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

Pattern Recognition Letters 26 (2005) 639–652
www.elsevier.com/locate/patrec
A <strong>novel</strong> <strong>fuzzy</strong> <strong>clustering</strong> <strong>algorithm</strong> <strong>based</strong> on a <strong>fuzzy</strong>
scatter matrix with optimality tests
Kuo-Lung Wu a , Jian Yu b , Miin-Shen Yang a, *
a Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li 32023, Taiwan, ROC
b Department of Computer Science, Beijing Jiaotong University, Beijing 100044, PR China
Received 16 January 2004
Available online 28 October 2004
Abstract
Most <strong>clustering</strong> <strong>algorithm</strong>s are <strong>based</strong> on a within-cluster scatter matrix with a compactness measure. In this paper we
propose a <strong>novel</strong> <strong>fuzzy</strong> <strong>clustering</strong> <strong>algorithm</strong>, called the <strong>fuzzy</strong> compactness and separation (FCS), <strong>based</strong> on a <strong>fuzzy</strong> scatter
matrix in which the FCS <strong>algorithm</strong> is derived using compactness measure minimization and separation measure maximization.
The compactness is measured using a <strong>fuzzy</strong> within-cluster variation. The separation is measured using a <strong>fuzzy</strong>
between-cluster variation. The proposed FCS objective function is a modification of the FS validity index proposed by
Fukuyama and Sugeno and also a generalization of the <strong>fuzzy</strong> c-means (FCM). The FCS <strong>algorithm</strong> assigns a crisp
boundary (cluster kernel) for each cluster such that hard memberships and <strong>fuzzy</strong> memberships can co-exist in the
<strong>clustering</strong> results. Thus, FCS can be seen as a <strong>clustering</strong> <strong>algorithm</strong> with a <strong>novel</strong> sense between the hard c-means and
<strong>fuzzy</strong> c-means. The FCS optimality tests and parameter selection are also investigated. Some numerical examples
are demonstrated to show its robust properties and effectiveness.
Ó 2004 Elsevier B.V. All rights reserved.
Keywords: Fuzzy <strong>clustering</strong> <strong>algorithm</strong>; Scatter matrix; Within-cluster variation; Between-cluster variation; Fuzzy compactness and
separation
1. Introduction
* Corresponding author. Tel.: +886 3 456 3171; fax: +886 3
456 3160.
E-mail address: msyang@math.cycu.edu.tw (M.-S. Yang).
Cluster analysis is a branch in statistical multivariate
analysis and unsupervised pattern recognition
learning. It is a method for <strong>clustering</strong> a data
set into most similar groups in the same cluster
and most dissimilar groups in different clusters.
0167-8655/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.patrec.2004.09.016

640 K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652
The <strong>clustering</strong> applications in various areas have
been well documented (Duda and Hart, 1973; Jain
and Dubes, 1988; Kaufman and Rousseeuw,
1990). In these <strong>clustering</strong> methods, the hard
c-means (or k-means) and <strong>fuzzy</strong> c-means (FCM)
<strong>clustering</strong> <strong>algorithm</strong>s are the most well-known
methods (Bezdek, 1981; Jain and Dubes, 1988;
Yang, 1993). Most of these methods are <strong>based</strong>
on minimizing the within-cluster scatter matrix
trace. The within-cluster scatter matrix trace can
be interpreted as a compactness measure with a
within-cluster variation. Because the <strong>clustering</strong> results
obtained using k-means and FCMare
roughly spherical with similar volumes, many <strong>clustering</strong>
<strong>algorithm</strong>s such as the Gustafson–Kessel
(G–K) <strong>algorithm</strong> (Gustafson and Kessel, 1979),
the sum of all normalized determinants (SAND)
<strong>algorithm</strong> (Rouseeuw et al., 1996), the minimum
scatter volume (MSV) and minimum cluster volume
(MCV) <strong>algorithm</strong>s (Krishnapuram and Kim,
2000), the unsupervised <strong>fuzzy</strong> partition-optimal
number of classes (UFP-ONC) <strong>algorithm</strong> (Gath
and Geva, 1989), etc. were proposed to accommodate
elliptical clusters with different volumes.
These <strong>algorithm</strong>s are all <strong>based</strong> on a within-cluster
scatter matrix with a compactness measure.
The concept of adopting a separation measure
in <strong>clustering</strong> is used widely in solving cluster validity
problems such as the separation coefficient proposed
by Gunderson (1978), the XB index
proposed by Xie and Beni (1991), the FS index proposed
by Fukuyama and Sugeno (1989), the SC
index proposed by Zahid et al. (1999), F HV (<strong>fuzzy</strong>
hyper-volume) and P D (partition density) indexes
proposed by Gath and Geva (1989), etc. Özdemir
and Akarun (2001) proposed an inter-cluster separation
(ICS) <strong>clustering</strong> <strong>algorithm</strong> that involves a
separation measure in the ICS objective function.
Because the between-cluster scatter matrix trace
can be interpreted as a separation measure with a
between-cluster variation, maximization of the between-cluster
scatter matrix trace will induce a result
with well-separated clusters (Yang et al., 2003).
In this paper, we propose a <strong>novel</strong> <strong>fuzzy</strong> <strong>clustering</strong>
<strong>algorithm</strong>, called the <strong>fuzzy</strong> compactness and
separation (FCS) <strong>algorithm</strong>. The FCS objective
function is <strong>based</strong> on a <strong>fuzzy</strong> scatter matrix. The
FCS <strong>algorithm</strong> is derived by minimizing the compactness
measure and simultaneously maximizing
the separation measure (Yang et al., 2003). The
compactness is measured using a <strong>fuzzy</strong> within-cluster
scatter matrix. The separation is measured
using a <strong>fuzzy</strong> between-cluster scatter matrix trace.
