Proceeding ofthe 6th International Symposium on Mechatronics and its Applications (ISMA09), Sharjah, UAE, March 24-26, 2009

RADIAL BASIS FUNCTION NETWORK BASED ON

MULTI-OBJECTIVE PARTICLE SWARM OPTIMIZATION

Sultan Noman Qasem

University Technology MalaysiaFaculty of Computer Science and Information

SystemSkudai, lohor, Malaysia

sultannoman@yahoo.com

ABSTRACT

The problem of unsupervised and supervised learning is discussedwithin the context of multi-objective optimization. In this paper, anevolutionary multi-objective selection method of RBF Networksstructure is discussed. The candidates of RBF Network structureare encoded into the particles in PSO. Then, they evolve towardPareto-optimal front defined by several objective functionsconcerning with model accuracy and model complexity. This studysuggests an approach of RBF Network training throughsimultaneous optimization of architectures and weights with PSO­based multi-objective algorithm. Our goal is to determine whetherMulti-objective PSO can train RBF Networks, and the performanceis validated on accuracy and complexity. The experiments areconducted on benchmark datasets obtained from the VCI machinelearning repository. The results show that our proposed methodprovides an effective means for training RBF Networks that iscompetitive with other evolutionary computational-based methods.

1. INTRODUCTION

A Multi-Layer Perceptron (MLP), which is a kind of ArtificialNeural Networks (ANNs), has good nonlinear approximationability. Its applications to the pattern classification and thesystem identification have been much studied over the past twodecade [1]. However, the inner processes of an MLP which usessigmoid functions as the activation functions in the hidden layerare hardly interpretable. Therefore it is difficult to reflect priorknowledge on its learning. Above problem can be improved byradial basis function (RBF) network since basis functions in itshidden layer respond locally.

Since the number of input and output layer neurons isdetermined from the dimension of data, the model designer canonly set the number of hidden layer neuron. Hence the modelstructure determination is to determine the number of hiddenlayer neuron. Although parameter estimation methods such as thebackpropagation method are well known, the method of modelstructure determination has not been established due to modelcomplexity [1]. If a model is complex, the generalization abilityis low because of its high variance error [2]. Conversely, if amodel is simple, then it can't represent better correlation betweeninput and output because of the high bias error. The learningmethod with regularization and the model structure selection

Siti Mariyam Hj. Shamsuddin

University Technology MalaysiaFaculty of Computer Science and Information

SystemSkudai, lohor, Malaysia

mariyam@utm.my

method using information criteria have been studied [3] to take atrade-off about the model complexity into account. The approachto obtain the set of trade-off models based on multi-objectiveoptimization has also been studied [4][5][6].

In this study, the construction of the Pareto RBF Network setobtained from the perspective of the multi-objective optimizationis considered. The set of Pareto optimal RBF Networks based onevaluation of approximation ability and structure complexityusing evolutionary computations which show good results onmulti-objective optimization is obtained. The optimal RBFNetwork is constructed from the Pareto set since the obtainedPareto-optimal RBF Networks are diverse on their structure andthe diversity. The Pareto optimal RBF Network is applied to thepattern classification problems.

The remainder of this paper is organized as follows: Section 2describes an outline of related work of Evolutionary Multi­Objective RBF Networks; Section 3 describes RBF Network andmulti-objective optimization; Section 4 introduces the concept ofmult.i-objective optimization and particle swarm optimization;SectIon 5 presents the proposed approach to train RBF Networksinvolving a PSO-based multi-objective algorithm. Theimplementation, experiments done and results are discussed inSection 6. The paper closes with the Summary and Conclusion inSection 7.

2. RELATED WORK OF EVOLUTIONARY MULTI-OBJECTIVE RBF NETWORK

Although there are few studies regarding to implement multi­objective RBF Network training, but research on training ofRBFNetwork with evolutionary multi-objective is still fresh. Thissection presents some existing work of training RBF Networkbased on Multi-Objective Evolutionary Algorithms (MOEAs)

In [7], multi-objective (MOBJ) optimization algorithm hasbeen applied to the problem of inductive supervised learningdepended on smoothness based apparent complexity measure forRBF Networks. A multi-objective genetic algorithm based designprocedure for the RBF Network has been proposed in [8]. AHierarchical Rank Density Genetic Algorithm (HRDGA) hasbeen developed to evolve both the neural network's topology andparameters simultaneously.

