Post on 20-Feb-2018
Distance Maintaining Compact Quantum Crossover Based Clonal Selection Algorithm
Hongwei Dai, Yu Yang, Cunhua Li
Distance Maintaining Compact Quantum Crossover Based Clonal
Selection Algorithm
*1Hongwei Dai,
1Yu Yang,
1Cunhua Li
*1,Corresponding Author School of Computer Engineering, Huaihai Institute of Technology,
hongweidai@hotmail.com
doi:10.4156/jcit.vol5. issue10.8
Abstract Premature convergence is one of the major difficulties with Clonal Selection Algorithm (CSA). It
has been observed that this problem is closely tied to the problem of losing diversity in the population.
In this paper, we propose a distance maintaining compact quantum crossover based CSA. The compact
quantum crossover is useful for information exchanging between different solutions, whereas uses
fewer antibodies. On the other hand, distance maintaining scheme permits the algorithm to fine tune its
solutions. Simulation results on Traveling Salesman Problems (TSP) show that the novel algorithm is
able to effectively balance exploration and exploitation of the search space and can also present the
optimal or near-optimal solutions.
Keywords: Clonal Selection Algorithm (CSA), Diversity,
Distance Maintaining Compact Quantum Crossover, Optimization Problem
1. Introduction
Over the last few decades, there has been a rapid increasing interest in the area of natural inspired
computing. such as Genetic Algorithm (GA) [21][29], Evolutionary Algorithm (EA) [3][28], Ant
Colony Optimization (ACO) [10][2] and Particle Swarm Optimization (PSO) [12]. Unlike other
traditional optimization algorithms, which emphasize accurate and exact computation at the cost of
spending much more computation time, these nature inspired algorithms have better convergence
speeds to the optimal or near optimal results in reasonable time.
More recently however, an artificial computational systems named artificial immune system (AIS),
exploits the immune system’s characteristics of learning and memory in order to develop adaptive
systems capable of performing a wide range of tasks in various engineering applications, such as patter
recognition [6],intrusion detection [13] [8], computer security [16], optimization problems [11][1], and
fault detection [4]. In particular, Clonal Selection Algorithm (CSA) [9], one of the most famous
artificial immune algorithms which was designed based on the clonal selection principle of adaptive
immunity, has been verified as having a great number of useful mechanisms from the viewpoint of
programming, controlling, information processing and so on [8], [11], [1].
As mentioned above, CSA is very useful for solving some difficult problems, however, suffers from
poor convergence properties and difficulties in reaching high-quality solutions in reasonable time [30].
Before analyzing this problem, the concept of diversity is introduced at first.
Diversity, a measure of a population independent of the evaluation function, typically represents the
relative distance between individuals with respect to their representation [25]. Population based
algorithms such as evolutionary algorithms and genetic algorithms have a tendency to converge to local
optima. CSA, a population based AIS, has been observed that its premature convergence problem is
closely tied to the problem of losing diversity [7].
In traditional CSA, according to the clonal selection theory, low affinity antibodies have to be
deleted and replaced by the new random antibodies during mutation process. On the other hand, initial
antibody repertoire is also generated randomly. These random operations clearly reduce the complexity
of algorithm. However, random antibodies have a bad influence on the diversity of repertoire. Hence,
the search within affinity landscape will become premature and can not produce better solutions.
Moreover, both somatic hypermutation of clonal selection theory and receptor editing generate not only
high affinities, but also low affinity antibodies usually [26]. As a result, the search efficiency is low.
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Journal of Convergence Information Technology
Volume 5, Number 10. December 2010
To solve this problem, a distance maintaining compact quantum crossover is constructed in this
paper.
The classical quantum interference crossover proposed by Narayanan [22] has been proved useful
for solving multicast routing problem [19]. An improved quantum crossover has also proposed for
information exchanging between different antibodies in our previous work [7]. In our paper submitted
to ICICEL, it has been also approved that the compact quantum crossover is able to exchange
information like the improved crossover in [7], whereas uses fewer antibodies. In addition, distance
maintaining scheme is expected to further improve the algorithm performance.
