Data Clustering using a Hybrid of Fuzzy C-Means and
Quantum-behaved Particle Swarm Optimization
Saptarshi Sengupta

Sanchita Basak

Richard Alan Peters II

Department of EECS
Vanderbilt University
Nashville, TN, USA
saptarshi.sengupta@vanderbilt.edu

Department of EECS
Vanderbilt University
Nashville, TN, USA
sanchita.basak@vanderbilt.edu

Department of EECS
Vanderbilt University
Nashville, TN, USA
alan.peters@vanderbilt.edu

Abstract – Fuzzy clustering has become a widely used data mining
technique and plays an important role in grouping, traversing and
selectively using data for user specified applications. The
deterministic Fuzzy C-Means (FCM) algorithm may result in
suboptimal solutions when applied to multidimensional data in
real-world, time-constrained problems. In this paper the
Quantum-behaved Particle Swarm Optimization (QPSO) with a
fully connected topology is coupled with the Fuzzy C-Means
Clustering algorithm and is tested on a suite of datasets from the
UCI Machine Learning Repository. The global search ability of
the QPSO algorithm helps in avoiding stagnation in local optima
while the soft clustering approach of FCM helps to partition data
based on membership probabilities. Clustering performance
indices such as F-Measure, Accuracy, Quantization Error,
Intercluster and Intracluster distances are reported for
competitive techniques such as PSO K-Means, QPSO K-Means
and QPSO FCM over all datasets considered. Experimental
results indicate that QPSO FCM provides comparable and in most
cases superior results when compared to the others.
Keywords—QPSO; Fuzzy C-Means Clustering; Particle Swarm
Optimization; K Means; Unsupervised Learning

I. INTRODUCTION
Clustering is the process of grouping sets of objects such that
objects in one group are more similar to each other than to those
in another group. Data clustering is widely used for statistical
analyses in machine learning, pattern recognition, image
analysis and the information sciences making it a common
exploratory data mining technique [1-2]. The K-Means
algorithm is one of the widely used partitioned data clustering
techniques, however its solution quality is sensitive to the initial
choice of cluster centres and it is susceptible to getting trapped
in local optima [1]. K-Means is NP-hard, thus approximation
algorithms have been used to obtain close to exact solutions [3].
Fuzzy C-Means (FCM) [4] algorithm is an unsupervised soft
clustering approach which uses a membership function to
assign an object to multiple clusters but suffers from the same
issue of stagnation using iterative gradient descent as in hard KMeans. This has led to several attempts to intelligently traverse
the search space and minimize the underlying cost, often at the
expense of increased time complexity. In the past two decades,
powered by increased computational capabilities and the advent

of nature-inspired algorithmic models of collective intelligence
and emergence, many studies have led to the application of
guided random search algorithms in cost optimization of
partitioned and soft clustering. Several metaheuristics
mimicking information exchange in social colonies of bird and
insect species are well known for their robust performances on
ill-structured global optimization problems, irrespective of the
continuity or gradient of the cost function. This paper makes a
comparative analysis of the performance of one such algorithm:
the Quantum-behaved Particle Swarm Optimization (QPSO)
[16], from both a hard, partitioned (QPSO K-Means) as well as
a soft, fuzzy clustering (FCM QPSO) point of view. The
literature suggests prior work on integrating Particle Swarm
Optimization (PSO) [8] into the deterministic K-Means
framework has led to improved clustering accuracy across
many datasets. This is evidenced by the works of Izakian et al.
[5], Emami et al. [6] and Yang et al. [7], among others. In [5]
the authors integrated FCM with a fuzzy PSO and noted the
efficiency and improvement in solution quality whereas the
authors of [6] hybridized FCM with PSO on one hand and an
Imperialist Competitive Algorithm (ICA) [24] on the other to
come to the conclusion that ICAPSO suited the clustering jobs
under consideration better than the competitor methods tested.
The work of Yang et al. in [7] used as metric the harmonic
average of distances between individual data points and cluster
centres summed over all points. The proposed PSO KHarmonic Means (PSOKHM) in [7] was found to outperform
K-Harmonic Means (KHM) and PSO in that it not only reduced
convergence time of PSO but also helped KHM escape local
minima. In this work, a detailed report of performance indices
for some popular datasets from the UCI Machine Learning
Repository [20] using FCM QPSO is made against QPSO KMeans, PSO K-Means and traditional K-Means. Subsequent
sections of the paper are structured as follows: Section II
elaborates on the FCM algorithm, Section III introduces the
variants of PSO used and Section IV describes the FCM QPSO
approach. Section V details the experimental setup while
Section VI reports and analyzes the results obtained. Finally,
Section VII makes concluding remarks.

