Journal of Signal Processing, Vol.18, No.6, pp.291-302, November 2014

PAPER

Network-Structured Particle Swarm Optimizer That Considers
Neighborhood Distances and Behaviors
Haruna Matsushita1 , Yoshifumi Nishio2 and Chi K. Tse3
1

Department of Electronics and Information Engineering, Kagawa University
2217–20 Hayashi-cho, Takamatsu, Kagawa 761–0396, Japan2
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan3
Department of Electronic and Information Engineering, Hong Kong Polytechnic University
Hung Hom, Kowloon, Hong Kong
E-mail: 1 haruna@eng.kagawa-u.ac.jp, 2 nishio@ee.tokushima-u.ac.jp, 3 encktse@polyu.edu.hk

Abstract

This study proposes a network-structured particle swarm optimizer (NS-PSO), which considers neighborhood distances. All particles of the NS-PSO are connected to adjacent particles in theneighborhood of topological space, and NS-PSO utilizes the connections between them not only to share
local best position but also to increase swarm diversification. Each NS-PSO particle is updated dependingon the positions of the local best and current best particles. In NS-PSO, the neighborhood distance in the
topological space from each particle to the current best position is also considered. This effect promotes thediversification of solutions and avoids the solutions from becoming trapped at local optima. Simulation results and comparisons with conventional particle swarm optimization show that the proposed NS-PSO caneffectively enhance the searching efficiency by measuring in terms of accuracy, robustness and parameterdependence. Furthermore, we consider various network topologies, grid, hexagonal, cylinder and toroidal.
We investigate their behaviors and evaluate the kind of topology that would be the most appropriate foreach benchmark.

Keywords: particle swarm optimization (PSO), swarm intelligent, neighborhood relationship, network topology

1.

Introduction

The particle swarm optimizer (PSO) [1] is an evolutionary algorithm used to simulate the movement of
the flocks of birds. Due to its simple concept, easy
implementation and quick convergence, PSO has attracted recent attention and finds wide applications invarious fields. In the PSO algorithm, the position ofeach particle is updated according to its personal best
position (pbest) and the global best position among
all the particles (gbest). However, the degrees of influence of both pbest and gbest are determined randomly in each iteration. In other words, the particles
of PSO [1] are fully connected and all the particles are
influenced by gbest in the same way, whichever particle becomes the global best. However, in real world,
various personal relationships exist, such as hierarchical relationships, those based on parent-child relationships, etc., and the degree of influence from respective
Journal of Signal Processing, Vol. 18, No. 6, November 2014

individuals depends on their relationship.
Various topological neighborhoods for PSO havebeen considered [2–6]. Researches have applied thering neighborhood, the von Neumann neighborhood,or some other topological neighborhood, and each particle shares its local best position among neighboring
particles in the topological space. In most PSOs using
local best, a connection between the particles is determined prior to the search, and the local neighbors aredefined not as particles located close to each other inthe search space, but as particles directly connectedto each other in the topological space. The neighborhoods of each particle do not change throughout
the searching but the particle distances in the search
space dynamically change according to the movementof the particles. This is because a huge amount ofcalculation is needed if the neighborhoods of the particles change depending on the particle distances inthe search space [7]. The connections between the291

particles increases convergence speed, and quick convergence helps the swarm to focus on the search in a
local region by exploiting the information of solutions
found in this region. Although it increases the searchability of PSO for unimodal functions that have no
local optima other than the global optimum, it causes
the solutions to be trapped at the local optima for
multimodal functions. To avoid this, a researcher proposed PSO [8] which tries to increase swarm diversification by changing connections between the particles
during the search. In PSO, however, we have to redetermine the local neighbors each time the connections
change. Furthermore, although various modified topological PSOs are certainly effective [9], the algorithms
are somewhat complicated, and it is hard for a person who is unfamiliar with PSO, to set appropriate
parameters and obtain the desired optimization performance.
This study proposes a new PSO algorithm with
topological neighborhoods—network-structured PSO(NS-PSO)—which considers neighborhood distances.
NS-PSO utilizes the connections between the particles
not only to share the local best but also to increaseswarm diversification. All NS-PSO particles are connected to adjacent particles in the neighborhood of
a topological space. The connected particles, namely
neighbors in the topological space, share their own
best position information. Furthermore, for each iteration, we find a winning particle whose fitness value isthe current best among all the particles and each particle is attracted to the winner depending on the neighborhood distance in the topological space. Therefore,
the nearer a particle is connected to the winner in the
topological space, the more strongly it is attracted to
the winner, even if its position in the search space isfar from the winner’s position. This effect promotes
the diversification of solutions and avoids the solutions
from becoming trapped at local optima without recalculating the neighborhood relationship during thesearch because the neighborhood distances between
all particles are uniquely determined and correspondto the topology. Although the neighborhood distanceis known as a learning tool in the self-organizing map(SOM) [10] which is an unsupervised neural network,
we can say that NS-PSO consists of a combination of
PSO and SOM.
We apply NS-PSO to various network topologies—
grid, hexagonal, cylinder, and toroidal. NS-PSO withvarious topologies are applied to seven benchmark
functions considering unimodal and multimodal functions. We confirm that NS-PSOs improve the optimization performance compared with the conventional
global best PSO [1] by measuring in terms of accuracy,
robustness, and parameter dependence. Furthermore,
we investigate their behaviors and evaluate the kind of
topology that would be the most appropriate for eachbenchmark. From the results, we find that the a circu292

lar topology is effective for simple unimodal functions
because this topology easily transmits the best position information of each particle to others. We also
confirm that a hexagonal topology is appropriate forcomplex multimodal functions because this topology
contains various kinds of particles and this effect promotes the diversification of solutions and avoids thesolutions from being trapped at local optima.
2.
2.1

Particle Swarm Optimizer (PSO)

Conventional PSO

PSO is a computational algorithm based on swarm
intelligence. In this algorithm, multiple potential
solutions called “particles” coexist. Each particle
has the following information: position and velocity. At each iteration, the particle flies toward its
own previous best position (pbest) and the best position among all the particles (gbest). The position vector of each particle i and its velocity vector are represented by X i = (xi1 , · · · , xid , · · · , xiD )and V i = (vi1 , · · · , vid , · · · , viD ), respectively, where(d = 1, 2, · · · , D), (i = 1, 2, · · · , M ).