A method in which RBF Network ensemble has beenconstructed from Pareto-optimal set obtained by multi-objectiveevolutionary computation has been proposed in [9]. Pareto-

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Proceeding ofthe 6th International Symposium on Mechatronics and its Applications (ISMA09), Sharjah, UAE, March 24-26, 2009

optimal set of RBF Networks has been obtained by multi­objective GA based on three criteria, Le. model complexity,representation ability, and model smoothness, and RBF Networkensemble has been constructed of this Pareto set. A newevolutionary algorithm, the RBF-Gene algorithm, has beenapplied to optimize RBF Networks [9]. Unlike other works, theiralgorithm can evolve both the structure and the numericalparameters of the network: it is able to evolve the number ofneurons and their weights.

Another study in [11], a problem of optimizing RBF Networkhas been presented from training examples as a multi-objectiveproblem and an evolutionary algorithm has been proposed tosolve it properly. This algorithm incorporates mutation operatorsto guide the search to good solutions. A method of obtainingPareto optimal RBF Network set based on multi-objectiveevolutionary algorithms has been proposed in [12]. RBFNetworks are widely used as a model structure for nonlinearsystems. The determination of its structure that is the number ofbasic functions, and the tradeoff between model complexity andaccuracy exists in this problem.

On the other hand [13], proposed a multi-objective geneticalgorithm to the identification of RBF Network couple models ofhumidity and temperature in a greenhouse. Two combinations ofperformance and complexity criteria were used to steer theselection of model structures, resulting in distinct sets ofsolutions.

Unlike previous studies that mentioned earlier, this researchshares the problem of unsupervised learning and supervisedlearning of RBF Network within the context of multi-objectivePSO to evolve toward Pareto-optimal front defined by severalobjective functions concerning with model accuracy and modelcomplexity.

3. RBF NETWORK AND MULTI-OBJECTIVEOPTIMIZATION

3.1. RBF Network

An ANN using Radial Basis Function (RBF) as activationfunction instead of sigmoid functions is RBF Network. The RBFis a function which outputs the maximum value at its center pointand decreases its output value as the input leaves from its center.Typically the Gaussian function is used for the activationfunction. RBF Network is used in function approximation, timeseries prediction, and control. The RBF Network is constructedof three layers which are the input layer, the hidden layer and theoutput layer, as shown in Fig.I. In the input layer, neurons whichnumber is same as the number of input dimension are placed. Theinput layer neuron has a role to transmit data to the hidden layer.The hidden layer neuron calculates a value of the basis functionby the received data from the input layer, and then transmits thevalue to the output layer. The output layer neuron calculates thelinear sum of values of the hidden neuron and outputs it. In thisstudy the Gaussian function is used as basis functions.

Let <I> j (x )be the j-th basis function, <I> j (x )is represented asfollows:

(1)

Here, x = (x}, X2••• ~)T is the input vector, Cj = (Clj , C2j, ••• ,Cdj)T

and (f] are the j-th center vector and the width parameter,respectively. The output of RBF network y which is the linearsum of basis function, as follows:

my = .L W j <t> j (x ) (2)

]=1

Here, y is the output of the RBF Network, m is the number of the

hidden layer neuron and Wj is the weight from jth neuron to theoutput layer.

Figure 1. Structure ofREF Network

To construct RBF Network, at first the number of the hiddenlayer neuron m must be set, and then the centers cj, the widths crjand the weights wj must be estimated. In the typical learning ofthe RBF Network, the network structure will be determined basedon prior knowledge or the experiences of experts, and then theparameters are estimated by methods such as the clustering andthe least square method. On the other hand, there are approachesin which the network structure and the parameters are estimatedby the evolutionary computation [14] [15]. Such a RBF Networkis called an evolutionary RBF Network.