II. FUZZY C-MEANS ALGORITHM (FCM)
The Fuzzy C-Means (FCM) algorithm aims to partition N
objects into C clusters. Essentially, this reduces to grouping the
object set D = {D1,D2,D3……..DN} into C clusters (1<C<N)
with Ω ={Ω1, Ω2, Ω3,…. ΩC} being the cluster centres. Each data
point belongs to a cluster with randomly initialized centroids,
according to a membership function μij defined as:

𝜇𝑖𝑗 =

1

(1)

2
𝑑𝑖𝑗
∑𝐶
𝑟=1(𝑑 )𝑚−1
𝑟𝑗

dij = || xi – yj || is the distance between i-th centre and j-th data
point, drj = || xr – yj || is that between r-th centre and j-th data
point and m ϵ [1, ∞) is a fuzzifier. FCM employs an iterative
gradient descent to compute centroids, which are updated as:
𝑥𝑖 =

𝑚
∑𝑁
𝑗=1 𝜇𝑖𝑗 𝑦𝑗

𝑓(𝑥𝑖 (𝑡 + 1)) < 𝑓(𝑝𝑖 (𝑡)) ⇒ 𝑝𝑖 (𝑡 + 1) = 𝑥𝑖 (𝑡 + 1)
else 𝑝𝑖 (𝑡 + 1) = 𝑝𝑖 (𝑡)
(6)
Here, f is the cost and pi is the personal best of a particle. The
global best (pg) is the minimum cost bearing element of the
historical set of personal bests pi of a particular particle. A
major limitation of the standard PSO is its inability to guarantee
convergence to an optimum as was shown by Van den Bergh
[14] based on the criterion established in [15].

(2)

𝑚
∑𝑁
𝑗=1 𝜇𝑖𝑗

B.

The objective function minimized by FCM can be formulated
as the sum of membership weighted Euclidean distances:
𝑚
𝜑 = ∑𝐶𝑖=1 ∑𝑁
𝑗=1 𝜇𝑖𝑗 (‖𝑥𝑖 − 𝑦𝑗 ‖)

2

(3)

By recursively calculating eqs. (1) and (2), FCM can be
terminated once a preset convergence criteria is met. Like many
algorithms which employ gradient descent, FCM can fall prey
to local optima in a multidimensional fitness landscape. To
avoid this, a stochastic optimization approach can be used.

III. VARIANTS OF PARTICLE SWARM OPTIMIZERS USED
A.

the position and velocity of 𝑖𝑡ℎ particle in 𝑗𝑡ℎ dimension
whereas 𝑝𝑖𝑗 (𝑡) and 𝑝𝑔𝑗 (𝑡) are the pbest and gbest positions. In
term 1 in the RHS of eq. (4), 𝜔 represents the inertia of the i-th
particle and terms 2 and 3 introduce guided perturbations
towards basins of attraction in the direction of movement of the
particle. The personal best (pbest) update follows a greedy
update scheme considering a cost minimization goal, as
discussed in the following equation.