In k-means, data points always have crisp membership
values of zero or one. Although, FCMallows
the data points to have <strong>fuzzy</strong> membership values
between zero and one, it does not exactly produce
a zero or one for the membership values. In the
proposed FCS <strong>algorithm</strong>, crisp and <strong>fuzzy</strong> membership
values could co-exist. These FCS properties
will be discussed. We will also show that, when the
weighting exponent m is large, the FCS <strong>algorithm</strong>
is more robust to noise and outliers than FCM.
The theoretical analysis on FCS will be investigated.
Yu et al. (2004) gave a theoretical upper
bound for the weighting exponent m in FCMin
which the grand sample mean x is a unique optimizer
of the FCMobjective function. In this paper,
we will show that FCS with the different cluster
kernel characteristic can avoid the situation in
which x is a unique optimizer of the FCS objective
function. We also studied the optimality tests.
These results will be used as the parameter selection
for FCS. The paper is organized as follows.
In Section 2, the (crisp) scatter matrix is extended
to the <strong>fuzzy</strong> scatter matrix. Some <strong>clustering</strong> <strong>algorithm</strong>s
<strong>based</strong> on the within-scatter matrix are then
reviewed. In Section 3, we propose the <strong>novel</strong> <strong>fuzzy</strong>
<strong>clustering</strong> <strong>algorithm</strong> <strong>based</strong> on the <strong>fuzzy</strong> withinand
between-scatter matrix. Section 4 gives our
theoretical analysis on the optimality tests and
the FCS parameter selection. Section 5 gives the
robust properties of FCS <strong>based</strong> on the gross error
sensitivity and influence function. Some numerical
examples are presented in Section 6. Conclusions
are made in Section 7.
2. Clustering <strong>algorithm</strong>s <strong>based</strong> on a within-cluster
scatter matrix
Let X ={x 1 ,...,x n } be a data set in an s-dimensional
Euclidean space R s and let c be a positive
integer larger than one. A partition of X into c
clusters can be presented using mutually disjoint
sets X 1 ,...,X c such that X 1 [[X c = X or

K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 641
equivalently using the indicator functions l 1 ,...,l c
such that l ij = l i (x j )=1 if x j 2 X i and l ij =
l i (x j )=0 if x j 62 X i . Let the sample mean of the
ith cluster be
a i ¼ X P n
x j j¼1
¼
l ijx j
P n
x j2X i
n i j¼1 l ;
ij
i ¼ 1; ...; c; j ¼ 1; ...; n;
ð1Þ
where n i is the number of data points in X i . Let the
grand mean be x ¼ P n
j¼1 x j=n. The total scatter matrix
S T for the data set X can then be decomposed
into a within cluster scatter matrix S W and a between-cluster
scatter matrix S B with S T = S W + S B
where
S T ¼ Xc X
ðx j xÞðx j xÞ t
x j2X i
i¼1
¼ Xc
i¼1
S W ¼ Xc
i¼1
¼ Xc
S B ¼ Xc
i¼1
i¼1
¼ Xc
i¼1
X n
j¼1
X
l ij ðx j xÞðx j xÞ t ; ð2Þ
x j2X i
ðx j a i Þðx j a i Þ t
X n
j¼1
X
l ij ðx j a i Þðx j a i Þ t ; ð3Þ
x j2X i
ða i xÞða i xÞ t
X n
j¼1
l ij ða i xÞða i xÞ t : ð4Þ
Duda and Hart (1973) noted that the determinant
jS W j of a within-cluster scatter matrix could
be a criterion function for <strong>clustering</strong>. jS W j can be
interpreted as the square of the scatter volume.
On the basis of jS W j, Rouseeuw et al. (1996) created
the ‘‘so-called’’ SAND <strong>algorithm</strong>.
Let tr(A) denote the trace of a matrix A. The
trace tr(S W ) of a within-cluster scatter matrix can
be used to measure the compactness with a within-cluster
variation. It is reasonable to use tr(S W )
as a <strong>clustering</strong> objective function. The k-means
and most <strong>clustering</strong> <strong>algorithm</strong>s are created by minimizing
the objective function <strong>based</strong> on tr(S W ). Because
tr(S T ) = tr(S W ) + tr(S B ) for a given data set
X and tr(S T ) is independent of the cluster center
a i , the minimization of tr(S W ) will be equivalent
to the maximization of tr(S B ). However, we know
that the quantity tr(S B ) can be used as a measure
of compactness. Thus, building an objective function
such that it can be optimized with minimizing
tr(S W ) and simultaneously with maximizing tr(S B )
should be sense work and the main goal of this
paper.
Since Zadeh (1965) introduced the <strong>fuzzy</strong> set
concept, researches on <strong>fuzzy</strong> <strong>clustering</strong> have
been widely investigated (Bezdek, 1981; Yang,
1993). In <strong>fuzzy</strong> <strong>clustering</strong>, FCMis a most used
<strong>clustering</strong>
P
<strong>algorithm</strong>. Suppose that l ij 2 [0, 1] with
c
i¼1 l ij ¼ 1 for all j and m > 1 is a given real value.