The remainder of this paper is organized as follows: In Section 2, we introduce the clonal selection
theory and quantum interference crossover. We then describe the distance maintaining compact
quantum crossover based clonal selection algorithm in Section 3. In Section 4, a number of benchmark
problems - traveling salesman problems are solved for validating the performance of the proposed
algorithm. Finally, we give some general remarks to conclude this paper in Section 5.
2. Background
This section will describe the clonal selection theory behind CSA. Also, a brief description of the
classical quantum crossover and the improved crossover in [7] will be provided.
2.1. Clonal Selection Theory[5]
Clonal selection theory is a form of natural selection and it explains the essential features which
contain sufficient diversity, discrimination of self and non-self and long-lasting immunologic memory.
This theory interprets the response of lymphocytes in the face of an antigenic stimulus.
When an animal is exposed to an antigen, some subpopulation of its bone marrow derived cells (B
lymphocytes) can recognize the antigen with a certain affinity (degree of match).B lymphocytes will be
stimulated to proliferate and eventually mature into terminal (non-dividing) antibody secreting cells,
called plasma cells. Proliferation of the B lymphocytes is a mitotic process whereby the cells divide
themselves, creating a set of clones identical to the parent cell. The proliferation rate is directly
proportional to the affinity level, i.e. the higher affinity levels of B lymphocytes, the more of them will
be readily selected for cloning and cloned in larger numbers.
According to the clonal selection theory, random point mutation is performed during the maturation
process. However, frequently, a large proportion of the cloned population becomes dysfunctional or
develops into harmful anti-self cells after the mutation. Moreover, the authors [23] interpret that clonal
deletion, previously regarded as the major mechanism of central B cell tolerance, has been shown to
operate secondarily and only when receptor editing is unable to provide a non-autoreactive specificity.
Recent investigations [24], [26], [31] indicate that receptor editing has played a major role in
shaping the lymphocyte repertoire. Both B and T lymphocytes that carry antigen receptors are able to
change specificity through subsequent receptor gene rearrangement.
2.2. Quantum Interference Crossover
Quantum computing, a research area including concepts such as quantum mechanical computers
and quantum algorithms, has gained much attention and wide applications because of its superiority to
classical computer on various problems. Some works on combining quantum computing with
evolutionary algorithms (EA), genetic algorithms (GA) and particle swarm optimization (PSO) for
some combinatorial optimization problems can be found in [14][15][27].
In addition, a quantum interference crossover based on quantum computing was first proposed by
Narayanan [22]. In paper [19], a quantum clonal algorithm (QCA) merging immune clonal selection
algorithm and quantum crossover was presented to deal with multicast routing problem.
The classical quantum crossover occurs as follows: take the 1st city of tour one, 2nd city of tour two,
3rd city of tour three, etc. Note that no duplicates are permitted within the same universes. If a city is
already presented in the visiting tour, choose the next alphabetical city not already contained. Based on
simulation results, it can be found that this classical quantum crossover, we call it position based
quantum crossover, is not suitable for TSPs. In [7], we proposed an improved quantum crossover based
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Distance Maintaining Compact Quantum Crossover Based Clonal Selection Algorithm
Hongwei Dai, Yu Yang, Cunhua Li
CSA for solving traveling salesman problem (TSP). This distance based quantum crossover works well
and presents better solutions.
It should be noticed that both the position based classical quantum crossover and the improved
distance based ones have to face the same problem: The number of universes required for crossover is
equal to the number of cities of the TSP instance. In order to overcome this problem, a distance
maintaining compact quantum crossover based CSA is proposed in this paper. The novel crossover is
able to exchange information like the improved crossover in [7] whereas uses fewer antibodies.
Furthermore, distance maintaining scheme is expected to further improve the algorithm performance.
3. Distance Maintaining Compact Quantum Crossover Based Clonal Selection
Algorithm
This section provides the full architecture of the distance maintaining compact quantum crossover
based clonal selection algorithm (DMCQCSA). Firstly, we introduce operation process of the distance
maintain compact quantum crossover. Then, we propose the DMCQCSA and list its procedure.
3.1. Distance Maintaining Compact Quantum Crossover
As discussed in the above section, the number of universes required for crossover is equal to the
number of cities of the TSP instance. What is more, simulation results on TSPs indicate that the low
affinity antibodies in crossover population have less contribution to improve fitness. As a result, a
distance maintaining compact quantum crossover is designed.