(PSO1) (Initialization) Let a iteration step t = 0.
Randomly initialize the particle position X i and its
velocity V i for each particle i, and initialize pbest
P i = (pi1 , pi2 , · · · , piD ) with a copy of X i . Evaluate the objective function f (X i ) for each particle iand find gbest P g = (pg 1 , pg 2 , · · · , pg D ) with the best
fitness value among all the particles;
g = arg min{f (P i )}i(1)

(PSO2) Evaluate the fitness f (X i ). For each particle i, update pbest as P i = X i , if f (X i ) < f (P i ).
Update gbest P g , if needed.
(PSO3) Update V i and X i of each particle i depending on its pbest and gbest according to the following:vid (t + 1) = wvid (t) + c1 rand(·) (pid − xid (t))
+ c2 Rand(·) pg d − xid (t) (2)xid (t + 1) = xid (t) + vid (t + 1)
where w is the inertia weight determining the amountof previous velocity of the particle is preserved. c1 andc2 are two positive acceleration coefficients, generally
c1 = c2 . rand(·) and Rand(·) are two uniform random
numbers from U (0, 1), and they are different values atrespective dimensions d.
(PSO4) Let t = t + 1, and go back to (PSO2).
2.2

PSO with grid topology

The most traditional PSO [1] is the fully connected
PSO whose particles are directly connected to each
Journal of Signal Processing, Vol. 18, No. 6, November 2014

(a)
Grid
topology(b)
Hexagonal
topology(c)
Cylindertopology(d)Toroidal
topology

Fig. 1 Different network topologies with 10×10 particles: Connections between the directly connected particles are represented by black lines and create “neighbors”. (a) Each particle has four neighbors, except
particles located at the edges of the topology, (b) Eachparticle has six neighbors, except particles located at
the edges of the topology, (c) In addition to the grid
topology, particles on the top and bottom edges areneighbors, (d) In addition to the cylinder topology,
particles on the left and right edges are neighbors.

other. However, PSOs having topological neighborhoods between the particles have been proposed. inwhich each particle is affected by the local best (lbest)in its local neighborhood.
We should define the meaning of neighborhood. In
some research, the neighboring particles were definedas those particles with similar cost values or as theparticles whose position are close to each other in the
search space. However, in this study, the particles
directly connected in the topological space are definedas neighborhood, even if their positions or their costs
are widely different. For an example we consider agrid topology shown in Fig. 1(a). Each particle has
four local neighbors, except the edge particles, whichhave two or three neighbors. A local best position lbest
Li = (li1 , li2 , · · · , liD ) of each particle i is the position
with the best fitness value among its neighbors.
In the PSO algorithm with grid topology, the particles are influenced by lbest instead of pbest;vid (t + 1) = wvid (t) + c1 rand(·)(lid − xid (t))(3)
+ c2 Rand(·)(pgd − xid (t))
3.

Network-Structured PSO Considering
the Neighborhood Distances

The particles in a conventional PSO are fully connected, however, the connections are used only forsharing each best position. Thus, a conventional PSO
has a star connection with gbest.
The NS-PSO algorithm is based on structures of
the conventional PSO and SOM. NS-PSO has the following four key features.
1) All the particles are connected to adjacent particles
in the neighborhood of a topological space.
Journal of Signal Processing, Vol. 18, No. 6, November 2014

2) The particles share the local best position (lbest)
between the directly connected particles (neighbors),
and lbest is used instead of pbest.
3) Instead of gbest which is the global best positionso far, NS-PSO uses the current best position cbest ineach iteration.
4) NS-PSO introduces a neighborhood function tocontrol the degree of attraction of each particle to thecurrent best particle, which is called the “winner.”
The neighborhood function depends on the distance
between each particle, and the winner in the topological space.
With these features, each NS-PSO particle is updated
depending on its local best position (lbest) and the current best position (cbest) in the search space as wellas the neighborhood distance between each particle
and the winner in the topological space. Differences
between conventional PSO, PSO with grid topology,
and NS-PSO are summarized in Table 1. In this study,
we use various topologies as shown in Fig. 1 and investigate their behaviors. The grid and the hexagonal
topologies in Figs. 1(a)–(b) are sheet shapes, and thecylinder and toroidal topologies in Figs. 1(c)–(d) arecircular in shape.
(NS-PSO1) (Initialization) Consider an iteration
step t = 0. Initialize X i , V i , P i , and P g according to (PSO1). Define the global best particle g as awinner c, which is a particle with P g . Find lbest Liof each particle i.
(NS-PSO2) Evaluate the fitness f (X i ) and find awinner c with the best fitness among all the particles
at current time t;
c = arg min{f (X i (t))}i(4)

The winner c changes dramatically with time. Notethat the winner is different from the particle with gbest
in the definition, namely X c = P g . cbest has the best
fitness value at the current time (Eq. (4)); however,gbest has the best fitness value so far (Eq. (1)).
(NS-PSO3) Update each pbest P i , if needed. Update
gbest as P g = X c , if f (X c ) < f (P g ).
Update each lbest Li which is the local best between the particle i and its directly connected neighbors, denoted as Ni ;Li = arg min{f (P j )}, j ∈ {i, Ni }j

(5)

(NS-PSO4) Update V i and X i of each particle idepending on its lbest, the winner’s position X c , andthe neighborhood distance in the topological space between i and the winner c, according to the following:vid (t + 1) =xid (t + 1) =wvid (t) + c1 rand(·) (lid − xid (t))+ c2 hc,i (xcd − xid (t))(6)
xid (t) + vid (t + 1)
293