The FCMupdate equation of <strong>fuzzy</strong> sample mean
is with
P n
j¼1
a i ¼
lm ij x j
; i ¼ 1; ...; c; j ¼ 1; ...; n: ð5Þ
P n
j¼1 lm ij
Thus, we define the <strong>fuzzy</strong> total scatter matrix
S FT , the <strong>fuzzy</strong> within-cluster scatter matrix S FW
and the <strong>fuzzy</strong> between-cluster scatter matrix S FB
on the basis of the <strong>fuzzy</strong> sample mean a i as
S FT ¼ Xc X n
l m ij ðx j xÞðx j xÞ t ; ð6Þ
i¼1
S FW ¼ Xc
S FB ¼ Xc
i¼1
i¼1
j¼1
X n
j¼1
X n
j¼1
l m ij ðx j a i Þðx j a i Þ t ; ð7Þ
l m ij ða i xÞða i xÞ t ; ð8Þ
where l ij 2 [0, 1], P c
i¼1 l ij ¼ 1 and m > 1. We know
that
S FT ¼ Xc X n
l m ij ðx j a i þ a i xÞðx j a i þ a i xÞ t
i¼1
¼ Xc
i¼1
j¼1
X n
j¼1
l m ij ½ðx j a i Þðx j a i Þ t þða i xÞða i xÞ t
þðx j a i Þða i xÞ t þða i xÞðx j a i Þ t Š
¼ S FW þ S FB þ Xc
ða i xÞ t Xn
l m ij ðx j a i Þ
þ Xc
i¼1
ða i
i¼1
xÞ Xn
j¼1
j¼1
l m ij ðx j a i Þ t :

642 K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652
According to Eq. (5), we have P n
j¼1 lm ij a i ¼
P n
j¼1 lm ij x j. Thus, we can create the property that
S FT = S FW + S FB where a i is defined by Eq. (5).
This property is exactly the same as S T = S W + S B
for the (crisp) scatter matrix.
Fuzzy <strong>clustering</strong>s including FCM(Bezdek,
1981), alternative FCM(Wu and Yang, 2002;
Yang et al., 2002), G–K (Gustafson and Kessel,
1979), SAND (Rouseeuw et al., 1996), MCV
(Krishnapuram and Kim, 2000), and UFP-ONC
(Gath and Geva, 1989), etc., are all <strong>based</strong> on
the <strong>fuzzy</strong> within-cluster scatter matrix S FW .
It is known that the FCM<strong>clustering</strong> <strong>algorithm</strong>
is created by minimizing the objective
function
J FCM ¼ trðS FW Þ¼ Xc
i¼1
X n
j¼1
with the membership update equation
l ij ¼ ðkx j a i k 2 Þ 1
m 1
P c
k¼1 ðkx j a k k 2 Þ 1
m 1
l m ij kx j a i k 2 ; ð9Þ
ð10Þ
and the cluster center update Eq. (5). Although
tr(S FT ) = tr(S FW ) + tr(S FB ) for a given data set X,
tr(S FT ) is not a fixed constant but depends on l ij .
Thus, to minimize tr(S FW ), it is not necessary to
maximize tr(S FB ). The trace tr(S FB ) of a <strong>fuzzy</strong> set
between the cluster scatter matrix can be interpreted
as a separation with a cluster variation in
between. A maximum value of tr(S FB ) will induce
a <strong>clustering</strong> result with separated (distinguishable)
clusters. In the next section, an <strong>algorithm</strong> that considers
tr(S FW ) and tr(S FB ) simultaneously is
introduced.
3. A proposed <strong>fuzzy</strong> <strong>clustering</strong> <strong>algorithm</strong> <strong>based</strong> on
tr(S FW ) and tr(S FB )
Fukuyama and Sugeno (1989), Sugeno et al.
(1993) had used tr(S FW ) and tr(S FB ) to create an
index FS(c) in the cluster validity problem where
Pal and Bezdek (1995) gave more discussions in
cluster validity for FCM. The validity index
FS(c) was formed with
FSðcÞ ¼trðS FW Þ
¼ Xc
i¼1
X c
i¼1
X n
j¼1
X n
trðS FB Þ
l m ij kx j a i k 2
j¼1
l m ij ka i xk 2 : ð11Þ
A small FS(c) value will induce a good <strong>fuzzy</strong>
<strong>clustering</strong> result with a small <strong>fuzzy</strong> within-cluster
variation tr(S FW ) and a large <strong>fuzzy</strong> between-cluster
variation tr(S FB ). This will help us find a good
cluster number estimate. It is reasonable to have
a <strong>clustering</strong> objective function containing a measure
of within- and between-cluster variations such
as the FS(c) index. However, there does not exist
an update equation for a i by differentiating FS(c)
with respect to a i . Thus, FS(c) cannot be a <strong>clustering</strong>
objective function.