Figure 1 illustrates the new crossover. S1, S2, S3, ..., SN−1, SN are N current solutions sorted in a
descending order of affinity. In traditional compact quantum crossover, antibody pairs S1 − S2, S2 −
S3, ..., SN−2 − SN−1, SN−1 − SN undergo crossover. However crossover operation based on these “similar”
antibodies always generates low fitness offspring. To solve this problem, a distance maintaining
mechanism is introduced. Compact quantum crossover occurs between S1−SN, S2−SN−1, S3−SN−2 etc.
Fig. 2 shows detail information of the new crossover.
Figure 1. Distance maintaining compact quantum crossover
Figure 2. Operation process of the new crossover
Fig. 2 (a) exhibits the crossover process and (b) shows the operation results of each step.
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Journal of Convergence Information Technology
Volume 5, Number 10. December 2010
Without loss of generality, assuming pair S1−SN is selected for crossover. The crossover procedure
can be explained as follows:
Step1: Randomly select a city in S1 (For example city A). Then compare two edges: edge A-B in S1
and edge A-C in SN. City C will be selected if the length of A-C is shorter than that of edge A-B. Now,
the offspring sequence includes A and C.
Step2: City C is the current city. Then compare two edges: edge C-E in SN and edge C-D in S1. City
D will be selected if the length of C-D is shorter than that of edge C-E. Now, the offspring sequence
includes A, C and D.
Step3: Now, city D is the current city. Then compare edge D-E in S1 and edge D-B in SN. City B
will be selected if the length of D-B is shorter than that of edge D-E. New city B is added into the
offspring sequence, that is A, C, D and B.
Step4: Now, city B is the current city. Then compare edge B-C in S1 and edge B-F in SN. Because
city C has been already selected, city F will be selected and added to the offspring sequence. Then we
get A, C, D, B, F sequence.
Step5: Finally, the only one remaining city E will be added into the former sequence. Then a close
visiting tour A − C − D − B − F − E − A is constructed.
Figure 3. Flowchart of the proposed algorithm
3.2. DMCQCSA
The new DMCQCSA by merging distance maintaining compact quantum crossover with CSA is
presented in Figure 3. The full produce of the proposed algorithm can be represented as follows:
Step 1 Randomly creating an initial pool of m antibodies (S1, S2, ..., Sm).
Step 2 Compute the affinity of all antibodies (A(S1),A(S2), ...,A(Sm)). Then sort them in a descending
order. Where A(.) is the affinity function.
Step 3 Select n (n ≤ m) best affinity elites from the m original antibodies.
Step 4 Place each of the n selected elites in n separate and distinct pools (EP1,EP2, ...,EPn). They
will be referred to as the elite pools.
Step 5 Clone the elites in each elite pool with a rate proportional to its fitness, i.e., the fitter the
antibody, the more clones it will have. The amount of clone generated for these antibodies is given by
Eq.(1):
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Distance Maintaining Compact Quantum Crossover Based Clonal Selection Algorithm
Hongwei Dai, Yu Yang, Cunhua Li
)( Mn
inroundpi
(1)
where i is the ordinal number of the elite pools, M is a multiplying factor which determines the scope
of the clone and round(.) is the operator that rounds its argument towards the closest integer. After this
step, we can obtain ip antibodies just as (EP1,1,EP1,2, ...,EP1,p1; ...;EPn,1,EPn,2, ...,EPn,pn).
Step 6 Subject the clones in each pool through either random point mutation or receptor editing
processes. Some of the clones in each elite pool undergo the random point mutation process and the
remainder of the clones pass the receptor editing process. After this step, we obtain ip mutated cells
just as (EP’1,1,EP’1,2, ...,EP’1,p1; ...;EP’n,1,EP’n,2, ...,EP’n,pn)
Step 7 According to the distance maintaining compact quantum interference crossover described
above, subject the selected n elites through crossover.
Step 8 Reselect the fittest cell from each elite pool and quantum crossover generated cells.
Step 9 Update the parent cells in each elite pool with the fittest cells selected in Step 8.
If the iteration number reaches a pre-specified maximal generation number Gmax, the produce will
be terminated. Otherwise, it returns to Step 3.