Table 1
MethodConnection relationships
between particles

Differences between PSO, PSO with grid topology and NS-PSO
PSO

NS-PSO (Grid)

Fully connected

Grid topology

previous velocity V i
inertia weight; w
pbest P ilbest Lirandom; c1 rand(·)
random; c1 rand(·)
gbest P g
cbest X crandom; c2 Rand(·)
neighborhood dist. between c and i; hc,i
Sharing lbest and using
Sharing gbest
Sharing gbest and lbest
neighborhood dist. between particles

Influenced factors
and its influence rate

Object of connections
between particles

topological space from cbest; i.e., NS-PSO considers
the current connection to be important. However, NSPSO has no problem using gbest instead of cbest andcan obtain better results.
(NS-PSO5) Let t = t + 1, and return to (NS-PSO2).

hc,i
1

0

Fig. 2||ri - rc||2

Neighborhood function defined by Eq. (7)where hc,i is the fixed neighborhood function shownin Fig. 2 and is defined byr i − r c 2
hc,i = exp −(7)2σ 2where r i − r c  is called the “neighborhood distance”
and is the distance between particles c and i with thegiven topology shown in Fig. 3. It is important to notethat the neighborhood distance r i − r c  is different
from the distance in the search space, represented by
X i − X c . Actually, we do not need to calculate
the neighborhood distance r i − r c  at each iteration
because the neighborhood distances between all theparticles are uniquely determined and correspond to
the topology. The fixed parameter σ corresponds tothe width of the neighborhood function. Therefore,a large σ strengthens the particles’ spreading force tothe whole space, and a small σ strengthens their convergent force toward the winner. NS-PSO uses lbest
and the neighborhood distance from the winner c. In
other words, if the particle is connected far from thewinner in the topological space, it is not highly attracted to the winner, even if its position is near it in
the search space. In addition, in the NS-PSO algorithm, we use not the global best gbest but also thecurrent best cbest, although we only use gbest to decode the final best solutions. This is because NS-PSO
does not use the distance in the search space fromgbest, but it uses the neighborhood distance in the294

NS-PSO
Grid, Hexagonal, Cylinder,
Toroidal topology

4.

Experimental Results

To evaluate the performance of NS-PSO with various topologies, we use seven benchmark optimization
problems summarized in Table 2. f1 , f2 , f3 and f4
are unimodal functions, and f5 , f6 and f7 are multimodal functions with numerous local minima. All thefunctions have D variables, and in this study we set
D = 50. The optimum solution X ∗ of Rosenbrock’s
function f2 is [1, 1, . . . , 1], and X ∗ of the other functions are all [0, 0, . . . , 0]. The optimum fitness valuesf (X ∗ ) of all the functions are 0.
We compare the proposed NS-PSOs with threekinds of PSO—the conventional PSO, PSO with grid
topology and NS-PSO using pbest instead of lbest.
PSO: This is the conventional PSO whose particles
are fully connected. Each particle information is updated depending on its pbest and gbest according to
Eq. (2).
PSO with grid topology: This is a PSO whose particles are connected depending on grid topology. Eachparticle information is updated depending on lbest andgbest, according to Subsection 2.2.
NS-PSO with various topologies: This is the
proposed method. The particles are connected toeach other according to four different topologies—grid,
hexagonal, cylinder, and toroidal. Each particle information is updated depending on lbest—the position of
the current best (namely, winner)—and the neighborhood distance in the topological space from the winner, according to Eq. (6). The neighborhood distancesand lbest are determined depending on each topology,
according to Fig. 3.
NS-PSO using pbest: This is NS-PSO with grid
topology; however, it does not use lbest but uses pbest.
Each particle information is updated depending on
Journal of Signal Processing, Vol. 18, No. 6, November 2014

1
1

2

3

1

i (1,1)2

13

i (1.5,)2

3i (1,1)

1

1

2

2

3

3

3

c (3,

4

)

4

4
c (3,4)c (3,4)(c) Cylinder √topology.
r i − r c  = 5.

(d) Toroidal √topology.
r i − r c  = 2.

c (3,4)(a) Grid √topology.
r i − r c  = 13.

(b)
Hexagonal
r i − r c  = 3.

topology.

3i (1,1)

1

2

2

Fig. 3 How to measure neighborhood distance between nodes i and c in case with different topologies: Theneighborhood distance r i − r c  between i and c is calculated as Euclidean distance, depending on the topologicalgrid coordinates of the NS-PSO particles based on the given topology. Consider, a 4 (= n rows) × 3 (= m columns)
network. (a) The network grid coordinates of the particles are simply
grid, (b) The x-coordinates of every other√row are shifted by +0.5, and the y-coordinates are multiplied by 0.75. This is executed to make the distances ofa unit to all its six neighbors equal, (c) The particles in the 1st and nth rows are neighbors, (d) In addition to thecylinder topology, the particles in the 1st and mth columns are neighbors.

Table 2Function nameSphere func.;

Seven test functions. D = 50

Test function
DX

f1 (X) =Initialization space2
xd ,[−5.12, 5.12]D

d=1

Rosenbrock’s func.;

f2 (X) =

D−1
X „
d=1

3rd De Jong’s func.;

f3 (X) =DX|xd |,

D
X

dxd ,

d=1

4th De Jong’s func.;

f4 (X) =

«
”2
“
2
2
+ (1 − xd ) ,100 xd − xd+1[−2.048, 2.048]D[−2.048, 2.048]D

4[−1.28, 1.28]D

d=1

Rastrigin’s func.;

D “
”
X
2
xd − 10 cos (2πxd ) + 10 ,f5 (X) =[−5.12, 5.12]D

d=1

Ackley’s func.;

f6 (X) =D−1
X

20 + e − 20ed=1Stretched V
sine wave func.;

f7 (X) =

D−1
X

2

2

(xd + xd+1 )

d=1

r
“
”
−0.2 0.5 x2 +x2
d
d+1

0.25

“

2

2

0.5(cos(2πxd )+cos(2πxd+1 ))−e

2

1 + sin (50(xd + xd+1 )lbest, the position of the winner and the neighborhood
distance in the topological space from the winner, isgiven as follows:vid (t + 1) = wvid (t) + c1 rand(·)(pid − xid (t))(8)
+ c2 hc,i (xcd − xid (t))
The population size M is set to 36 in PSOs, and the
network size is 6 × 6 in NS-PSO with each topology.
For PSOs and NS-PSOs, the parameters are set as
w = 0.7 and c1 = c2 = 1.6 because they were themost favorable parameters to the conventional PSO ina preliminary experiment. The neighborhood radius
σ of all NS-PSOs is 1.5. We perform the simulations
100 times repeatedly for all the optimization functions
Journal of Signal Processing, Vol. 18, No. 6, November 2014

0.1

”
) ,

!