3.2. Multi-objective Optimization ofRBF Network

Generally a mathematical model has a dilemma about modelcomplexity [1]. If a model is complex, then it can fit to data wellbecause it has low bias error, but the variance error become high,so the model generalization ability becomes worse. Moreover, acomplex model is not desirable since it is not easy to treat.Conversely, if a model is simple then variance error is small. Sosuch a model can prevent overlearning and the treatment of it willbe easy. The learning method with regularization and the modelstructure selection method using information criteria have beenstudied [3] to take a trade-off about the model complexity intoaccount. The trade off about model complexity is expressed byadding a penalty term about model complexity to the error termand using it as evaluation criterion of model construction ormodel selection. On the other hand, there is an approach to solvethe above problem by considering the model construction as themulti-objective optimization problem about the modelrepresentation ability and the model complexity [4][5][6]. Theevolutionary RBF network needs to evolve its structure with

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4.1. Multi-objective Optimization Problem

4.2. Multi-objective Particle Swarm Optimization Algorithm

4. MULTI-OBJECTIVE OPTIMIZATION ANDPARTICLE SWARM OPTIMIZATION

consideration of model complexity. By utilizing characteristics ofevolutionary algorithms that they are methods of multi-pointsearch, the method obtaining trade-off models has been studied[5] [16].

ii. Compute the new velocity:

V[i] = W x V[i] + Rl x (PBESTS[i] - P[i]) +R2 x (A[GBEST] - P[i])(W is the inertia weight equal to 0.4)(Rl and R2 are random numbers in the range[0.. 1])(PBESTS[i] is the best position that theparticle i have reached)(A[GBEST] is the global best guide for eachnon-dominated solution)

iii. Calculate the new position of P[i]:

P[i] = P[i] + V[i]iv. If P[i] goes beyond the boundaries, then it is

reintegrated by having the decision variabletake the value of its corresponding lower orupper boundary and its velocity is multipliedby -1 so that it searches in the oppositedirection.

1. For i = 1 to M (M is the population size)

a. Initialize P[i] randomly (P is the population ofparticles)

b. Initialize V[i] = 0 (V is the speed of each particle)

c. Evaluate P[i]

d. Initialize the personal best of each particle

PBESTS[i] = P[i]e. GBEST = Best particle found in P[i]

2. End For

3. Initialize the iteration counter t = 0

4. Store the non-dominated vectors found in P into A

(A is the external archive that stores non-dominatedsolutions found in P)

5. Repeat

a. Compute the crowding distance values of each non­dominated solution in the archive A

b. Sort the non-dominated solutions in A in descendingcrowding distance values

c. For i = 1 to M

i. Randomly select the global best guide for P[i]from a specified top portion (e.g. top 10%) ofthe sorted archive A and store its position toGBEST.

has a constraint handling mechanism for solving constrainedoptimization problems.

(3)

(4)

j=I, ...,N

k=I, ...,K

Many real-world problems involve simultaneous optimizationof several objective functions. Generally, these functions are non­commensurable and often conflicting objectives. Multi-objectiveoptimization with such conflicting objective functions gives riseto a set of optimal solutions, instead of one optimal solution. Thereason for the optimality of many solutions is that no one can beconsidered to be better than any other with respect to all objectivefunctions. These optimal solutions are known as Pareto-optimalsolutions.

A general multi-objective optimization problem consists of anumber of objectives to be optimized simultaneously and isassociated with a number of equality and inequality constraints. Itcan be formulated as follows:

Minimize Ii (x) i = 1, ..., M

{g .(x) =0

Subject to: }

hk(X) 5: 0

Where fi is the ith objective function, x is a decision vectorthat represents a solution, and M is the number of objectives.

For a multi-objective optimization problem, any two solutionsxl and x2 can have one of two possibilities- one dominates theother or none dominates the other. In a minimization problem,without loss of generality, a solution xl dominates x2 iff thefollowing two conditions are satisfied:

't/i E {1,2, , M} : f i (Xl) 5: f i (x2),

3j E {1,2, , M} : f j (xl) -< f j (x2)

The solutions which are not dominated by any other solutions arecalled the Pareto-optimal solution or non-dominated solution.Generally many Pareto-optimal solutions exist. The set of thePareto-optimal solutions is called Pareto optimal front. A non­dominated set is demanded to be near to the true Pareto front andto be distributed uniformly.

The MOPSO extends the algorithm of the single-objectivePSO to handle multi-objective optimization problems. Itincorporates the mechanism of crowding distance computationinto the algorithm of PSO specifically on global best selectionand in the deletion method of an external archive of non­dominated solutions. The crowding distance mechanism togetherwith a mutation operator maintains the diversity of non­dominated solutions in the external archive [17]. MOPSO also

v. If(t < (MAXT * PMUT),

Then perform mutation on P[i].(MAXT is the maximum number of iterations)(PMUT is the probability ofmutation)

vi. Evaluate P[i]

d. End For

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Figure 2. REF Network-MOPSO Procedure

Yes

Maximum

GenerationReached?

