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International Journal of Computer Applications in Engineering Sciences
[VOL I, ISSUE III, SEPTEMBER 2011]
[ISSN: 2231-4946]
313 | P a g e
Hybrid Evolutionary Clonal Selection for
Parameter Estimation of Biological Model
Afnizanfaizal Abdullah1, Safaai Deris2, Sohail Anwar3
1,2Artificial Intelligence and Bioinformatics Group (AIBIG)
1,2Faculty of Computer Science and Information Systems, Universiti Teknologi Malaysia,
81310 UTM, Johor
3Pennsylvania State University, 3000 Ivyside Park Altoona, Pennsylvania, USA
1afnizanfaizal@utm.my
2safaai@utm.my
3sxa15@psu.edu
Abstract— The Clonal Selection Algorithm (CSA) is a
widely used Artificial Immune Optimization (AIO)
approach that tends to mimic the immune response when
the pathogenic pattern is detected by the immune cells.
However, this method, in its standard form, shows slow
convergence and frequently traps in one of the local
optima, especially for high dimensional problems. Hence,
in this paper, an improved CSA method is introduced by
integrating evolutionary operations adopted from the
Differential Evolution (DE) method. The proposed method,
called Differential Clonal Evolution (DICE) method,
utilizes the mutation and crossover operation to exploit the
information of different antibodies in the population.
Furthermore, antibodies that yield trivial fitness value are
relocated randomly so that the method can escape from the
local optima in a more straightforward manner. To show
the effectiveness of this method, the method is used to
estimate parameters of a bacterial lactose production
model using noisy and incomplete time series data. The
statistical results suggest that the proposed method has
better speed and accuracy performance compared to the
standard CSA, Particle Swarm Optimization (PSO) and
Genetic Algorithm (GA) techniques.
Keywords— Clonal Selection; Differential Evolution;
Hybrid Optimization; Parameter Estimation; Bacterial
Model
I. INTRODUCTION
For the past few years, global optimization
problems have received significant attention, which led
to the implementation of a variety of optimization
methods [1]. Among these methods, stochastic
population-based approach offers a number of
advantages including freedom from derivative
constraints, improved accuracy and robustness
performance, as well as, utilizing a wide range of search
space [6, 8]. Due to these reasons, studies have been
carried out to solve many optimization problems in both
scientific and industrial fields. As a result, many
varieties of optimization methods are proposed,
including Particle Swarm Optimization (PSO) [2],
Genetic Algorithms (GA) [3], Ant Colony Optimization
(ACO) [4], and Artificial Bee Colony (ABC) [5].
However, depending on single method can be very
restrictive for certain problems. This is due to the fact
that every method has its limitations, especially in terms
of searching accuracy and convergence speed. Several
recent studies have shown that the hybridization of
different methods may considerably improve the
searching capability [6-10]. In particular, the
hybridization usually overcomes the limitations of the
standard methods by exploiting the advantages of the
other methods [6]. Hence, this provides a promising
opportunity to enhance the accuracy and speed
performance of the standard methods.
The Clonal Selection Algorithm (CSA) method is
one of the most widely used Artificial Immune
Optimization (AIO) approaches [12, 14]. The method is
basically motivated by the concept of clonal selection
principle that describes the immune response when the
pathogenic pattern is identified [15]. Despite of the
advantage to perform good approximation for finding
global optimum in multimodal problems [16], the main
shortcoming of this method consists of premature and
slow convergence. For the problem of prematurity, the
CSA method usually fails to explore new possible
solutions. This is main reason is that the method got
trapped in one of the local optima. On the other hand, it
is shown that the CSA method frequently converges
slowly, particularly when searching for better solutions
in high dimensional problems.
Recently, many research investigations have been
made to overcome these limitations. They include the
introduction of elimination feature to remove the oldest
candidate solutions [17] and employ the chaos-based
mutation strategy [20] for improving the diversity of the
possible candidate solutions. More recently, local search
technique [18] and immune memory encoding [19] are
incorporated into the the standard CSA method to
enhance the exploitation of the population. Alternatively,
the evolutionary algorithms such as Differential
Evolution (DE) [13] method have been used to improve
Abdullah et. al.
314 | P a g e
the searching capability of the CSA method. The
evolutionary operations of the DE method are commonly
utilized to enhance the proliferation process in the CSA
method, thereby substantially utilizing the information
regarding the adjoining clones [15-16].