,

[−30, 30]D[−10, 10]Dwith 3000 iterations.
4.1 Compared performances depending on problems
The performances with the minimum and mean
fitness values, and their standard deviations over 100
independent trials on seven functions are listed inTable 3. The best results of the mean fitness values among all the algorithms are shown in bold. All
the NS-PSOs with their various topologies surpassed
the conventional PSO on all seven functions. In fact,
apart from NS-PSO with grid topology on f3 , the conventional PSO never obtained better results than any
proposed NS-PSOs, which consider neighborhood distance and use lbest. Furthermore, because the stan295

Table 3

Mean
Minimum
Std
Mean
Minimum
Std
Mean
Minimum
Std
Mean
Minimum
Std

PSO
Grid
pbest
lbest
1.94e-19
8.35e-16
1.56e-26
1.13e-20
1.44e-18
4.10e-15
51.61
59.17
19.35
4.56
25.40
34.03
2.05e-04
6.64e-08
7.89e-122.00e-11
1.05e-03
3.18e-07
2.77e-35
2.95e-25
9.24e-42
1.34e-30
2.39e-34
1.81e-24

pbest
8.69e-03
5.77e-07
0.020
47.85
40.37
9.94
0.17
9.05e-04
0.26
1.71e-05
5.28e-14
7.87e-05

Mean
Minimum
Std
Mean
Minimum
Std
Mean
MinimumStd

149.7666.66
26.83
231.79100.49
64.00
64.65
40.18
13.18

94.88
59.70
23.30
300.30187.76
55.27
50.48
29.83
8.85

f
f1
f2
f3
f4

f5
f6
f7

Comparison results of PSOs and NS-PSOs on seven test functions

170.8788.55
37.85
152.46
10.32
69.76
76.15
41.31
16.54

Table 4f

Best
Result

f1
f2
f3
f4

2.94e-25
39.55
5.74e-08
1.68e-44

f5
f6
f7

87.26
152.46
34.86

62.50
79.33
29.79

Toroidallbest
3.03e-25
6.19e-31
2.23e-24
39.55
26.85
6.22
5.74e-08
3.95e-122.42e-07
1.68e-44
1.73e-52
6.07e-44

112.9948.75
22.71
228.77
84.88
69.89
39.49
12.79
11.06

90.82
52.73
19.42
161.93
45.63
57.81
34.86
16.62
9.59

87.26
41.79
22.68
193.78
76.80
55.56
39.11
13.39
11.97

109.8833.83
82.06
216.03
68.44
61.57
38.54
17.22
11.16

Grid
lbest
8.35e-16
19.62
9.01e-09
2.95e-25

Cylinder
lbest
0
2.69
1.30e-08
1.50e-41

Toroidallbest
9.38e-27
0
0
0

83.62
0
41.29

0
41.32
4.26

22.62
63.56
3.69

Difference from the best result
NS-PSO
Grid
Hexagonal
pbest
lbest
lbest
8.69e-03
5.69e-25
4.91e-22
8.29
1.30
2.71
1.70e-01
2.05e-02
1.21e-05
1.71e-05
1.67e-41
2.23e-39
7.62
147.8415.62

dard deviations of NS-PSOs are better than the conventional PSO on all functions, we can say that therobustness of NS-PSO was improved. PSO with grid
topology highly improved the performance of conventional PSO on f3 and f6 , and compared with theproposed NS-PSOs with various topologies, the performances were much better. However, because theresults of PSO with grid topology were worse thanthe conventional PSO on most other problems, we
can say that PSO with grid topology greatly depends
on the problem. On the other hand, NS-PSO whichuses pbest instead of lbest, obtained better results thanconventional PSO on f2 , f4 , f5 and f7 , although theperformances were much worse than PSO on otherproblems. Furthermore, in most cases, the proposed
NS-PSOs using lbest were better than NS-PSO using
pbest. From these results, we can conclude that using
only the network topology (lbest) or the neighborhood
function (hi,c ) does not make the performance better,but using both is effective.
Next, we consider relationships between performance and topologies. Table 4 shows the best result
from the seven algorithms and the differences between
the best result and the results of the respective algorithms. NS-PSO with grid topology, hexagonal topol296

Cylinderlbest
2.94e-25
1.58e-31
1.42e-24
42.24
28.19
11.62
7.04e-08
5.86e-21
2.23e-07
1.50e-41
3.01e-49
1.28e-40

Differences from the best result

PSO
pbest
1.94e-19
12.06
2.05e-04
2.77e-35

lbest
8.63e-25
4.85e-31
4.97e-24
40.85
24.96
7.20
2.05e-02
4.01e-120.20
1.67e-41
5.17e-49
1.55e-40

NS-PSO
Hexagonal
lbest
4.92e-22
2.51e-28
3.58e-21
42.27
33.33
9.83
1.21e-05
3.19e-12
1.19e-04
2.23e-39
2.56e-46
1.32e-38