Output non­

dOininated

solutions

No'-------------'

!Evaluate()

(Objective Functions)

Detennine the

particle's

dimension

Set number of

hidden neurons

and m.aximmn

generation

the two objective functions and after the maximum generation isreached the algorithm outputs a set of non-dominated RBFNetworks (Pareto optimal set). Figure 2 illustrates this procedure.

PBESTS[i] = P[i]g. Increment iteration counter t

6. Until maximum number of iterations is reached

e. Insert all new non-dominated solution in P into A ifthey are not dominated by any of the stored solutions.All dominated solutions in the archive by the newsolution are removed from the archive. If the archiveis full, the solution to be replaced is determined bythe following steps:

i. Compute the crowding distance values of eachnon-dominated solution in the archive A

ii. Sort the non-dominated solutions in A indescending crowding distance values

iii. Randomly select a particle from a specifiedbottom portion (e.g. lower 10%) whichcomprise the most crowded particles in thearchive then replace it with the new solution

f. Update the personal best solution of each particle inP. If the current PBESTS dominates the position inmemory, the particles position is updated using

5. RBF NETWORK TRAINING USING Multi-ObjectivePSO 5.3. Objective Functions

The proposed algorithm called RBFN-MOPSO is a multi­objective optimization approach to RBF Network training withMOPSO as the multi-objective optimizer. The algorithm willsimultaneously determine the set of connection weights and itscorresponding architecture by treating this problem as a multi­objective minimization problem. In this study, a particlerepresents a one-hidden layer RBF Network and the swarmconsists of a population ofone-hidden layer RBF Networks.

5.1. Parameters and Structure Representation

The RBF Network is represented as a vector with dimension Dcontains the connections. The dimension of a particle is:

D =(I x H) +(H x0) + H + 0 (5)

Where I, H and 0 are refer to the number of input, hidden andoutput neurons respectively. The centers of RBF are initializedfrom k-means clustering algorithm and the connection weights ofRBF Network are initialized with random values. The number ofinput and output neurons is problem-specific and there is no exactway of knowing the best number of hidden neurons. We set thenumber of hidden neurons (RBFs) depended on the problem to besolved.

5.2. RBFN-MOPSO

RBFN-MOPSO starts by reading the dataset. This is followedby setting the desired number of hidden neurons and themaximum number of generation for MOPSO. The next step isdetermining the dimension of the particles and initializing thepopulation with fully-connected feed-forward RBF Networkparticles. In each generation, every particle is evaluated based on

Two objective functions are used to evaluate the RBFNetwork particle's performance. The two objective functions are:

1. Mean-squared error (MSE) on the training set.

1 N 2

11 = M ~(tkj -Okj) (6)

2. Complexity based on the sum of the squared weightswhich is based on the concept of regularization andrepresents the smoothness of the model.

I 2 =.!.t w J (7)2 j=l

6. EXPERIMENTS

6.1. Experimental Setup

The goal is to use a PSO-based multi-objective algorithm tosimultaneously optimize architectures and connections of RBFNetworks. To test its effectiveness, RBFN-MOPSO had beenapplied to the two datasets.

For each dataset, the experiments were implemented tominimize the influence of random effects. Each experiment usesa different randomly generated initial population. In addition, thenumber of input and output nodes is problem-dependent but thereis no exact method to determine the best number of hidden nodes.The parameters of the MOPSO algorithm are shown in Table 1and the characteristics of these datasets are summarized in Table2. The number of iterations are the same for all gorithms.

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Table 1. Execution parameters ofRBFN-MOPSO Table 3. Results ofMOPSO and MOGA ofRBFN on Iris

Parameter RBFN-MOPSOOptimization Type Minimization

Population Size 100Archive Size 100

Objective Functions 2Constraints 0

Lower Limit of Variable -0.5Upper Limit of Variable 0.5

Probability ofMutation (pM) 0.5

Table 2. Description ofthe datasets used in the experiments

RBFN-MOPSO RBFN-MOGA

Training Testing Training Tesingt

Number of Iterations 664 1 1000 1Error Rate (MSE) 0.049852 0.014109 0.37351 0.668619Classification Rate 95.82 97.21 55.74 55.91

Table 3 shows that RBFN-MOPSO is better than RBFN­MOGA with convergence to Pareto optimal set, error andaccuracy of classification. Figure 3 shows that the RBFN­MOPSO converge to Pareto optimal set with minimum iterationsand also improves the generalization on unseen data compared toMOGA.