4. Simulation on TSPs
In this section we first provide a description of the benchmark problems - TSP used for our
simulation. Then we present the simulation results, as well as comparison between the new method and
other algorithms.
4.1. Traveling Salesman Problem
Through the years the TSP, one of the most famous NP-hard combinatorial optimization problems,
has gathered a lot of attentions from academic researchers and industrial practitioners. The TSP is very
easy to describe, yet very difficult to solve.
Paper [18] introduced the TSP problem whose objective is to find the shortest route for a traveling
salesman who, starting from his home city has to visit every city on a given list precisely once and then
return his home city. The main difficulty of this problem lies in the immense number of possible tours.
No polynomial time algorithm is known with which can be solved.
In simulation, the gene sequence of immune cell can be expressed as a set of coordinates in a N-
dimensional shape-space. That is X = (x1, x2, ..., xN). Naturally, we can use different gene sequences
denote the solutions of a N-city TSP. If d(i, j) denotes the distance between city i and j which is
symmetric and known, the object of TSP is to find a permutation of the set {1, 2, 3, ...,N} that
minimizes the quantity [18]:
1
1
))1(),(())1(),(()(N
i
NdiidC (2)
4.2. Simulation Results
In this section, we present the simulation results from the application of the distance maintaining
compact quantum crossover based CSA and some other traditional algorithms to a number of TSP[17]
problems. All these simulations are implemented in 10 replications of C++ on a Pentium Dual-Core
2.2GHz CPU with 2.0GB memory. Table 1 lists the TSP instances to be tested.
The meaning of the parameters used in the proposed algorithm and their values are illustrated in
Table 2.
Premature convergence is one of the main difficulties with the CSA. It has been observed that this
problem is closely related to the problem of losing diversity in the population. Here, the diversity is
defined as the mean edge-distance between the best tour and all other tours. Edge-distance means
different edges between two tours [20].
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Journal of Convergence Information Technology
Volume 5, Number 10. December 2010
In order to provide insight into the relationship between the diversity and the actual results, we
present the experimental results of population diversity and tour distance for eil51, rd100 and pr124
respectively.
Table 1. Problems to be solved
Table 2. The meaning of the user-defined parameters and their values
Figure 4. (a)Convergence curve of CQCSA and DMCQCSA for eil51 instance (b)Improvement of
DMCQCSA comparing with CQCSA for eil51
Figure 5. Comparison of population diversity of CQCSA and DMCQCSA on eil51 problem
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Distance Maintaining Compact Quantum Crossover Based Clonal Selection Algorithm
Hongwei Dai, Yu Yang, Cunhua Li
Figure 4 (a) shows the experimental results of CQCSA and DMCQCSA for applying eil51 instance.
In Figure 4 (b), the improvements of DMCQCSA comparing with CQCSA for eil51 is presented. The
plotted lines represent that the DMCQCSA has outperforms better than CQCSA when line segments
are under the axis, which means, the negative value indicates the DMCQCSA has improvement
(shorter route) comparing with CQCSA.
As we know population diversity is a key issue in the performance of CSA. A common hypothesis
is that high diversity is important for the process of algorithm to avoid premature convergence and
escaping from local optimal solution.
Figure 5 presents the population diversity of CQCSA and DMCQCSA for eil51 during the
maturation process. In this figure, we can easily discover that the CQCSA has much higher population
diversity than DMCQCSA. It seems that the diversity is not the only key. Naturally, the same question
presented in [7] will be asked again: Is it possible for a lower population diversity algorithm to
generate better solutions than higher diversity ones?
Figure 6. (a)Convergence curve of CQCSA and DMCQCSA for rd100 instance (b)Improvement of
DMCQCSA comparing with CQCSA for ed100
Figure 7. Comparison of population diversity of CQCSA and DMCQCSA on rd100 instance
To answer this question, simulations on rd100 instance are also performed. Figure 6 (a) shows the
experimental results of CQCSA and DMCQCSA for applying rd100 instance. In Figure 6 (b), the
improvements of DMCQCSA comparing with CQCSA for rd100 is illustrated.
Figure 7 presents the population diversity of CQCSA and DMCQCSA for rd100 during the
maturation process.