In this section, we propose a <strong>novel</strong> <strong>fuzzy</strong> <strong>clustering</strong>
objective function which is a modification
of the FS(c) index. This could be also a generalization
of the FCMobjective function by combining
<strong>fuzzy</strong> within- and between-cluster variations. Our
goal is to minimize a <strong>fuzzy</strong> within-cluster variation
tr(S FW ) and also simultaneously maximize a <strong>fuzzy</strong>
between-cluster variation tr(S FB ). We call this a
<strong>fuzzy</strong> compactness and separation (FCS) <strong>algorithm</strong>,
because the compactness is measured using
a <strong>fuzzy</strong> within variation and the separation is
measured using a <strong>fuzzy</strong> between variation. Thus,
the FCS objective function J FCS is defined as
J FCS ¼ Xc X n
l m ij kx j a i k 2
i¼1 j¼1
X c X n
i¼1
j¼1
g i l m ij ka i xk 2 ; ð12Þ
where g i P 0. Note that, J FCS = J FCM when g i =0
and J FCS = FS(c) when g i = 1. By minimizing J FCS
we have the following update equations:
ðkx j a i k 2 g
l ij ¼
i ka i xk 2 m
Þ 1
1
P c
k¼1 ðkx ð13Þ
j a k k 2 g k ka k xk 2 m
Þ 1
1
and
P n
j¼1
a i ¼
lm ij x j
P n
j¼1 lm ij
P n
g i j¼1 lm ij
P x
g n ; ð14Þ
i j¼1 lm ij

K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 643
where the parameter g i could be set up with
g i ¼ ðb=4Þmin i 0 6¼ika i a i 0k 2
; 0 6 b 6 1:0: ð15Þ
max k ka k xk 2
In <strong>fuzzy</strong> <strong>clustering</strong>, we restrict l ij 2 [0,1]. Because
l ij in Eq. (13) might be negative for some data
point x j , we make some restrictions on it. For a
given data point x j ,ifkx j a i k 2 6 g i ka i xk 2 , then
l ij = 1, and l i 0 j ¼ 0, for all i 0 5 i. That is, if the distance
between the data points and the ith cluster
center are smaller than g i ka i xk 2 , these data
points will then belong exactly to the ith cluster with
membership value of one. Each cluster in FCS will
have a crisp boundary such that all data points inside
this boundary will have a crisp membership
value l ij 2 {0,1} and other data points outside this
boundary will have <strong>fuzzy</strong> membership values
l ij 2 [0, 1]. Each crisp boundary will form a hyperball
for the corresponding cluster and can be seen
as a cluster kernel. Fig. 1 shows a two-cluster data
set in which each cluster contains a cluster center
and a cluster kernel. The volume of each cluster kernel
is decided by the term g i ka i xk 2 . Data points
outside the kernel will have <strong>fuzzy</strong> memberships.
Note that Özdemir and Akarun (2002) proposed
the partition index maximization (PIM) <strong>algorithm</strong>
that uses a fixed volume of cluster kernels for each
cluster. In our FCS, the volume of the kernel for
each cluster is different. This FCS characteristic
could catch more information from data according
to different volumes with different shapes.
In the k-means <strong>algorithm</strong>, each data point has a
crisp membership value with l ij 2 {0,1}. Although
cluster center
<strong>fuzzy</strong> boundary
cluster kernel
Fig. 1. Clusters obtained by FCS.
crisp boundary
FCMallows data points to have membership values
l ij in the interval [0,1], it has fewer crisp membership
values (i.e. zero or one) in FCM, even
when the data points are very close to any one of
these c cluster centers. The memberships seem to
be too <strong>fuzzy</strong> in FCM. In our FCS, crisp and <strong>fuzzy</strong>
membership values co-exist. Data points that fall
inside any one of c-cluster kernels (i.e. close to
any one of c cluster centers) will have crisp memberships
and those outside the cluster kernels (i.e.
far away from all cluster centers) will have <strong>fuzzy</strong>
membership values. To guarantee that no two of
these c cluster kernels will overlap, g i is chosen
as Eq. (15) such that the parameter b will control
the size of each kernel. Since, for all i,
0
1
g i ka i
xk 2 ¼ðb=4Þðminka i
i 0 6¼i
6 b min
i 0 6¼i
a i 0k 2 ka i xk 2
Þ@
A
maxka k xk 2
k
ka i a i 0k
! 2
2
ka i a i 0k
2
6 min
i 0 6¼i 2
for all 0 6 b 6 1;
we have, for a given data point x j with l ij =1,
2
kx j a i k 2 6 g i ka i xk 2 ka i a i 0k
6 min
:
i 0 6¼i 2
Note that the value of ka i a i 0k=2 is a half the
distance between cluster centers a i and a i 0. Thus,
Eq. (15) should guarantee that no two of these c
cluster kernels will overlap. If b = 1.0, the FCS
<strong>algorithm</strong> will cluster the data set using the largest
kernel for each cluster. If b = 0 (i.e. g i = 0), the
FCS <strong>algorithm</strong> will cluster the data set with no
cluster kernel. This will be equivalent to the
FCM<strong>clustering</strong> <strong>algorithm</strong>.
The cluster center update Eq. (14) in FCS can
be interpreted as a weighted mean of the cluster
center update equation of FCMand the grand
mean x. We can rewrite Eq. (14) as
P n
j¼1
a i ¼
lm ij x j= P n
j¼1 lm ij
g i x
: ð16Þ
1 g i
The weights of the FCMcluster center and the
grand mean x are 1 and g i , respectively. To maximize
tr(S FB ), the cluster centers obtained by FCS

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.

K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 645
Table 1
Cluster centers obtained by FCS
Beta
Cluster centers
0.900 0.366 3.667 12.319
0.500 0.281 3.687 11.221
0.200 0.164 3.714 10.557
0.100 0.111 3.726 10.356
0.050 0.083 3.733 10.256
0 (FCM) 0.052 3.740 10.156
Although the FCMis a popular <strong>clustering</strong> <strong>algorithm</strong>,
the cluster centers obtained by it will be closer
to the grand mean x when the data set heavily
overlaps. According to the FCS properties, the
cluster centers obtained by it can be more accurate
than the FCMin some situations. This will
be illustrated in Section 5. In the next section,
more details about the FCS parameters are
presented.