Dataset Train Test Features Class MaxGenSet Set

Iris 120 30 4 3 1000Cancer 349 175 9 2 1000

6.2. Experimental Results

This section presents results of study on PSO-based multi­objective algorithm of RBF Network. The experiments areconducted by using two datasets: Iris and Cancer. The results foreach dataset are compared to NSGA-II [18] which is the lastversion of multi-objective Genetic Algorithms and analysedbased on the convergence to Pareto optimal set, error andclassification performance.

6.2.2. Cancer Data

The purpose of the breast cancer data set is to classify atumour as either benign or malignant based on cell descriptionsgathered by microscopic examination. It contains 9 attributes and699 examples of which 485 are benign examples and 241 aremalignant examples. The first 349 of the whole data set wereused for training, the following 175 examples for validation, andthe final 175 examples for testing [19]. The ending conditions ofRBFN-MOPSO and RBFN-MOGA are set to minimum error of0.05 or maximum iteration of 1000.

Cancer Training

90 1-",~---'------'----'------;:I=="'+~=RB=F=N:::!-M=O=PS=O=I;r

80 RBFN-MOGA

6.2.1. Iris Data 70

The Iris dataset is used for classifying all the information intothree classes which are iris setosa, iris versicolor, and irisvirginica. The classification is based on its four input patternswhich are sepal length, sepal width, petal length and petal width.Each class refers to type of iris plant contain 50 instances. Forthis dataset, the minimum error of MOPSO and MOGA is set to0.05 or maximum iteration of 1000.

60

~ 50

~ 40

30

20

Iris Training

Figure 3.Pareto optimal set on iris data

10

Figure 4.Pareto optimal set on cancer data

..........1IIi+ ................. ..>l<..>IL-->Io.I<...a..

o0!---=-0..l:-05=----0:L.1:--~0~.1~5~-0~.2~~0.~25=----0:-l.3::---0~.3.5Error

Table 4. Results ofMOPSO and MOGA ofRBFN on Cancer

In Cancer learning process, Table 4 shows the RBFN-MOPSOtakes 5 iterations compared to 1000 iterations in RBFN -MOGAto converge and illustrates that MOPSO-RBFN converges toerror with minimum error compared to RBFN-MOGA with anaccuracy of classification and error rate. Figure 4 shows that

RBFN-MOPSO RBFN-MOGA

Training Tesingt Training Testing

Number of Iterations 5 1 1000 1Error Rate (MSE) 0.039253 0.206486 0.1416 0.220508Classification Rate 89.67 68.09 57.73 58.29

I

... RBFN-MOPSO I+ RBFN-MOGA

1200

1000 ......

800 ....s=. ...C)

~600

.............oo!--o:-l-.1------=-o.l..:-2--0..L.3~t.:IIlIO.~411t+t1~0~.5 ....-0.~6-'++loMl-.7H-H-~0.8

Error

400

200

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MOPSO-RBFN is better than RBFN-MOGA in convergence toPareto optimal set.

7. CONCLUSIONS AND FUTURE WORK

This paper introduces PSO-based multi-objective approach toRBF Network design called Multi-objective PSO-RBF NetworkOptimizer to concurrently optimize the architectures andconnections of network. The optimal RBF network is constructedfrom Pareto-optimal set obtained by multi-objective PSO. RBFnetwork structure and its parameters are encoded to the particles,and Pareto-optimal set of RBF Networks is obtained by multi­objective PSO based on two criteria, Le. model complexity,representation ability. The benchmark of pattern classificationindicates that our proposed method can get better or comparableresults to other evolutionary computation-based methods found inexisting literature to date. Further improvement of MOGAalgorithm will be the automatic parameters tuning and a nextapplication of these algorithms will be on other data.

8. Acknowledgment

This work is supported by Ministry of Higher Education(MOHE) under Fundamental Research Grant Scheme (FRGSVOT 78243). Authors would like to thank Research ManagementCentre (RMC) Universiti Teknologi Malaysia, for the researchactivities and Soft Computing Research Group (SCRG) for thesupport and incisive comments in making this study a success.

9. REFERENCES

[1] O. Nelles, "Nonlinear System Identification", Springer,2001.