In this work, the research that is relevant to the
improvement of the searching capability of standard
CSA method is extended by using the evolutionary
operations of DE method. In this variant of the CSA
method, the crossover and mutation operation are
implemented to exploit the information regarding
different antibodies in the population. Simultaneously,
the antibodies providing insignificant solutions are
relocated randomly to enhance the fitness values. By
doing so, the method can efficiently improve the
searching quality as well as utilizing the computational
time. The effectiveness of the proposed method is tested
to estimate parameter values in a biological model and
the statistical results are then compared with the
standard CSA, PSO and GA methods. The rest of the
paper is organized as follows. Sections II introduce the
standard CSA, standard DE methods, and the proposed
Differential Clonal Evolution (DICE) method.
Subsequently, Section III presents the experimental
results. Section IV discusses the contribution of the
work and Section V presents the conclusion and future
works.
II. METHODS
A. Standard Clonal Selection Algorithm (CSA) Method
The clonal selection principle [21] describes the
reaction of immune system to pathogens and the process
of improving the capability to identify these unintended
agents. In particular, the theory illustrates that a number
of immune cells that identify the pathogens will
proliferate. Some of them will become the effector cells
while the others maintain their role as memory cells [18].
In general, the CSA method employs three main phases:
cloning, mutation and selection. The method starts with
a population of d-dimensional search vectors, called
antibodies. The ith antibody, X, of the whole population
at a specific generation t is given by:
(1)
In CSA method, the fitness value of each antigen is
represented as affinity, which implies the goodness of the
antibody to generate antigen for the specific pathogen.
Initially, the population of antibody is initiated
randomly and the affinity of each antibody is evaluated.
The antibodies that produce good affinity values are
selected to undergo cloning phase. As a result, a new set
of population is created. Next, the mutation process is
performed to every clone, based on the mutation
constant. Hence, the mutated clones are formed with
new components and the affinity values are then been
evaluated to measure the fitness. In the last phase, the
mutated clones are selected to replace the original
antibodies. Eventually, the population is built with the
new improved antibodies. The overall procedure of the
standard CSA method is outlined in Figure 1:
Fig. 1. The standard CSA
B. Standard Differential Evolution (DE) Method
The DE method is also a stochastic population-
based optimization method. The method is proposed
based on the evolutionary operations of the GA method
[13]. Compare to GA, this method employs a mutation
operation to produce a trivial chromosome from the
original chromosome. Then, this trivial chromosome is
crossovered with its original counterpart to generate an
offspring chromosome. A simple selection operation is
performed to select the chromosome with a better fitness
value. In each generation, a range of search space is
specified to find a good solution. Thus, at initial
generation or t = 0, each chromosome is initialized, with
a lower and an upper bound,
and
respectively [13]:
(2)
where R is a random number generated between 0 and 1
and j is the dimension size.
In order to produce the trivial chromosome, Vi, the
mutation operation is executed according to the
differentiation of neighborhood chromosomes, given as
following: (3)
where xbest(t) denotes the current best chromosome, F is
the scaling factor, while xr1(t) and xr2(t) are randomly
chosen chromosomes [13]. Using this chromosome, an
1: Begin
2: Initiate population, X
3: // evaluate fitness of each antibody
4: While max number of generation is not met
5: // Select m best antibodies
6: For i = 1 to m antibody
7: // cloning selected antibodies
8: // mutate clones
9: // select improved clones to
10: replace old antibodies
11: End For
12: //include improved best
13: antibodies to population
14: End While
15: End Begin
Hybrid Evolutionary Clonal Selection for Parameter Estimation of Biological Model
315 | P a g e
offspring chromosome, Yi, is created by performing a
crossover operation between the new and the parent
chromosomes:
(4)
where CR is the crossover constant and R is a random
number between 0 and 1 [13]. As another population of
chromosomes is produced, a selection operation is
needed to keep the population size constant. The
selection is performed based on the calculated fitness
value of each chromosome:
(5)
This implies that if the offspring chromosome produces a
better fitness value, the current parent chromosome will
be replaced. Otherwise, this chromosome will remain in
the population of the next generation.