4. Optimality tests and parameter selection of FCS
In general cases, when the data set is clustered
into c (c > 1) subsets, each subset is often expected
to have a different prototype (or cluster center)
than the others. However, the grand sample center
x of the data set is always a fixed point in the FCS
<strong>algorithm</strong>. The FCS output will be x with a great
probability if x is one stable solution in the FCS
<strong>algorithm</strong>. To avoid such cases, we hope that x is
not an attracted point in the FCS. How do we
judge if x is attracted or a stable point of FCS?
The Hessian matrix of the FCS objective function
(12) must be studied. In order to simplify
the calculations, substituting (13) into (12) yields
(17)
J ¼ Xn
j¼1
X c
i¼1
ðkx j a i k 2 g i ka i xk 2 Þ 1
m 1! 1 m
:
ð17Þ
It can be proved that J = min l J FCS . Therefore,
it is enough to judge whether or not x is attracted
or is a stable point in FCS using the Hessian
matrix of (17). Let us set
qðx j ; a i Þ¼kx j a i k 2 g i ka i xk 2 ;
S j ¼ Xc
i¼1
m
qðx j ; a i Þ 1
1
; l ij ¼ qðx m
j; a i Þ 1
1
S j
and
h ij ¼ l m ij ½ð1 g iÞa i ðx j g i xÞŠ:
We know that
ð1Þ oJ
oa i
¼ 2 Xn
ð2Þ
j¼1
o 2 J
¼
4m
oa i oa k m 1
þ 2d ik
l m ij ½ð1 g iÞa i ðx j g i xÞŠ:
X n
j¼1
X n
j¼1
ðS j Þ m
l m ij ð1
1 h ij ðh kj Þ t
!
g iÞ I ss
4m
m 1 d X n
ik ½qðx j ; a i ÞŠ 1 h ij ½ð1
j¼1
8i; 8j
g i Þa i
ðx j g i xÞŠ t : ð18Þ
Therefore, the second-order term of TaylorÕs
series expansion of (17) cab be expressed as
follows:
o 2
!
J
u
oa i oa a ¼ 4m
2
X n X
ðS j Þ m 1 c
l m ij
k m 1
ut a i
½ð1 g i Þa i ðx j g i xÞŠ
j¼1
i¼1
!
þ2 Xc
ð1 g i Þ Xn
l m 4m X c X n
ij ut a i
u ai
u t a
m 1
i
l m ð1 g i Þa i ðx j g i xÞð1 g i Þa i ðx j g i xÞ t
ij
u
qðx
i¼1
j¼1
i¼1 j¼1
j ;a i Þ
ai
:
ð19Þ
u t a

646 K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652
ðx j xÞðx j xÞ t
If "i, a i ¼ x, C b X ¼ P n
j¼1
, and "j,
nkx j xk 2 Þ
kx j xk > 0, then we get the following equation
u t o 2
J a¼ða1
a
u
oa i oa a
k
;a 2 ;...;a cÞ where a
i¼x;8i
P
¼
4m
c
X n i¼1 u t 2
a i
ðx j xÞ
cðm 1Þ kx j xk 2
þ 2n
c m X c
i¼1
j¼1
ð1 g i Þu t 2m C X
a i
I ss u
ðm 1Þ ð1 g i Þ
ai
:
ð20Þ
For simplifying the analysis, we assume that "i,
g i = g and "j, "k, q(x j ,a k ) > 0. In this way, we can
ignore the update equations for g i to reduce the
complexity of the analysis on FCS. Thus, Eq.
(19) turns into (21) as follows:
A good <strong>clustering</strong> method should have the robust
ability to tolerate noise and outliers. In this
section, we use the gross error sensitivity and influence
function (Huber, 1981) to show that our
weighted cluster center update equation is robust
to noise and outliers. Let {x 1 ,...,x n } be an
observed data set of real numbers and h is an unknown
parameter to be estimated. An M-estimator
(Huber, 1981) is generated by minimizing the
form
X n
j¼1
qðx j ; hÞ;
ð22Þ
where q is an arbitrary function that can measure
the loss of x j and h. Here, we are interested in a
location estimate that minimizes
X n
j¼1
qðx j hÞ ð23Þ
u t a
o 2 J
oa i oa k
u a ¼
! 4m
2
X n X
ðS j Þ m 1 c
l m ij
m 1
ut a i
½ð1 gÞa i ðx j gxÞŠ
j¼1
i¼1
þ 2ð1
gÞ Xc
i¼1
X n
j¼1
l m ij ut a i
I
2m X n
m 1
j¼1
l m ij ½ða !
i x j Þ gða i xÞŠ½ða i x j Þ gða i xÞŠ t
ð1 gÞqðx j ; a i Þ P n
u
j¼1 lm ai
:
ij
ð21Þ
From Eq. (21), we know that if g approaches
negative infinity, then any FCS solution will be stable.
This is an unacceptable result. Similarly, if g
approaches positive infinity, any FCS solution will
be unstable. This is also unacceptable. Therefore, we
can roughly set the range for g with 1

K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 647
uðx hÞ
ICðx; F ; hÞ ¼R u0 ðx hÞdF X ðxÞ ; ð25Þ
where F X (x) denotes the distribution function of
X. If the influence function of an estimator is unbounded,
a noise or outliers might cause trouble.
Identically, if the u function of an estimator is unbounded,
noise and outliers will cause trouble.
Many important robustness measures can be observed
from the influence function. One of the
important measures is the gross error sensitivity
c , defined by
c ¼ sup jICðx; F ; hÞj:
ð26Þ
x
This quantity can interpret the worst approximate
influence that the addition of an infinitesimal point
mass can have on the value of the associated
estimator.