[2] L. Yu, K. K. Lai, S. Wang and W. Huang, "A BiasVariance-Complexity Trade-Off Framework for ComplexSystem Modeling", Springer-Verlag, 2006, pp. 518 - 527

[3] T. Hatanaka, K. Uosaki and T. Kuroda, "Structure Selectionof RBF Neural Network Using Information Criteria",Proceedings of Fifth International Conference onKnowledge-Based Intelligent Information EngineeringSystems and Allied Technologies, 2001.

[4] H. A. Abbass, "A Memetic Pareto Evolutionary Approach toArtificial Neural Networks", Australian Joint Conference onArtificial Intelligence, pp.I-12, 2001.

[5] G. G. Yen and H. Lu, "Hierarchical Rank Density GeneticAlgorithm for Radial-Basis Function Neural NetworkDesign", International Journal ofComputational Intelligenceand Applications, vo1.3, No.3, pp.213-232, 2003.

[6] Y. Jin, T. Okabe and B. Sendhoff: "Neural NetworkRegularization and Ensembling Using Multi-objectiveEvolutionary Algorithms", Proceedings of Congress onEvolutionary Computation, pp.I-8, 2004.

[7] I. Kokshenev and A. P. Braga, "A multi-objective approachto RBF network learning", Neurocomputing, Vo1.71,p. 1203­1209,2008.

[8] G. G. Yen, "Multi-Objective Evolutionary Algorithm forRadial Basis Function Neural Network Design", Springer­Verlag, Vo1. 16, pp. 221-239, 2006.

[9] N. Kondo, T. Hatanaka and K. Uosaki, "PatternClassification by Evolutionary RBF Networks EnsembleBased on Multi-objective Optimization", International JointConference on Neural Networks, pp. 2919- 2925, 2006.

[10] V. Lefort, C. Knibbe, G. Beslon and 1. Favrel, "SimultaneousOptimization of Weights and Structure of an RBF NeuralNetwork", Springer-Verlag, LNCS 3871, pp. 49-60, 2006.

[11] J. Gonzalez, I. P. H. Rojas and j. Ortega, "RBF NeuralNetworks, Multiobjective Optimization and Time SeriesForecasting, Springer-Verlag, LNCS 2084, p.498-505, 2001.

[12] N. Kondo, T. Hatanaka and K. Uosaki, "Nonlinear DynamicSystem Identification Based on Multiobjectively SelectedRBF Networks", Proceedings of the 2007 IEEE Symposiumon Computational Intelligence in Multicriteria DecisionMaking, pp. 112-127.

[13] Ferreira, P. M. Ruano., A.E. Fonseca and C.M."EVOLUTIONARY MULTIOBJECTIVE DESIGN OFRADIAL BASIS FUNCTION NETWORKS FORGREENHOUSE ENVIRONMENTAL CONTROL", Proc. ofthe 16 thIFAC World Congress, 2005.

[14] Y. Bai and L. Zhang,"Genetic Algorithm Based Self­Growing Training for RBF Neural Networks", ProceedingsofInternational Joint Conference on Neural Networks 2002,pp. 840-845, 2002.

[15] Z. Wenbo, H. De-Shuang and Y. Ge,"The StructureOptimization of Radial Basis Probabilistic Neural NetworksBased on Genetic Algorithms", Proceedings ofInternationalJoint Conference on Neural Networks 2002, pp. 1086-1091,2002.

[16] N. Kondo, T. Hatanaka and K. Uosaki,"Pareto RBFNetworks Based on Multiobjective EvolutionaryComputation", SICE Annual Conference in Sapporo,pp.2177-2182, 2004.

[17] C. R. Raquel and P. C. Naval, "An Effective Use ofCrowding Distance in Multiobjective Particle SwarmOptimization". Proceedings of the 2005 Conference onGenetic and Evolutionary Computation, pp. 257-264, 2005.

[18] K. Deb, A. Pratap, S. Agarwal and T. Meyarivan,"A Fast andElitist Multiobjective Genetic Algorithm: NSGA-II", IEEETransactions on Evolutionary Computation, vo1.6, No.2,pp.182-197, 2002.

[19] B. Al-kazemi and C. K. Mohan,"Training FeedforwardNeural Network Using Multi-phase Particle SwarmOptimization", Proceeding of the 9th InternationalConference on Neural Information Processing, 2002, p.2615-2619.

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