C. Differential Clonal Evolution (DICE) Method
In this paper, a new hybrid method is introduced
based on the standard CSA method. The proposed
method, that is, the Differential Clonal Evolution (DICE)
method, employs the evolutionary operations of DE
method to enhance the utilization of information from
different clones [15-16]. In the standard CSA method,
the standard mutation operation considers only the single
clone and its original antibody. Conversely, in this new
variant, the mutation and crossover operations are used to
include the information of neighboring clones. Different
to [15], the DICE method completely replaces the
standard mutation of CSA method with the evolutionary
operations of DE method. At the same time, DICE
method differs with [16] as the proposed method exploits
the antibodies that give poor fitness values. Thus, this
provides a mechanism that permits the method to
increase the possibility of escaping the local optima more
effectively.
Firstly, the population of antibodies is initialized
randomly and the affinity value of each antibody is
evaluated. Then, the population is sorted and a number of
m antibodies with potential affinity values are selected.
These antibodies undergo the cloning phase. The
mutation and crossover operations using Eq. 3 and Eq. 4
are performed to these clones and a new population of
offspring antibodies is produced. Next, a selection
operation is executed between the original antibody and
its offspring using Eq. 5. Simultaneously, the antibodies
that produced poor affinity values are chosen to endure a
randomization process using Eq. 2. Then, these improved
antibodies are combined with the selected antibodies to
form a new population and the antibody that produces
best affinity value is chosen as the current best antibody.
The procedure is iterated until the maximum number of
generations is met. The overall procedure of DICE
method is outlined in Figure 2:
Fig. 2. The DICE algorithm
III. EXPERIMENTAL RESULTS
The effectiveness of the proposed DICE method is
tested using a biological model of Escherichia coli
bacterium. The model is basically described as an
interaction network of the regulation of induction in the
lac operon in the bacterium [22]. The accuracy and speed
performance of the DICE method are compared with the
standard CSA, PSO and GA methods. Furthermore,
statistical analyses are performed to measure the
reliability of the proposed method compared to other
methods.
A. Lac Operon Regulation Model
The networks of interacting biomolecules usually
accomplish several fundamental functions in the cells.
However, frequently, the processes are difficult to be
extracted as the interactions commonly involve complex
behaviors. Hence, these networks are reconstructed using
mathematical modeling to represent the actual processes.
Unfortunately, the modeling of such networks typically
involves several parameters that explicitly represent the
entire processes. To determine these parameters, the
experimental data are usually fitted with model so that
these parameters can be estimated computationally.
Therefore, optimization methods are utilized to perform
this parameter estimation procedure. Yildirim and
1: Begin
2: Initiate population, X
3: // evaluate fitness of each antibody
4: While max number of generation is not met
5: // sort antibodies
6: // Select m best antibodies
7: // Select n poor antibodies
8: For i = 1 to m best antibody
9: // cloning best antibodies
10: End For
11: For j = 1 to p clones
12: // DE mutation (Eq. 3)
13: // DE crossover (Eq. 4)
14: // selection (Eq. 5)
15: End For
16:
17: // Poor antibodies (randomize operation)
18: For i = 1 to n poor antibody
19: //randomize (Eq. 2)
20: End For
21: //combine
22: //select current global best
23: End While
24: End Begin
Abdullah et. al.
316 | P a g e
Mackey [22] introduced a mathematical model for the
regulation of induction process in the lac operon that
considered the dynamics of the permease enabling the
internalization of several biomolecules such as lactose
and β-galactosidase. The model is important for the
observation of the conversion of lactose to allolactose,
glucose and galactose; the allolactose interactions with
the lac repressor; and the mRNA [22]. The model is
formed through the following equations:
(6)
(7)
where A, B and L are the concentrations of allolactose, β-
galactosidase and lactose, respectively; M is the mRNA
translation; is time; , and are the production
rate constants; and are the loss rate constants; is
the dilution rate constant; and are the equilibrium
constants of allolactose and lactose, respectively [22].
Thus, in this work, the values of , , , , ,
and parameters are tended to be estimated. The
experimental values of these parameters are given in
Table 1 [22].