Grid

25.73
76.31
4.63

3.56
9.46
0

ogy, cylinder topology and toroidal topology achieve
the best values 0, 1, 2 and 3 times, respectively. For
the unimodal functions of f1 , f2 , f3 and f4 , NS-PSO
with toroidal topology obtained the best results mostfrequently, and the cylinder topology delivered a verysmall difference from the best results. For multimodal
functions f5 , f6 and f7 , NS-PSO with hexagonal topology obtains the best result on f7 and second-best results, which are small differences from the best results,
on f5 and f6 . Therefore, we can say that the hexagonal topology is the most effective for the multimodal
functions. NS-PSO with grid topology achieves goodstable results for both unimodal and multimodal functions, even if it cannot obtain the best results among
NS-PSOs for any benchmarks. From these results, we
can conclude that the circular-shaped NS-PSO withtoroidal topology is more suitable for unimodal functions, and the sheet-shaped NS-PSO with hexagonal
topology is more effective for multimodal functions.
4.2

Comparison of convergence curves

Figure 4 shows the mean gbest values of every iteration over 100 trials for seven test functions. We consider unimodal functions f1 –f4 . All the NS-PSOs conJournal of Signal Processing, Vol. 18, No. 6, November 2014

10

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroidal)

0

10

−10

10

Function Value log(f(x))

Function Value log(f(x))

10

−20

10

−30

10

0

500

1000
1500
2000
Number of Generation t

2500

3000

10

5

10

4

10

3

10

2

10

1

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroidal)

0

500

1000
1500
2000
Number of Generation t(a) f1
Function Value log(f(x))

Function Value log(f(x))

0

10

−5

−10

10

0

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroidal)

500

1000
1500
2000
Number of Generation t

2500

3000

10

20

10

0

10

−20

10

−40

10

−60

0

500

1000
1500
2000
Number of Generation t

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)101NS−PSO (toroidal)

0

500

1000
1500
2000
Number of Generation t

2500

3000

10

2.8

10

2.6

10

2.4

10

2.2

0

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroidal)

500

1000
1500
2000
Number of Generation t(e) f52500

Function Value log(f(x))

2

3000

(d) f43

Function Value log(f(x))

Function Value log(f(x))

10

2500

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroidal)(c) f310

3000(b) f2

5

10

10

2500

3000

10

3

10

2

10

1

0

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroidal)

500(f) f6

1000
1500
2000
Number of Generation t

2500

3000

(g) f7

Fig. 4 Convergence curves of PSO and four NS-PSOs on seven 50-dimensional functions: The curves show themean f (P g ) of 100 trials, corresponding to each iteration.

verged faster than the conventional PSO and, in particular, NS-PSO with toroidal topology was fastest.
This is attributed to a difference in the average neighborhood distance between the particles. The particlesof the toroidal topology are connected to each other at
a short distance, thus all the particles of the toroidal
topology are connected via fewer particles than inother topologies. This effect makes communication
between the particles faster than in other topologies,
therefore the particles of the toroidal topology canconverge quickly. This indicates that the circularshaped NS-PSO with toroidal or cylinder topology issuitable for unimodal functions that have no local optima other than the global optimum.
Next, we consider multimodal functions f5 –f7 . Almost all NS-PSOs converged faster than the conventional PSO, and all the NS-PSOs obtained better results than the conventional PSO for all multimodal
functions. This is because NS-PSO uses lbest andthe neighborhood distance according to the connections between particles. It increases the diversification of the particles and avoids the NS-PSO solutions
from becoming trapped at local optima. In particular, NS-PSO with hexagonal topology obtained thebest or second-best results for all multimodal functions. This is attributed to the fact that the diversityof the hexagonal topology is highest because its particles have various number of neighbors and various
average neighborhood distances between other particles. This indicates that the hexagonal topology is
Journal of Signal Processing, Vol. 18, No. 6, November 2014

suitable for multimodal functions.
The effectiveness of using neighborhood functions
and relationships between different topologies and performance are discussed in detail in Section 4.4.
4.3 Parameter dependence
To investigate the ease of use of NS-PSO, we consider the degree of parameter dependence. First, weinvestigate the effects of the parameters—the inertia
weight w and the acceleration coefficients c1 and c2 on
performance quality, and their sensitivity.
Figure 5 shows the mean fitness values of gbest,f (P g ), with different w over 100 trials for all the test
functions with 50 dimensions. The fixed parameters
are the same as that in the above simulations. We cansee that all the NS-PSOs achieved better performances
than the conventional PSO, even if the inertia weight
w is varied. If we decrease or increase w by just 0.1,
the performance of conventional PSO turns drasticallyworse, as shown in Figs. 5(a)–(e). However, NS-PSOs
obtained better results and were more robust than theconventional PSO if w is set as a small value (w ∈[0.1, 0.6]) [11], particularly with multimodal functions,as shown in Figs. 5(e)–(g).
Figure 6 shows the mean fitness values of gbest,f (P g ), with different c1 (= c2 ) over 100 trials for all
the test functions with 50 dimensions. Note that for f3
and f5 , the best gbest values of the conventional PSO
among all the c1 (= c2 ) are smaller than NS-PSOs, as
297

10

10

10

0

4

10

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

10

−10

10

−20

10

−30

10

−40

log(f(x))

log(f(x))

3

10

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

2

10

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10

0.8

0.1

0.2

0.3

0.4

(a) f1
10

5

10

0

0.5

0.6

0.7

0.8

0.5

0.6

0.7

0.8

w

w

(b) f2
20

10

0

10

10

10

log(f(x))

log(f(x))−20

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)−5

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

−60

−10

−80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10

0.8

0.1

0.2

0.3

0.4

w

w

(c) f3(d) f4

3

3

3

2

10

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)
1

10

log(f(x))

10log(f(x))

log(f(x))

−40

10
10

10

10

10

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

2

10

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

10

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

2

10

1

0.1

0.2

0.3

w

(e) f5

0.4

0.5
w

0.6

0.7

0.8

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

w

(f) f6(g) f7

Fig. 5 Relationship between the inertia weight w and the performances of PSO and NS-PSOs: These results
show the mean f (P g ) of 100 trials. Fixed parameters; c1 = c2 = 1.6, σ = 1.5.