Based on the above simulation results, it is apparent that higher population diversity is useful for a
greater exploration of the search space during the first half process, however a lower diversity which
permits the algorithm to fine-tune its results during the latter stages is also necessary.
In order to confirm the effectiveness and the robustness of the new algorithm, we perform
simulations on TSPs from eil51 to kroA200 and also compare our method with RECSA, TQCSA and
CQCSA. (RECSA: basic clonal selection without crossover; TQCSA: RECSA with traditional
quantum interference crossover; CQCSA: RECSA with compact quantum crossover.)
Table 3 shows the experimental results. Parameters PDM and PDB which indicate the percentage
deviation from the optimal tour length Dopt of the mean distance Dm and the best distance Db by the
following formula.
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Journal of Convergence Information Technology
Volume 5, Number 10. December 2010
100
opt
optm
D
DDPDM
(3)
100
opt
optb
D
DDPDB (4)
Table 3. Experimental results of RECSA, TQCSA, CQCSA and DMCQCSA for TSPs from eil51 to
kroA200
From Table 3, we can observe that the proposed distance maintaining compact quantum crossover
based CSA (DMCQCSA) has a superior ability to search better solutions than that of the RECSA,
TQCSA and CQCSA.
5. Conclusions
In this paper, we have proposed a distance maintaining compact quantum crossover based cloanal
selection algorithm (DMCQCSA) for overcoming premature convergence. The compact quantum
crossover is useful for information exchanging between different solutions, whereas uses fewer
antibodies than the traditional quantum interference crossover. What is more, the distance maintaining
mechanism permits the algorithm to fine tune its solutions. Experimental results of population diversity
and solutions on eil51 and re100 instances confirm that higher population diversity is useful for a
greater exploration of the search space during the first half process, and a lower diversity which
permits the algorithm to further exploit its results during the latter stages is also necessary. Simulation
results on a wide range of TSP instances demonstrate that the performance of the proposed DMQCSA
is superior as compared to other related CSAs.
6. Acknowledgements
This work was supported by the Ministry of Science and Technology of China under Grant No.
2009GJC10043, the Natural Science Foundation of Jiangsu Province under Grant No. BK2008190, the
Natural Science Foundation of Huaihai Institute of Technology under Grant No. KX08013, No.
KQ09014.
7. References
[1] E. Ahmed and M. El-Alem. “Immune-motivated optimization”. International Journal of Theoretical
Physics, 41(5):985–990, 2002.
- 63 -
Distance Maintaining Compact Quantum Crossover Based Clonal Selection Algorithm
Hongwei Dai, Yu Yang, Cunhua Li
[2] S. Janakiraman, V. Vasudevan. “ACO based distributed intrusion detection system”, JDCTA:
International Journal of Digital Content Technology and its Applications, Vol. 3, No. 1, pp. 66-72,
2009.
[3] T. Bäck. “Evolutionary Algorithms in Theory and Practice”. New York: Oxford Univ. Press, 1996.
[4] D. W. Bradley and A. M. Tyrrell. “Immunotronics: Hardware fault tolerance inspired by the
immune system”. In LNCS 1801, pages 11–20, 2000.
[5] F. M. Burnet. “The Clonal Selection Theory of Acquired Immunity”. Cambridge Press, 1959.
[6] J. H. Carter. “The immune system as a model for pattern recognition and classification”. Journal of
the American Medical Informatics Association, 7(1):28–41, 2000.
[7] H. W. Dai, Y. Yang, C. H. Li, J. Shi, S. C. Gao, and Z. Tang. “Quantum interference crossover-
based clonal selection algorithm and its application to traveling salesman problem”. IEICE Trans. Inf.
& Syst., E92-D(1):78–85, 2009.
[8] D. Dasgupta and N. S. Majumdar. “Anomaly detection in multidimensional data using negative
selection algorithm”. In CEC2002-2002 Congress on Evolutionary Computation, pages 1039–1044,
2002.
[9] L. N. de Castro and F. J. Von Zuben. “The clonal selection algorithm with engineering
applications”. In GECCO’00, Workshop on Artificial Immune Systems and Their Applications, pages
36–37, 2000.
[10] M. Dorigo and L. Gambardella. “Ant colony system: a cooperative learning approach to the
traveling salesman problem”. IEEE Transactions on Evolutionary Computation, 1(1):53–66, 1997.