TABLE I
EXPERIMENTAL VALUES OF THE REGULATION MODEL
Parameter
Experimental Value
1.76 × 104 min-1
1.66 × 10-2 min-1
2.15 × 104 min-1
5.20 × 10-1 min-1
8.33 × 10-4 min-1
2.26 × 10-2 min-1
1.95 × 10-3 M
9.70 × 10-4 M
In this work, the experimental data is obtained in
silico by generating noisy and sparse version of the
model data. Firstly, the model is simulated and the values
at several randomly chosen time points are evaluated.
Then, the Gaussian noise is added to the values so that it
will simulate the measurement noise [23]. The model
data and the generated noisy and sparse experimental
data of β-galactosidase and allolactose are illustrated in
Figure 3 and Figure 4, respectively.
B. Parameter Estimation
Generally, the parameter estimation problem is
formulated in the following way. Suppose that a system
is formed by the d-dimensional state variable, x, at time t,
which is the distinctive solution of the initial value
problem:
(8)
where p is the parameters [24]. So, let y signify the
observation of experimental value, i, corresponding to
the measurement, j, and represented by the following
equation:
(9)
where σij > 0 and ɛij is a Gaussian distributed random
variable [24]. Thus, the parameter estimation problem of
a biological system consists of finding the optimal
parameter p such that the difference of the experimental
data and the simulated data is minimized:
(10)
where is the trajectory at time t, n is the total
number of parameters and m is the total number of
observed values [24].
The results obtained from the proposed method are
compared with those from the standard CSA, PSO and
GA methods. For each method, a population size of 50
particles or chromosomes is initiated and the maximum
number of generations is set to 200. Furthermore, each
method is executed 100 times independently to observe
its reliability and consistency. Table 2 shows the average
fitness values and the corresponding standard deviation
for each method. In general, the proposed DICE method
has outperformed the standard methods. Hence, the
accuracy of the proposed method is better compared to
the other methods, as the overall fitness value obtained is
the lowest among those from the other methods.
TABLE II
ACCURACY AND SPEED PERFORMANCE
Method
GA
PSO
CSA
DICE
Average
3.72×10-3
3.56×10-3
4.64×10-4
1.93×10-9
Standard
Deviation
3.07×10-3
3.00×10-3
7.94×10-4
4.15×10-9
Average
Speed
(second)
0.358
6.240
0.483
0.452
Hybrid Evolutionary Clonal Selection for Parameter Estimation of Biological Model
317 | P a g e
Fig. 3. Comparison of the model data and the experimental data for concentration of β-galactosidase
Fig. 4. Comparison of the model data and the experimental data concentration of allolactose
To address the performance of the proposed
method in terms of convergence speed, Figure 5
illustrates the graph of convergence for all methods.
Obviously, the standard GA and PSO methods
converged prematurely while the standard CSA method
successfully finds better fitness values compared to the
GA and PSO methods. However, the method was
eventually trapped in one of the local optima starting at
the 165th generation. This problem has been effectively
solved by the proposed method as the values are kept
decreasing until the maximum number of generations is
reached.
In addition, a statistical analysis of the observed
measurements and the fitted data produced by the
proposed DICE method is conducted. In this analysis,
confidence interval estimates using chi-squared (χ2)
distribution is used. The result of this analysis is
presented in Table 3. The result shows that the proposed
method is reliable for the estimation of the parameter
values as the mean error is substantially small for both
components of the model. Moreover, the variance point
lies between the interval estimates. Thus it is confirmed
that the estimate obtained using the DICE method can
be generally considered as valid.
Abdullah et. al.
318 | P a g e
Fig. 5. Convergence behaviours of each method
Parameter estimation of complex biological models
is usually presented as an optimization problem [23, 24].
The approximation of the parameter values is always
hindered by the noise and incompleteness of the
experimental data. Thus, optimization methods such as
the GA and PSO methods have always been considered
for this problem because they are capable of fitting the
experimental data with the model prediction effectively.
However, a substantial number of studies have shown
that these methods are frequently trapped in one of the
local optima [6]. Moreover, these methods always
involve a huge search space that requires a large amount
of computational time. Hence, a significant number of
studies have been conducted to merge several methods to
overcome this challenge [6-11]. Nevertheless, this
approach shows potential in improving the accuracy and
speed of the standard methods.