Let the loss between the data point x j and ith
cluster center a i be
qðx j a i Þ¼l m ij kx j a i k 2 g i l m ij ka i xk 2 ð27Þ
and
uðx j a i Þ¼ o qðx j a i Þ
oa i
¼ 2l m ij ðx j a i Þ 2g i l m ij ða i xÞ: ð28Þ
By solving the equation P n
j¼1 uðx j a i Þ¼0, we
have the result shown in Eq. (14). Thus, the FCS
cluster center is an M-estimator with the loss function
(27) and u function (28). Note that, the u
function of our estimator is a function of l m ij which
depends on the fuzzifier m. We will show that the
FCS cluster center is robust to noise and outliers
when m is large.
For a given data set, we will produce the FCS
cluster centers {a 1 ,...,a c }, the parameters
{g 1 ,...,g c } and the sample mean x by processing
the FCS <strong>clustering</strong> <strong>algorithm</strong>. The relative influence
(the u function) of an individual observation
x toward the ith cluster center can be defined as
uðx a i Þ¼ 2ðl i ðxÞÞ m ½ðx a i Þ g i ða i xÞŠ;
ð29Þ
where
l i ðxÞ ¼
ðkx a i k 2 g i ka i xk 2 m
Þ 1
1
P c
k¼1 ðkx a : ð30Þ
kk 2 g k ka k xk 2 m
Þ 1
1
Note that, l i (x)=1ifkx a i k 2 6 g i ka i xk 2 . Suppose
that x 2 R. For an extremely large or small x,
we will have kx a i k 2 P g i ka i xk 2 and hence
l i (x) 2 (0, 1). In general, the extremely large or
small x will fall outside any one of c cluster kernels
and have <strong>fuzzy</strong> membership value l i (x) 2 (0, 1) or
more precisely, l i (x) will be very closed to 1/c
and hence l i (x) m will be very closed to zero when
m is large. Although ðx a i Þ g i ða i xÞ is a monotone
increasing function of x, ðl i ðxÞÞ m ½ðx a i Þ
g i ða i xÞŠ will be very closed to zero for an extremely
large or small x when m is large. Thus, the u
function of the FCS cluster center will be bounded
for an extremely large or small x when m tends to
infinity. Therefore, for a large m case, only the
data point x inside the ith cluster kernel with
l i (x) = 1 will have an influence on the ith FCS
cluster center a i and the data point x falls on the
crisp boundary of the ith cluster center will have
the gross error sensitivity c < 1. The above discussions
give a theoretical foundation to asseverate
the FCS <strong>algorithm</strong> to be robust to noise and
outlier when m is large. The following is a simple
example.
For the data set shown in Fig. 2(a), we implement
FCS with b = 0.1 and m = 2 to produce the
set of {a 1 ,a 2 ,a 3 }, {g 1 ,g 2 ,g 3 }andx. For a given
data point x, the membership function l 1 (x) (Eq.
(30)) is illustrated in Fig. 3(a) and the u function
uðx a 1 Þ¼ 2ðl 1 ðxÞÞ 2 ½ðx a 1 Þ g 1 ða 1 xÞŠ
is illustrated in Fig. 3(d). When m = 2, an extremely
large or small x will have a large influence
on a 1 as shown in Fig. 3(d). Equivalently, we
implement FCS with m = 3 and m = 4 to have
other sets of {a 1 ,a 2 ,a 3 } and {g 1 ,g 2 ,g 3 } to illustrate
the l 1 (x) inFig. 3(b) and (c), respectively. The corresponding
u functions are illustrated in Fig. 3(e)
and (f). Fig. 3 shows that the influence of an extremely
large or small x will become small when m increases.
These numerical results coincide to our
theoretical analysis.
6. Numerical examples
We implement the Normal-4 data set which was
proposed by Pal and Bezdek (1995). Normal-4 is a
four-dimensional data set with the sample size

648 K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652
FCS
beta=0.1, m=2
FCS
beta=0.1, m=3
FCS
beta=0.1, m=4
1.0
1.0
1.0
membership
0.5
membership
0.5
membership
0.5
0.0
0.0
0.0
0
10
20 30 40 50
x
0 10 20 30 40 50
x
0 10 20 30
x
(a) (b) (c)
40
50
4
The phi function of FCS
beta=0.1, m=2
4
The phi function of FCS
beta=0.1, m=3
4
The phi function of FCS
beta=0.1, m=4
2
2
2
phi function
0
-2
-4
phi function
0
-2
-4
phi function
0
-2
-4
-6
-6
-6
-8
0 10 20 30 40 50
x
-8
0 10 20 30 40 50
x
-8
0 10 20 30 40 50
x
(d) (e) (f)
Fig. 3. Membership functions (Eq. (30)) and phi functions (Eq. (29)) for a given data point x when m = 2, 3 and 4.
n = 800 points consisted of 200 points from each
of four clusters. The population mean vectors
are l 1 = (3,0,0,0), l 2 = (0, 3,0,0), l 3 = (0, 0,3,0)
P
and l 4 = (0, 0,0,3). The covariance matrices are
i ¼ I 4, i = 1, 2, 3, 4. We use the mean vectors
as the initial values for both FCMand FCS <strong>algorithm</strong>s.