shown in Figs. 6(c) and (e). However, the conventional PSO is extremely sensitive to small changes in
c1 (= c2 ), but the performances of NS-PSOs are stable. For other functions, NS-PSOs can obtain better
results than PSO, even if c1 (= c2 ) is varied. In particular, 0.4 ≤ w < 0.7 and 1.5 ≤ c1 (= c2 ) < 1.6 areconsidered appropriate for NS-PSO regardless of the
topologies. From these results, we can conclude thatthe parametric dependence of NS-PSOs is not stronger
than that of the conventional PSO.
Next, we consider σ which is an important parameter for NS-PSO and corresponds to the width of theneighborhood Gaussian function. Figure 7 shows themean fitness values of gbest f (P g ) with different σ over
100 trials for all test functions with 50 dimensions.
From these figures, σ seems to be more sensitive thanthe other two parameters for NS-PSO. However, all
the benchmark functions obtained satisfactory results,
regardless of topologies, when σ √∈ [1, 1.5]. Therefore,
we can presume that σ ≤ 1.5(= M /2) is consideredappropriate for NS-PSO regardless of the topologies.
It is also important to investigate the relationship between the number of particles M and its appropriate
value of σ; however, these could form important topics
for further research.
The results of this investigation show that the parameter dependence of NS-PSO is not stronger thanthat of the conventional PSO, and its degree is not sodepending on differences between topologies. We can298

conclude that we can easily use NS-PSO without complicated parameter settings even though NS-PSO has
a network structure. It shows that even if a person is
unfamiliar with PSO, they can obtain better results
than the conventional PSO by using NS-PSO.
4.4

Behaviors of NS-PSO with various topologies

4.4.1 Effect of the neighborhood function
The convergence speed of NS-PSO is almost sameor slower than the conventional PSO. In the conventional PSO, according to Eq. (2), the particles movein the direction of gbest and pbest; however, the respective degrees of movement are decided randomlyon each iteration. On the other hand, as can be seenfrom Eq. (6), NS-PSO uses the neighborhood Gaussian function which controls the degree of attraction ofeach particle to the winner. Then, the particles moveaccording to the neighborhood distance between thewinner and themselves. In the winner’s neighborhood
particles, the neighborhood distance ||r i −r c || is small
and a value of the neighborhood function hc,i is large,
as shown in Fig. 2. Therefore, they are strongly attracted by the winner beyond the winner, as shown in
Fig. 8(a), so that they spread out through the whole
space. For the particles that are not 1-neighbors of thewinner but are connected near the winner in the topological space, the degree of attraction to the winner isstrong (Fig. 8(b)). The other particles fly toward their
Journal of Signal Processing, Vol. 18, No. 6, November 2014

10

10

0

10

−10

10

−20

10

−30

5

10
PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

410
log(f(x))

log(f(x))

10

3

10

2

10

1

1.5

1.55

1.6

1.65

1.7

1.75
c1=c2

1.8

1.85

1.9

1.95

10

2

1.5

1.55

1.6

1.65

1.7

(a) f1
5

0

10

−5

1.85

1.9

1.95

2

1.8

1.85

1.9

1.95

2

20

log(f(x))

10

1.8

10
PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

010log(f(x))

10

1.75
c1=c2

(b) f2

−20

10

−40

10

10

−10

−60

1.5

1.55

1.6

1.65

1.7

1.75
c1=c2

1.8

1.85

1.9

1.95

10

2

1.5

1.55

1.6

1.65

(c) f3

10

10

2

PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

1

1.5

1.55

1.6

1.65

1.7

1.75
c1=c2

1.8

1.85

1.9

1.95

2

1.75
c1=c2

3

10

2

10

1

3

10

log(f(x))

10

3log(f(x))

log(f(x))10

1.7(d) f4PSO
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

PSO
NS−PSO (grid.)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

2

10

1

1.5

1.55

1.6

1.65

(e) f5

1.7

1.75
c1=c2

1.8

1.85

1.9

1.95

10

2

1.5

(f) f6

1.55

1.6

1.65

1.7

1.75
c1=c2

1.8

1.85

1.9

1.95

2

(g) f7

Fig. 6 Relationship between the acceleration coefficients c1 = c2 and the performances of PSO and NS-PSOs:
These results show the mean f (P g ) of 100 trials. Fixed parameters; w = 0.7, σ = 1.5.

lbest, as shown in Fig. 8(c), because c2 hc,i is small. In
other words, the roles of the NS-PSO particles are
determined by the connection, and the neighborhood
function easily produces particle diversity. These effects avert premature convergence, and the NS-PSO
particlescan easily escape from the local optima.
4.4.2 Discussion on the evaluation of each topology
Let us consider the topology and its behavior interms of the average neighborhood distance L, whichis also known as the average node-to-node distance,
and the average number of neighbors N which is also
called “average node degree.” The average neighborhood distance L is measured by
1
r i − r j (9)L=
M (M − 1)i=jwhere M is the number of nodes, namely particles,
and r i − r j  is the distance between node i and node
j in the topological space and is calculated according
to Fig. 3. The node degree of a node i, namely the
number of neighboring particles of the particle i, ismeasured by
Ni =

M


aij

(10)

j=1


1, node i and j are directly connectedaij =0, otherwise
Journal of Signal Processing, Vol. 18, No. 6, November 2014

therefore, the average node degree of the entire network is given as follows:
M

N=

1 
Ni
M i=1

(11)

On a 6×6 map, the average neighborhood distance
L of respective topologies—grid, hexagonal, cylinder,
and toroidal, are 3.17, 2.98, 2.80, and 2.40, respectively, and its histogram-based number of particles isshown in Fig. 9(a). The average node degree N of
respective topologies—grid, hexagonal, cylinder andtoroidal, are 3.33, 4.72, 3.67, and 4, and its histogramis shown in Fig. 9(b). From the simulation resultssummarized in Table 3 and the convergence curves
shown in Fig. 4, the circular-shaped NS-PSO is suitable for unimodal functions that have a unique optimum, and the hexagonal topology is suitable for multimodal functions. Because the cylinder and toroidal
are both circular topology, the individuality of eachparticle is almost the same, as shown in Fig. 9. In
the cylinder topology, the averages of the neighborhood distances between respective particles are of only
three kinds and the node degrees of respective particles are only two kinds. Especially in the toroidal
topology, Li and Ni are completely the same for anyof the particles, and they are 2.4 and 4, respectively.
This fact means that the features of the particles of
the circular-shaped NS-PSO are similar and without
much difference in their behavior, no matter which
299