TABLE III
STATISTICAL ANALYSIS OF FITTED DATA BY DICE METHOD
Component
β-galactosidase
Allolactose
Error
0.21%
0.40%
Variance Point
4.65×10-8
2.49×10-1
Variance Interval
[3.74×10-8,
6.26×10-8]
[2.00×10-1,
3.35×10-1]
Real Variance
4.64×10-8
2.48×10-1
χ2 Test
Pass
IV. DISCUSSION
In this work, the proposed DICE method has
presented another prospective alternative for enhancing
the quality of the parameter estimation results. As shown
in Table 2, the method has outperformed all the
competitive methods efficiently, in terms of both,
accuracy and speed performance. The accuracy
performance of the proposed DICE method has shown
remarkable improvement compared to the results
produced by the other methods. This is because of the
two main reasons. Firstly, the DICE method employs
evolutionary operations to the antibodies that yield
potentially good fitness values. As the operations are
performed to these antibodies, the fitness values are
improved significantly at each generation as the
information regarding different antibodies is utilized to
produce more significant fitness values. Secondly, the
antibodies that produced insignificant fitness values are
subjected to undergo randomization operation. By doing
so, the method can enhance the fitness values of these
antibodies, thus allowing the method to escape the local
optima more effectively. This is shown by the
convergence behavior of the DICE method in Figure 4.
Nonetheless, there is only a small difference of
speed performance between the proposed method and its
standard counterpart. This is due to the fact that the
proposed method uses the computational time
extensively for each antibody to exchange information
between its neighbors. Even though finding the possible
best values can be achieved more effectively, this
requires numerous runtimes to execute the evolutionary
operation on every antibody in the population. Hence, the
scalability of the problem dimension may affect the
speed performance of the method. However, statistical
analysis performed on the results produced by the
proposed DICE method show that the method is capable
of estimating the parameter values accurately. The
method passed the χ2 test, indicating that the values
estimated by the proposed method are very close to the
actual values.
V. CONCLUSION
Global optimization problems present a major
challenge in both scientific and industrial fields. Thus, a
significant number of optimization methods have been
Hybrid Evolutionary Clonal Selection for Parameter Estimation of Biological Model
319 | P a g e
developed to overcome these problems. In most cases,
global optimization methods are always chosen due to
the capability to handle nonlinearity of the problems.
However, these methods are usually hampered by some
limitations including huge computational time
consumption and getting stuck in one of the local optima.
This led to the development of hybrid optimization
methods, which mainly tends to combine several
different methods to improve the limitations by utilizing
the advantages of the combined methods.
This paper presented a new hybrid optimization
method based on the CSA method and the evolutionary
operations adopted from the DE method. The
effectiveness of the new method is tested using noisy and
incomplete experimental data of a bacterial lactose
production model. The results are compared to the
standard CSA, PSO and GA methods. The comparison
suggests that the accuracy and the speed performance of
the proposed method are better than that can be obtained
from other methods. Despite of this achievement, there
are several limitations which need to be addressed. The
computational time constitutes one such limitation.
Hence, research is needed to overcome this challenge.
The future research work may involve the improvement
of the proposed DICE method through a use of local
optimization approach and adaptive features. In addition,
this study only considered one nonlinear model, which
may ponder the restriction of the actual performance of
the proposed method. Therefore, in the future, the
performance of the method will be verified by using a
number of different models to show the reliability and
robustness of the method.
REFERENCES
[1] N. Noman and H. Iba, ―Accelerating differential evolution
using an adaptive local search,‖ IEEE Transactions on
Evolutionary Computation, pp.107-125, volume 12, no. 1, 2008.
[2] J. Kennedy and R. Eberhart, ―Particle swarm optimization,‖ in
Proc. IEEE International Conference on Neural Networks,
1995, pp.1942-1948.
[3] D.E. Goldberg and J.H. Holland, ―Genetic algorithms and
machine learning‖, Machine Learning, pp. 95-99, volume 3, no.
2, 1988.
[4] M. Dorigo and G. Di Caro,"The ant colony optimization meta-
heuristic," New ideas in optimization, pp.11-32, 1999.
[5] D. Karaboga and B. Basturk, "A powerful and efficient
algorithm for numerical function optimization: artificial bee
colony (ABC) algorithm," Journal of Global Optimization,
pp.459-471, volume 39, no. 3, 2007.
[6] S. Das, A. Abraham and A. Konar, ―Particle swarm
optimization and differential evolution algorithms: technical
analysis, applications and hybridization perspective,‖ Studies in
Computational Intelligence, pp. 1-38, 2008.
