
Frequency
10
5
membership
0.5
membership
0.5
0
0.0
0.0
0 5 10 15
0 5 10
0 5 10
data set
data set
data set
(a) (b) (c)
FCS
beta=0.5
FCS
beta=0.1
FCS
beta=0.05
1.0
1.0
1.0
membership
0.5
membership
0.5
membership
0.5
0.0
0.0
0.0
0 5 10
0 5 10
0 5 10
data set
data set
data set
(d) (e) (f)
Fig. 2. Membership functions of FCM and FCS.