Particle Swarm Optimization (PSO)

PSO proposed by Eberhart and Kennedy [8] is a stochastic
optimization strategy that makes no assumptions about the
gradient of the objective function. It has been able to effectively
produce promising results in many engineering problems where
deterministic algorithms fail. Although PSO is widely
considered a universal optimizer there exist numerous issues
with the standard PSO [8], most notably a poor local search
ability (Angeline et. al) [9]. This has led to several subsequent
studies on improvements of the same [10-13]. The particles in
PSO update their position through a personal best position pbest and a global best - gbest. After each iteration their
velocity and position are updated as:
𝑣𝑖𝑗 (𝑡 + 1) = 𝜔𝑣𝑖𝑗 (𝑡) + 𝐶1 𝑟1 (𝑡) (𝑝𝑖𝑗 (𝑡) − 𝑥𝑖𝑗 (𝑡))
+𝐶2 𝑟2 (𝑡) (𝑝𝑔𝑗 (𝑡) − 𝑥𝑖𝑗 (𝑡))
𝑥𝑖𝑗 (𝑡 + 1) = 𝑥𝑖𝑗 (𝑡) + 𝑣𝑖𝑗 (𝑡 + 1)

(4)
(5)

𝐶1 and 𝐶2 are social and cognitive acceleration constants, 𝑟1 and
𝑟2 are i.i.d. random numbers between 0 and 1, 𝑥𝑖𝑗 , 𝑣𝑖𝑗 represent

Quantum-behaved PSO (QPSO)

Sun et al. proposed a delta potential well model for PSO,
leading to a variant known as Quantum-behaved Particle
Swarm Optimization (QPSO) [16]. A detailed analysis of the
derivation of particle trajectories in QPSO may be found in [1619]. The state update equations of a particle in a fully connected
QPSO topology is described by the following equations:
1

𝑁

𝑚𝑏𝑒𝑠𝑡𝑗 = ∑𝑖=1 𝑝𝑖𝑗

(7)

𝛷𝑖𝑗 = 𝜃𝑝𝑖𝑗 + (1 − 𝜃)𝑝𝑔𝑗

(8)

𝑥𝑖𝑗 = 𝛷𝑖𝑗 + 𝛽 |𝑚𝑏𝑒𝑠𝑡𝑗 − 𝑥𝑖𝑗 (𝑡)| ln (1⁄𝑞 ) ∀ 𝑘 ≥ 0.5
= 𝛷𝑖𝑗 − 𝛽 |𝑚𝑏𝑒𝑠𝑡𝑗 − 𝑥𝑖𝑗 (𝑡)| ln (1⁄𝑞 ) ∀ 𝑘 < 0.5

(9)

𝑁

𝑚𝑏𝑒𝑠𝑡 is mean of pbest of the swarm across all dimensions and
𝛷𝑖𝑗 is the local attractor of particle i. θ, q and k are i.i.d. uniform
random numbers distributed in [0,1]. β is the contractionexpansion coefficient which is varied over the iterations as:
𝛽 = (1 − 0.1) (

𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑚𝑎𝑥 −𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑐𝑢𝑟𝑟𝑒𝑛𝑡
𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑚𝑎𝑥

) + 0.1

(10)

Eq. (6) updates the pbest set and its minimum is set as gbest.
IV. FUZZY C-MEANS QPSO (FCM QPSO)
In this approach, each particle is a D dimensional candidate
solution in one of the C clusters that can be formally
represented as the matrix X:
𝑥11
𝑋=[ ⋮
𝑥𝐶1

…
⋱
⋯

𝑥1𝐷
⋮ ]
𝑥𝐶𝐷

(11)

A population of particles is randomly initialized and personal
as well as global best positions are determined. Subsequently