[7] P. Kaelo and M.M. Ali, ―Differential evolution algorithms using
hybrid mutation,‖ Computational Optimization Application, pp.
231-246, volume 37, no.2, 2007.
[8] S. Das, P. Koduru, M. Gui, M. Cochran, A. Wareing, S.M.
Welch and B.R. Rabin, ―Adding local search to particle swarm
optimization,‖ in Proc. in IEEE Congress on Evolutionary
Computation, 2006, pp. 428-433.
[9] C. Zhang, J. Ning, S. Lu, D. Ouyang and T. Ding, ―A novel
hybrid differential evolution and particle swarm optimization
algorithm for unconstrained optimization,‖ Operational
Research Letters, pp. 117-122, volume 37, 2009.
[10] Z.H. Zhan, J. Zhang, Y. Li, H.S.H. Chung, ―Adaptive Particle
Swarm Optimization,‖ IEEE Trans. Syst. Man Cybr., pp.1362-
1381, volume 39, no.6, 2009.
[11] W. Fu, M. Johnston, M. Zhang, ―Hybrid particle swarm
optimization algorithms based on differential evolution and
local search,‖ AI 2010:Adv. Artif. Intel., pp.313-322, 2010
[12] L.N. De Castro and F.J. Von Zuben, ―The clonal selection
algorithm with engineering applications,‖ in Proc. of
GECCO'00, workshop on artificial immune systems and their
applications, volume 3637, 2000, pp. 36-37.
[13] R. Storn and K. Price, "Differential evolution—a simple and
efficient heuristic for global optimization over continuous
spaces." J. Glob. Optim, pp.341–359 , volume 11, 1997.
[14] L.N.de Castro, F.J.von Zuben, ―Learning and optimization
using the clonal selection principle,‖ IEEE Transactions on
Evolutionary Computation, pp.239–251, volume 6, no. 3, 2002.
[15] X.Z. Gao, X. Wang and S.J. Ovaska, ―Fusion of clonal selection
algorithm and differential evolution method in training cascade-
correlation neural network,‖ Neurocomputing, pp. 2483-2490,
volume 72, no. 10-12, 2009,
[16] M. Gong, L. Zhang, L. Jiao and W. Ma, ―Differential immune
clonal selection algorithm,‖ Int. Symp. on Intelligent Signal
Processing and Communication Systems (ISPACS 2007), 2007,
pp.666-669.
[17] V. Cutello, G. Nicosia and M. Pavone, ―Real coded clonal
selection algorithm for unconstrained global optimization using
a hybrid inversely proportional hypermutation operator,‖ in
Proc. ACM symposium on Applied computing, 2006, pp. 950-
954
[18] J. Yang, M. Gong, L. Jiao and Zhang, ―Improved clonal
selection algorithm based on lamarckian local search technique,‖
IEEE Congress on Evolutionary Computation (CEC 2008),
2008,pp.535-541
[19] W. Dong, G. Shi and L. Zhang, ―Immune memory clonal
selection algorithms for designing stack filters,‖
Neurocomputing, pp.777-784, volume 70, no. 4-6, 2007.
[20] M. Gong, L. Jiao, L. Zhang and W. Ma, ―Improved real-valued
clonal selection algorithm based on a novel mutation method,‖
Int. Symp. on Intelligent Signal Processing and Communication
Systems (ISPACS 2007), 2007, pp. 662-665.
[21] G.L. Ada and G.J.V. Nossal, ―The Clonal Selection Theory,‖
Scientific American, pp. 50-57, volume 257, no. 2, 1987.
[22] N. Yildirim and M.C. Mackey, ―Feedback regulation in the
lactose operon: a mathematical modeling study and comparison
with experimental data,‖ Biophysical Journal, pp.2841-2851,
volume 84, no. 5, 2003.
[23] G. Lillacci and M. Khammash, "Parameter estimation and
model selection in computational biology," PLoS
Computational Biology, e1000696, volume 6, no.3, 2010.
[24] E. Balsa-Canto, M, Peifer, J.R. Banga, J. Timmer and C. Fleck:,
―Hybrid optimization method with general switching strategy
for parameter estimation, ― BMC Systems Biology, pp.26-35,
volume 2, no. 1, 2008.