We compare FCMand FCS with the mean
squared error (MSE) criterion. The MSE is calculated
by P 4
i¼1 ka i l i k 2 . We implement the FCS
<strong>algorithm</strong> with different values combinations of b
for (0,0.005, 0.01, 0.05,0.1, 0.2) and m for
(1.5,2,2.5,3,3.5). For 100 repeated Normal-4 data
sets, the MSE values are shown in Fig. 4(a). The
result of b = 0 is equivalent to FCM. As b increases,
the volume of the cluster kernel grows.
When the <strong>fuzzy</strong> index m is small, the results of
FCS (b > 0) and FCM(b = 0) are similar. As m becomes
larger, FCMgives larger MSE values than
FCS. Thus, the FCS result is more insensitive to
the <strong>fuzzy</strong> index m than FCM, especially for
b = 0.05, 0.1 and 0.2. In general, the MSE values
will increase when m increases. Yu et al. (2004)
gave a theoretical upper bound for m for FCM
that the grand sample mean x will be a unique
optimizer. For each combination of m and b, the
range between worst and best MSE values among
these 100 repeated MSE values are shown in Fig.
4(b). The cases of m = 3.5 and b = 0,0.05, 0.01
are the examples of the grand sample mean x being
the unique optimizer. The range of MSE values for
FCS with b = 0.05, 0.1 and 0.2 are still insensitive
to the <strong>fuzzy</strong> index m. One may argue that we can
process FCMwith a small m value, say 1.5 or 2,
to avoid the above defective of FCM. However,
FCS are not only insensitive to the <strong>fuzzy</strong> index m
than FCM, but also more robust to noise and outliers
than FCMwhen m is large. We will show this
robust property of FCS in the following examples.
Fig. 5 shows a two-cluster data set with unequal
sample sizes. The <strong>clustering</strong> results of FCS with
m = 2 and b = 0,0.1,0.2 are shown in Fig. 5(a)–
(c). Three figures show the similar results. The
cluster centers are presented by the solid circle
points. Note that, the distance between two cluster
centers increases as b increases. This phenomenon
can be explained by the update Eq. (17) of FCS.

K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 649
MSE
10
5
0
00.005
0.01 0.05
beta
0.1
0.2
(a)
1.5
2.5
m
3.5
range of MSE
25
20
15
10
5
0
00.005
0.01 0.05 0.1
beta
0.2
(b)
1.5
2.5
m
3.5
Fig. 4. MSE values for different combinations of beta and m.
When m becomes larger (m = 6), the <strong>clustering</strong> results
of FCS with b = 0, 0.1 and 0.2 are shown in
Fig. 5(d)–(f). The FCS with cluster kernels
(b = 0.1 and 0.2) obtains better performance than
FCM(b = 0) which clusters the data set without
a cluster kernel. This shows that FCS with a large
and suitable m value can detect unequal sample
size clusters or is robust to the noise. Fig. 6 shows
a two-cluster data set with one outlying point
whose coordinate is (100, 0). When m is large
(m = 6), the results of FCS with b = 0.1 and
0.2 are more robust to the outlier than FCM
(b = 0). These robust properties of FCS can be
explained using the FCS update equations. Let
^l ¼ maxfl i1 ; ...; l in g, l 0 ij ¼ l ij=^l, j =1,...,n. We
have
80
FCM
m=2
80
FCS
m=2, beta=0.1
80
FCS
m=2, beta=0.2
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
10 20 30 40 50 60 70 80 90 100
(a)
10 20 30 40 50 60 70 80 90 100
(b)
10 20 30 40 50 60 70 80 90 100
(c)
80
FCM
m=6
80
FCS
m=6, beta=0.1
80
FCS
m=6, beta=0.2
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
(d) (e) (f)
Fig. 5. FCMand FCS <strong>clustering</strong> results for unequal sample size data sets.

650 K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652
2
FCM
m=6
2
FCS
m=6, beta=0.1
2
FCS
m=6, beta=0.2
1
1
1
0
0
0
-1
-1
-1
-2
-2
-2
-3 -2 -1 0 1 2 3 4 5 6 7
-3 -2 -1 0 1 2 3 4 5 6 7
-3 -2 -1 0 1 2
(a) (b) (c)
3
4
5
6
7
Fig. 6. FCMand FCS <strong>clustering</strong> results for the two-clusters data set with one outlier in the coordinate (100,0).
P n
lim fa j¼1
ig¼ lim
lm ij x P n
j g i j¼1 lm ij
P x
n P
m!1 m!1
j¼1 lm ij g n
i j¼1 lm ij
P n P n
j¼1
¼ lim
ðl0 ij Þm x j g i j¼1 ðl0 ij Þm x
P n m!1
j¼1 ðl0 ijÞ m P n
g i j¼1 ðl0 ijÞ m
P
Pl
¼
0 ¼1x j g
ij i l 0 ¼1x ij
ð1 g i Þ P l 0 ¼11 ij
P
l
¼
0 ¼1x j= P
l
g
ij ij1 0 i x
1 g
i
P
l ij ¼^l x j= P
l ij ¼^l 1 g i x
¼
: ð31Þ
1 g i
In FCM, when m is large, l ij =1/c for all i, j and
hence l ij ¼ ^l for all i, j. This is why FCMcould
obtain results in which the sample mean x will be
a unique optimizer when m is large. However,
the data points inside the cluster kernels in FCS
will have l ij 2 {0,1} and l ij 2 (0,1) for those data
points outside cluster kernels. When m is large,
the ith cluster center update Eq. (22) will give a
large ðl 0 ij Þm ¼ðl ij =^lÞ m ¼ 1 for the data points inside
the ith cluster kernel and will give a small
ðl 0 ij Þm 0 for the data points outside the ith cluster
kernel. When m is large, the ith cluster center update
Eq. (22) will be the weighted mean of the sample
mean of the data points inside the ith cluster
kernel and the grand mean x. The sample mean
data point weights inside the ith cluster kernel
and the grand mean x are 1 and g i , respectively.