membership values are computed and a cost is assigned to each
particle. The QPSO algorithm minimizes the cost associated
with the particles through recursively calculating the mean best
position using eq. (7), the membership values and cost function
through eqs. (1) and (3) and updating the candidate cluster
centre solution X. The algorithm is terminated if there is no
improvement in the global best and the algorithm stagnates or
if the preset number of iterations is exhausted. By using the
stochastic and non-differentiable objective function handling
capabilities of QPSO within the FCM algorithmic framework,
the problem of stagnation in a local minima within a
multidimensional search space is mitigated to an extent better
than that possible with only the traditional FCM. The
pseudocode of FCM QPSO is outlined below:
Algorithm 1 FCM QPSO
1: for each particle xi
2:
initialize position
3: end for
4: Evaluate membership values using eq. (1)
5: Evaluate cost using eq. (3) and set pbest, gbest
6: do
7: Compute mean best (mbest) position using eq. (7)
8:
for each particle xi
9:
for each dimension j
10:
Calculate local attractor Φij using eq. (8)
11:
if k ≥ 0.5
12:
Update xij using eq. (9) with ‘+’
13:
else Update xij using eq. (9) with ‘-’
14:
end if
15:
end for
16:
Evaluate cost using eq. (3) and set pbest, gbest
17: end for
18: while max iter or convergence criterion not met
V. EXPERIMENTAL SETUP
A. Parameter Settings
Learning parameters C1 and C2 are chosen as 2.05 and inertia
weight ω in PSO is decreased linearly from 0.9 to 0.1 over the
course of iterations to facilitate global exploration in the early
stages and exploitation in the latter stages. The contractionexpansion parameter β is varied according to eq. (10) for QPSO.
B. Datasets
Five well-known real datasets from the UCI Machine Learning
Repository were used in analysis. These are:
1) Fisher’s Iris Dataset consisting of three species of the Iris
flower (Setosa, Versicolour and Virginica) with a total of
150 instances with 4 attributes each.

2) Breast Cancer Wisconsin (Original) Dataset consisting of
a total of 699 instances with 10 attributes and can be
classified into 2 clusters: benign and malignant.
3) Seeds Dataset consisting of 210 instances with 3 different
varieties of wheat (Kama, Rosa and Canadian), each with
70 instances and 7 attributes.
4) Mammographic Mass Dataset consisting of 961 instances
with 6 attributes and classified into two clusters: benign
and malignant based on BI-RADS attributes and patient’s
age.
5) Sonar Dataset with 208 instances with 60 attributes and can
be classified into either of 2 objects: mines or rocks.
C. Performance Indices
The performance indices which provide insight into the
clustering effectiveness are outlined below:
(a) Intercluster Distance: The sum of distances between the
cluster centroids, larger values of which are desirable and
imply a greater degree of non-overlapping cluster
formation.
(b) Intracluster Distance: The sum of distances between data
points and their respective parent cluster centroids, smaller
values of which are desirable and indicate greater
compactness of clustering.
(c) Quantization Error: The sum of distances between data
points in a particular cluster and that parent cluster
centroid, divided by the total data points belonging to that
cluster, subsequently summed over all clusters and
averaged by the number of data clusters.
Indices such as F-Measure and Accuracy for the datasets under
test are calculated. The clustering algorithms are implemented
in MATLAB R2016a with an Intel(R) Core(TM) i7-5500U
CPU @2.40GHz. Experimental results for 10 trials are
tabulated and are thereafter analyzed. Table 1 lists the datasets
used in this paper.
VI. RESULTS AND ANALYSIS
Tables 2 through 6 contain results on clustering performance
reporting mean and standard deviations for the performance
indices for QPSO FCM, QPSO K-Means and PSO K-Means
and Figures 1 through 5 compare the accuracy of each
algorithm over all datasets.