[11] O. Engin and A. Doyen. “A new approach to solve hybrid flow shop scheduling problems by
artificial immune system”. Future Generation Computer Systems, 20:1083–1095, 2004.
[12] P. Radhakrishnan, V. M. Prasad and M. R. Gopalan. “Extensive analysis and prediction of optimal
inventory levels in supply chain management based on Particle Swarm Optimization algorithm”. JCIT:
Journal of Convergence Information Technology, 4(3):25-33, 2009.
[13] F. González and D. Dasgupta. “Anomaly detection using real-valued negative selection”. Genetic
Programming and Evolvable Machines, 4:383–403, 2003.
[14] K. H. Han and J. H. Kim. “Quantum-inspired evolutionary algorithm for a class of combinatorial
optimization”. IEEE Transactions on Evolutionary Computation, 6:580–593, 2002.
[15] K. H. Han and J. H. Kim. “Quantum-inspired evolutionary algorithms with a new termination
criterion, HЄ gate, and two-phase scheme”. IEEE Transactions on Evolutionary Computation,
8(2):156–169, 2004.
[16] S. Hofmeyr and S. Forrest. “Architecture for an artificial immune system”. Evolutionary
Compuiation,
8(4):443–473, 2000.
[17] http://www.iwr.uni heidelberg.de/groups/comopt/software/TSPLIB95/.
[18] E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys. “A Guided Tour of Combi-
natorial Optimization”. New York: Wiley and Sons, 1985.
[19] Y. Y. Li and L. CH. Jiao. “Quantum clonal algorithm for multicast routing problem”. Journal of
Software(In Chinese), 18(9):2063–2069, 2007.
[20] K. Maekawa, N. Mori, H. Kita, and H. Nishikawa. “A genetic solution for the travelling salesman
problem by means of a thermodynamical selection rule”. In Proc. IEEE Int. Conf. Evolutionary
Computation, Nagoya, Japan, pages 529–534, May, 1996.
[21] M. Mitchell. “An Introduction to Genetic Algorithms”. MIT Press, 1996.
[22] A. Narayanan and M. Moore. “Quantum-inspired genetic algorithm”. In Proc. IEEE Int. Conf.
Evolutionary Computation, Nagoya, Japan, pages 61–66, May,1996.
[23] R. Pelanda and R. M. Torres. “Receptor editing for better or for worse”. Current Opinion in
Immunology, 18:184–190, 2006.
[24] S. L. Tiegs, D. M. Russell, and D. Nemazee. “Receptor editing in self-reactive bone marrow B
cells”. J Exp Med, 177:1009–1020, 1993.
[25] M. Ventresca and H. R. Tizhoosh. “A diversity maintaining population-based incremental learning
algorithm”. Information Science, 178:4038–4056, 2008.
[26] L. K. Verkoczy, A. S. Martensson, and D. Nemazee. “The scope of receptor editing and its
association with autoimmunity”. Current Opinion in Immunology, 16:808–814, 2004.
- 64 -
Journal of Convergence Information Technology
Volume 5, Number 10. December 2010
[27] Y. Wang, X. Y. Feng, Y. X. Huang, D. B. Pu, W. G. Zhou, Y. CH. Liang, and CH. G. Zhou. “A
novel quantum swarm evolutionary algorithm and its applications”. Neurocomputing, 70:633–640,
2007.
[28] X. Yao and Y. Xu. “Recent advances in evolutionary computation”. J. Comput. Sci. & Techno.,
21(1):1–18, 2006.
[29] P. Th. Zacharia and N. A. Aspragathos. “Optimal robot task scheduling based on genetic
algorithms”. Robotics and Computer-Integrated Manufacturing, 21:67–79, 2005.
[30] Y. F. Zhong, L. P. Zhang, and P. X. Li. “Multispectral remote sensing image classification based
on simulated annealing clonal selection algorithm”. In IGARSS’05, volume 6, pages 3745–3748, 2005.
[31] D. Gay, T. Saunders, S. Camper, and M. Weigert. “Receptor editing: an approach by antoreactive
B cells to escape tolerance”. J Exp Med, 177:999–1008, 1993.
- 65 -