3

10

0

10

log(f(x))

log(f(x))

NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

−10

10

NS−PSO (rect.)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

2

10

−20

10

1

−30

10

1

1.2

1.4

1.6

1.8

2
σ

2.2

2.4

2.6

2.8

10

3

1

1.2

1.4

1.6

(a) f10log(f(x))−410log(f(x))

NS−PSO (grid.)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

10

−6

10

−8

10

−10

10

1

1.2

1.4

1.6

1.8

2
σ

2.2

2.4

2.6

2.8

3

10

20

10

0

3

10

−20

10

−40

10

−60

1

1.2

1.4

1.6

NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)
1.4

1.6

1.8

2
σ

2.2

2.4

2.6

2.8

3

2.2

2.4

2.6

2.8

3

3

10
NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)

2.5

10

NS−PSO (grid)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)
2

10

2.3

2.1

1.2

2.8

10

1

1

2.6

(d) f4log(f(x))

log(f(x))

log(f(x))

2

2.4

2
σ

2.9

2.7

10

1.8

10

10

10

2.2

NS−PSO (grid.)
NS−PSO (hex.)
NS−PSO (cyl.)
NS−PSO (toroid)(c) f3102σ(b) f2

10

−2

1.8

3

10

1

1

1.2

1.4

1.6

(e) f5

1.8

2
σ

2.2

2.4

2.6

2.8

3

10

1

1.2

1.4

(f) f6

1.6

1.8

2
σ

2.2

2.4

2.6

2.8

3

(g) f7

Fig. 7 Relationship between the width of the neighborhood function σ and the performances of NS-PSOs: These
results show the mean f (P g ) of 100 trials. Fixed parameters; c1 = c2 = 1.6, w = 0.7.

(a)

(b)

(c)

Fig. 8 Examples of updating the behaviors of NS-PSO depending on the neighborhood distance between thewinner and each particle: The degree of attraction to cbest X c is determined by c2 hc,i according to Eq. (6). Notethat rand(·) is [1, 1, · · · , 1]D for easy understanding and c2 > 1. (a) Case in which a particle i is directly connectedto the winner c in the topological space. Since hc,i is large value close to 1 and c2 hc,i > 1, the particle i is strongly
attracted by X c and goes beyond it, (b) Case in which a particle i is not directly connected but is connected near
the winner c in the topological space. The particle i is attracted by X c , (c) Case in which a particle i is located
far from the winner c in the topological space. The particle i is little influenced by X c since hc,i is a small value.

any particle becomes the winner. Furthermore, L of
the toroidal topology is the smallest in the four NSPSOs. From these effects, it is easy to transmit the
information of lbest to all particles; therefore, circular topology is effective for simple unimodal function.
However, premature communication produces premature convergence; then, the toroidal topology easily
goes into the local optima in multimodal functions.
On the other hand, Fig. 9 shows that NS-PSO withhexagonal topology contains various kinds of particles
that have different neighborhood distances and dif300

ferent sizes of local neighbors. These effects produce
diversity of particles and avert premature convergencealthough the communication speed of the particles is
kept in moderation because N is big. Therefore, theparticles of NS-PSO with hexagonal topology avoid
becoming trapped in the local optima. Furthermore,
Fig. 7 shows that the parameter dependence on σ ofthe hexagonal topology is weaker than that in other
topologies. From these considerations, we can conclude that the sheet-shaped topology as the hexagonal topology, in which the value of Li and Ni of each
Journal of Signal Processing, Vol. 18, No. 6, November 2014

40
grid
hex.
cyl.
tor.

Number of particles

35
30
25
20
15
10

30
25
20
15
10
5

5
0
2

grid
hex.
cyl.
tor.

35
Number of particles

40

2.5
3
3.5
4
Average neighborhood dist. L
(a)

0
2

3
4
5
Node degree N

6(b)Fig. 9 Topology features. (a) Average neighborhood
distance L, (b) Node degree Ni

particle is less bias and the average node degree N is
high, is suitable for multimodal functions and obtains
better results in unimodal functions than the conventional PSO.
5.

Conclusions

This study has proposed a new PSO with topological neighborhoods—NS-PSO, which considers neighborhood distances. All the particles of NS-PSO areconnected to adjacent particles in the neighborhoodof a topological space. NS-PSO utilizes the connections between the particles not only to enhance thelocal search, but also to increase the diversificationof the solutions. In NS-PSO, the directly connectedparticles share the information of their own best position. Furthermore, each particle moves dependingon the neighborhood distance in the topological space
between it and a current best particle called the “winner.” Therefore, the nearer the particle is connected to
the winner in the topological space, the more strongly
the particle is attracted to the winner, even if its position in the search space is far from the winner’s position.
In the simulation results, we have confirmed the accuracy of PSO and NS-PSO by applying seven benchmark functions, and it is clear that the overall performance of NS-PSO is efficient because the optimization
results and their standard deviations improved fromthe conventional PSO on all the benchmarks. In addition, the simulation results showed that by using only
the network topology (lbest) or the neighborhood function does not make the performance better, but using
both is effective. Because the parameter-dependenceof NS-PSO is weaker than that of the conventional
PSO, we can easily use NS-PSO without complicatedparameter settings. It means that, even if a person is
not familiar with PSO, they can obtain better results
than the conventional PSO by using NS-PSO.
Furthermore, we have applied NS-PSO to various
Journal of Signal Processing, Vol. 18, No. 6, November 2014

topologies and confirmed that NS-PSOs with various
topologies are more effective than other PSOs that
do not use neighborhood distance. In addition, we
have found that toroidal and hexagonal topologies are
suitable for unimodal and multimodal functions, respectively.
The utilization of neighborhood distance has possibilities for application to the updating of equations forother PSOs with topological neighbors since the NSPSO updating rule is not complicated. Moreover, various simulation results and investigations into NS-PSO
behaviors mean that their optimization performance
may be improved by using neighborhood functions.
This could form an important topic for further research. In addition, other important future studies are
as follows: an investigation of the relationship between
the number of particles and the appropriate topology
and a comparison of NS-PSO with other PSOs with
topological neighbors.