For a suitable b value, noise and outliers will be
outside the cluster kernels and their influences on
the <strong>clustering</strong> results will be small when m is large.
This explains the <strong>clustering</strong> results shown in Figs.
5 and 6 and also coincide to the theoretical analysis
in Section 5. This property also provides a
method to avoid the sample mean x being a unique
optimizer in FCM.
We know that when the sample mean x is the
unique optimizer of a <strong>fuzzy</strong> <strong>clustering</strong> <strong>algorithm</strong>,
the partition coefficient (PC) (Bezdek, 1974)
defined by
PCðCÞ ¼ 1 n
X c
i¼1
X n
j¼1
l 2 ij
ð32Þ
will be equal to 1/c or equivalently the non-<strong>fuzzy</strong>
index (NFI) (Pal and Bezdek, 1995) defined by
c
NFIðcÞ ¼1 ð1 PCðCÞÞ ð33Þ
c 1
equals to zero. Note that Dave (1996) also proposed
a modification of the PC index which is
equivalent to the NFI index. According to the
above analysis, we hope the FCS with cluster ker-
NFI
0.25
0.20
0.15
0.10
0.05
0.00
m=10
0 0.05 0.1 0.15 0.2 0.5 0.99
beta
m=20
Fig. 7. NFI(2) values for the unequal sample size data set
shown in Fig. 5.

K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 651
0.5
m=1.5
0.12
m=2
NFI
0.4
PIM
NFI
0.10
0.08
0.06
FCS
0.04
0.3
FCS
0.02
PIM
0.00
0 0.05 0.1 0.15 0.2 0.5 0.99
beta, delta
(a)
0 0.05 0.1 0.15 0.2 0.5 0.99
beta, delta
(b)
Fig. 8. NFI(11) values for the normalized Vowel data set in which both PIMand FCS <strong>algorithm</strong>s are processed with the same
parameter values.
nels can avoid the situation in which the sample
mean is a unique optimizer of the FCS objective
function. Fig. 7 presents the NFI (2) values of
the data set shown in Fig. 5. The NFI values of
FCS with cluster kernels (b > 0) are always larger
than the NFI values of the FCM(b = 0) which is
the case of the sample mean x being the unique
optimizer with NFI = 0 when m = 10 and 20. This
shows that the FCS <strong>algorithm</strong> can avoid the case
of NFI = 0 and is robust to the noise and outliers
than FCMwhen m is larger. Because the sample
mean x of the data set shown in Fig. 6 will not
be the unique optimizer of FCMand FCS when
m is larger, we do not show their NFI values.
Note that some properties of FCS discussed
above can also be achieved by the partition index
maximization (PIM) <strong>algorithm</strong> (Özdemir and
Akarun, 2002) which used a fixed volume for all
cluster kernels. The radius of each cluster volume
in PIMis defined by
a ¼ d minfmin ka i a 0
i6¼i 0 ik=2g; 0 6 d 6 1: ð34Þ
The NFI values of the normalized Vowel data set
in the UCI Machine Learning Repository (Blake
and Merz, 1998) of PIMand FCS are shown in
Fig. 8. Yu et al. (2004) showed that when
m > 1.7787, the sample mean x will be the unique
optimizer of FCMfor the normalized Vowel data
set in Blake and Merz (1998). InFig. 8(a), when
m = 1.5, both PIMand FCS with different d and
b values have the NFI index values larger than
0.3. However, when m = 2 as shown in Fig. 8(b),
the PIMgive the same NFI values as FCM
(d =0 or b = 0). The use of the same volumes of
the cluster kernels do not help PIMto have a larger
NFI values than FCM. The same situation
when m = 2 in FCS as shown in Fig. 8(b), the
NFI values of FCS are always larger than FCM
and PIM. Using the different cluster kernel volumes
in FCS produces these good merits.
7. Conclusions
We proposed a <strong>novel</strong> <strong>clustering</strong> <strong>algorithm</strong>
called the FCS <strong>algorithm</strong> which attempts to minimize
the <strong>fuzzy</strong> within-cluster scatter matrix trace
and simultaneously maximize the <strong>fuzzy</strong> betweencluster
scatter matrix trace. Each cluster obtained
by the FCS will have a cluster kernel. Data points
that fall inside any one of the c cluster kernels will
have crisp memberships and be outside all of the
cluster kernels that have <strong>fuzzy</strong> memberships.
The volume of each cluster kernel is decided by
the parameter g i which is a function of b. The crisp
and <strong>fuzzy</strong> memberships co-exist in the FCS. The
cluster center update equations in the FCS can
be interpreted as a weighted mean of the FCM
cluster centers and the grand mean x. Numerical
examples show that the FCS can have more accurate
results in the parameter estimation than the
FCM. It also shows that FCS can help avoid the
situation where the sample mean x is a unique
optimizer of FCMand is more robust to noise

652 K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652
and outliers than FCMwhen m is large. A theoretical
analysis of FCS was also investigated. Overall,
the proposed FCS is recommended as a good
<strong>clustering</strong> <strong>algorithm</strong> when the most compact kernels
are in the same cluster and most separated
are in different clusters.
Acknowledgement
This work was supported in part by the National
Science Council of Taiwan, ROC, under
grant NSC-91-2118-M-033-001.
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