Table 1. Data Set Information

Iris
Breast Cancer
Seeds
Mammographic Mass
Sonar

No. of Data Points
150
699
210
961
208

No. of Attributes
4
10
7
6
60

No. of Clusters
3
2
3
2
2

Table 2. Comparison of Various Performance Indices for Iris Data Set

QPSO FCM
QPSO K-Means
PSO K-Means

Inter Cluster Distance
5.7312±0.0067
6.1476±0.0330
6.1411±0.0934

Intra Cluster Distance
9.4608±0.0109
8.9548±0.0075
9.0478±0.0927

Quantization Error
0.6414±0.0035
0.6123±0.0360
0.6418±0.0176

F Measure
0.9133±0.0000
0.9030±0.0021
0.8937±0.0063

Table 3. Comparison of Various Performance Indices for Breast Cancer Data Set

QPSO FCM
QPSO K-Means
PSO K-Means

Inter Cluster Distance
13.3462±1.2050
14.2993±0.4993
14.1413±0.7667

Intra Cluster Distance
146.0102±5.2887
142.9908±1.3635
141.7168±1.2193

Quantization Error
3.8394±0.0410
5.3048±0.0514
5.2737±0.0415

F Measure
0.9641±0.0024
0.9627±0.0028
0.9616±0.0038

Table 4. Comparison of Various Performance Indices for Seeds Data Set

QPSO FCM
QPSO K-Means
PSO K-Means

Inter Cluster Distance
10.0939±0.3889
9.8110±0.0051
9.8460±0.2415

Intra Cluster Distance
25.5266±0.5904
24.3534±0.0059
24.5255±0.2069

Quantization Error
0.6677±0.0063
1.4835±0.0162
1.4949±0.0020

F Measure
0.8953±0.0124
0.8995±0.0000
0.8979±0.0044

Table 5. Comparison of Various Performance Indices for Mammographic Mass Data Set

QPSO FCM
QPSO K-Means
PSO K-Means

Inter Cluster Distance
21.6798±5.8964
21.5716±0.0174
21.3634±0.4812

Intra Cluster Distance
275.2412±17.1193
261.0328±0.0361
261.8542±1.1712

Quantization Error
6.2366±0.0386
7.3978± 0.0000
7.4319± 0.0153

F Measure
0.6910±0.0070
0.6855±0.0000
0.6851±0.0011

Table 6. Comparison of Various Performance Indices for Sonar Data Set

QPSO FCM
QPSO K-Means
PSO K-Means

Inter Cluster Distance
0.7381±0.0916
1.3264±0.0446
1.2750±0.0081

Intra Cluster Distance
19.2274±0.6306
17.0267±0.1275
16.8552±0.0181

Quantization Error
0.6851±0.0077
0.9329±0.2072
1.0002±0.1073

F Measure
0.5989±0.0345
0.5702±0.0025
0.5421±0.0145

Figure 1. Accuracy of Algorithms on Iris

Figure 5. Accuracy of Algorithms on Sonar

Figure 2. Accuracy of Algorithms on Breast Cancer
Figure 6. Clustering using FCM QPSO on Iris Dataset

Figure 3. Accuracy of Algorithms on Seed

Performance indicators such as Intercluster Distance,
Intracluster Distance and Quantization Error computed from
the results obtained in Tables 2-6 and that in Figures 1-5 imply
that FCM QPSO has promising performance. Accuracy
improvements of 1.556%, 0.151%, 0.825% and 5.061%
respectively over QPSO K-Means are obtained on the Iris,
Breast Cancer (Original), Mammographic Mass and Sonar data
using FCM QPSO. On the Seed data, the accuracy drops by
0.371% and 0.212% for FCM QPSO when compared to QPSO
K-Means and PSO K-Means, while recording an improvement
of 18.447% over traditional K-Means.

Figure 4.Accuracy of Algorithms on Mammographic Mass

The improvements in clustering accuracy and F-Measure
obtained in case of FCM QPSO are at the expense of increased
time complexity with respect to traditional K-Means based
implementations. For instance, a typical FCM QPSO
implementation with cluster numbers fixed a priori with the
fuzzifier m set as 2 results in approximately six times the
computational cost as compared to QPSO K-Means when run
on the Sonar dataset. Figure 6 shows a three dimensional
partially representative classification of Iris Dataset into three
distinct clusters along with the optimized cluster centres
computed using FCM QPSO.

VII. CONCLUSIONS AND FUTURE SCOPE
This paper makes an effort to compare and contrast the
accuracy of hard and soft clustering techniques such as KMeans and Fuzzy C-Means upon hybridization with the
standard, fully-connected quantum-behaved versions of the
swarm intelligence paradigm of PSO on a number of datasets.
FCM QPSO utilizes fuzzy membership rules of FCM and the
guaranteed convergence ability of QPSO, thus avoiding
stagnation in local optima in the multidimensional fitness
landscape. Future work will analyze supervised approaches to
mitigate the initial solution quality sensitivity in high
dimensional datasets and aim at developing automatic
techniques for detection of optimal cluster numbers and cluster
centres in search spaces with reduced dimensionality.
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