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particle swarm: simpler, maybe better, IEEE Trans. Evolut. Comput., Vol. 8, No.3, pp. 204–210, 2004.
[5] J. Lane, A. Engelbrecht and J. Gain: Particle swarm optimization with spatially meaningful neighbours, Proc. ofIEEE Swarm Intelligence Symposium, pp. 1–8, 2008.
[6] S. B. Akat and V. Gazi: Particle swarm optimization
with dynamic neighborhood topology: Three Neighborhood
Strategies and Preliminary Results, Proc. of IEEE SwarmIntelligence Symposium, pp. 1–8, 2008.
[7] M. Iwamatsu: Multi-species particle swarm optimizer for
multimodal function optimization, IEICE Trans. Inf. andSyst., Vol. E89-D, No. 3, pp. 1181–1187, 2006.
[8] W. H. Lim and N. A. Mat Isa: Particle swarm optimization
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fixed-structure robust controllers using a constrained particle swarm optimizer with cyclic neighborhood topology, Expert Systems with Applications, Vol. 40, Issue 9, pp. 3595–
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1461, 2002.

301

Haruna Matsushita
received
the B.E., M.E., and Ph.D. degrees
in electrical and electronic engineering from Tokushima University,
Tokushima, Japan, in 2005, 2007
and 2008, respectively. During October 2008–March 2010, she was a
Postdoctral Resarch Fellow (PD) of
the Japan Society for the Promotion
Science (JSPS) at the same university. From April 2010 to December 2010, she worked as a Postdoctral Resarch Fellow (PD) of Japan
Society for the Promotion Science(JSPS) at Hosei University. Since January 2011, she has been
currently working at Kagawa University as Assistant Professor.
Her research interests include nonlinear optimization, theory
and application of swarm intelligence, and self-organizing maps.
She is a member of the IEEE, the IEICE, and the RISP.

Yoshifumi Nishio
received
the B.E., M.E., and Ph.D. degrees inelectrical engineering from Keio University, Yokohama Japan, in 1988,
1990, and 1993, respectively. In
1993, he joined the Department
of Electrical and Electronic Engineering at Tokushima University,
Tokushima Japan, where he is currently a Professor. From May 2000
he spent a year in the Laboratory of
Nonlinear Systems (LANOS) at the
Swiss Federal Institute of Technology Lausanne (EPFL) as a Visiting Professor. His researchinterests are in the areas of nonlinear circuits engineering, including analysis and application of chaos in electrical circuits,
analysis of synchronization in coupled oscillatory circuits, development of analyzing methods for nonlinear circuits, theory andapplication of cellular neural networks, neural network architecture, evolutionary computation, and biosignal processing. He
was the Chair of the IEEE CAS Society Technical Committee on
Nonlinear Circuits and Systems (NCAS) during 2004–2005 andthe Steering Committee Secretary of the IEICE Research Society of Nonlinear Theory and its Applications (NOLTA) during
2004–2007, and is currently the Chair of the IEEE CAS Society Shikoku Chapter (from 2011) and a member of the IEEE
CAS Society Board of Governors (from 2012). He was an Editor of the IEICE Fundamentals Review during 2007–2012 andan Associate Editor of the IEEE Transactions on Circuits andSystems–I: Regular Papers during 2004–2005, the IEEE Transactions on Circuits and Systems–II: Express Briefs during 2012–2013, the IEEE CAS Magazine during 2008–2009, and the RISP
Journal of Signal Processing during 2005–2012. He is currently
serving as a Secretary for the NOLTA, IEICE (from 2009) andan Associate Editor for the IEEE CAS Society Newsletter (from2007). He is also a member of the Editorial Board of International Journal of Circuit Theory and Applications. He is asenior member of the IEEE, and a member of the IEICE andthe RISP.

302

Chi K. Tse
graduated fromthe University of Melbourne, Australia, with BEng (First Class Honors) and PhD degrees in 1988 and1991, respectively, both in electrical engineering.
He is presently
Chair Professor at Hong Kong Polytechnic University, with which heserved as Head of the Department
of Electronic and Information Engineering from 2005 to 2012. He is author/coauthor of 7 books, 15 bookchapters and over 500 papers inresearch journals and international
conference proceedings, and holds 5 US patents. He wasawarded a number of research and industry awards, including
Best Paper Award by IEEE Transactions on Power Electronics 2001, Best Paper Award by International Journal of Circuit
Theory and Applications in 2003, two Gold Medals at the International Inventions Exhibition in Geneva in 2009 and 2013,
and a number of recognitions by the academic and researchcommunities, including honorary professorship by several Chinese and Australian universities, IEEE Distinguished Lectureship, Distinguished Research Fellowship by the University ofCalgary, Gledden Fellowship and International Distinguished
Professorship-at-Large by the University of Western Australia.
He serves and has served as Editor-in-Chief for IEEE Circuits
and Systems Magazine, Editor-in-Chief for IEEE Circuits andSystems Newsletter, Associate Editor for two IEEE Transactions, Editor for Int J Circuit Theory and Applications, andis on the editorial boards of a few other journals. Back home,he serves as panel member of Hong Kong Research Council
and NSFC, and member of several professional and government
committees. He is a Fellow of IEEE and a Fellow of IE Australia.
(Received June 19, 2014: revised August 15, 2014)

Journal of Signal Processing, Vol. 18, No. 6, November 2014