Archimedean copula estimation ofdistribution algorithm based on arti fi cialbee colony algorithm

Schoolof Information Science and Engineering,Shandong University,Jinan 250100,China

1.Introduction

Archimedean copula estimation ofdistribution algorithm based on arti fi cialbee colony algorithm

Haidong Xu,Mingyan Jiang*,and Kun Xu

Schoolof Information Science and Engineering,Shandong University,Jinan 250100,China

The arti fi cial bee colony(ABC)algorithm is a competitive stochastic population-based optimization algorithm.However,the ABC algorithm does not use the social information and lacks the knowledge of the problem structure,which leads to insuf fi ciency in both convergent speed and searching precision. Archimedean copula estimation of distribution algorithm(ACEDA) is a relatively simple,time-economic and multivariate correlated EDA.This paper proposes a novelhybrid algorithm based on the ABC algorithm and ACEDA called Archimedean copula estimation of distribution based on the arti fi cial bee colony(ACABC) algorithm.The hybrid algorithm utilizes ACEDA to estimate the distribution model and then uses the information to help arti fi cial bees to search more ef fi ciently in the search space.Six benchmark functions are introduced to assess the performance of the ACABC algorithm on numericalfunction optimization.Experimentalresults show that the ACABC algorithm converges much faster with greater precision compared with the ABC algorithm,ACEDA and the globalbest(gbest)-guided ABC(GABC)algorithm in most ofthe experiments.

arti fi cial bee colony(ABC)algorithm,Archimedean copula estimation of distribution algorithm(ACEDA),ACEDA based on arti fi cialbee colony(ACABC)algorithm,numericalfunction optimization.

1.Introduction

Population-based optimization algorithms are biologicalinspired optimization algorithmswhich are capable of fi nding the near-optimal solutions to complicated numerical and real-valued problems.Some classic algorithms such as the genetic algorithm(GA)[1],the estimation of distribution algorithm(EDA)[2]and the arti fi cialbee colony (ABC)algorithm[3]are allpopulation-based optimization algorithms.

The ABC algorithm was proposed by Karaboga in 2005 [3],which is inspired by the foraging behavior of bees. The ABC algorithm is simple and uses only common control parameters such as the population size and the max cycle number.Experimentalresults on function optimization have shown that the ABC algorithm outperforms,or atleast performs as well as other well known populationbased algorithms like GA,PSO[4,5],etc.

However,there are still insuf fi ciencies in the ABC algorithm.The population reproduction mechanism of the ABC algorithm reveals that this algorithm does not take advantage of the socialinformation such as the globalbest value ofthe currentswarm,and lacks the knowledge ofthe problem structure when searching for the optimal value, which leads to insuf fi ciency in both convergentspeed and searching precision.Moreover,the solution search equation reveals thatthe ABC algorithm is good atexploration butpoor atexploitation.To conquer this problem,various improved strategies have been proposed.Zhu and Kwong [6]proposed global best(gbest)-guided ABC(GABC)by incorporating the information of the gbest solution into the solution search equation to improve the exploitation. Leietal.[7]proposed an improved ABC algorithm,which introduced an inertialweightto the originalABC iteration equation to balance local and global searching processes. Akay and Karaboga[8]modi fi ed the original ABC algorithm by employing two new solution search strategies,including frequency and magnitude of the perturbation.Gao and Liu[9]improved the ABCalgorithm by combining the mutation scheme of the differentialevolution(DE).Experimental results[6-9]show that these variant ABC algorithms improved the convergent speed and the searching precision to a certain extent.

Estimation of distribution algorithm(EDA)is an evolutionary algorithm which derives from GA[10],and it combines intelligence computation and the knowledge of statistics.EDA uses the probability modeling technique toguide the generation of new population.Population-based incrementallearning(PBIL)proposed by Baluja[11],the Bayesian optimization algorithm(BOA)proposed by Pelikan[12]are allpopular EDAs.

The copula theory was introduced into EDA by Fabrizio and Carlo[13].In copula EDA(CEDA),joint distribution of all variables is used to describe the correlation among variables,which can estimate the problem structure in a simple way.Archimedean copula EDA(ACEDA)is an importantbranch in CEDA,and a number of researches have been done in this fi eld[14,15].

In this paper,we propose a new hybrid algorithm based on the ABC algorithm and the ACEDA,which is called Archimedean copula ABC(ACABC)algorithm.This hybrid algorithmutilizes the ACEDAto estimate the distribution model(thatis,to learn the problem structure)and then uses the information to guide arti fi cialbees to search more ef fi ciently in the search space.This hybrid algorithm combines the advantages of both ABC and ACEDA.On one hand,ACABC keeps learning the problem structure during the searching process by incorporating the ACEDAmechanism,and uses the information to generate new population. This mechanism can guide bees to search directly,which accelerates the convergent speed and improves the exploration ability.On the otherhand,an improved gbest-guided mechanism is introduced into ACABC to improve the exploitation ability.The experimentalresults tested on six numericalbenchmark functions show thatthe ACABC algorithm converges much faster with greater precision compared with the ABC algorithm,ACEDA and the GABC algorithm in mostof the experiments.

The rest of the paper is organized as follows.Section 2 summarizes the ABC algorithm.Section 3 introduces the basic copula theory and ACEDA.The hybrid ACABC algorithm is described in Section 4 and experimentalsettings and results are given in Section 5.Finally,the conclusion is drawn in Section 6.

2.ABC algorithm

As mentioned above,ABC is a swarm intelligence algorithm by simulating the foraging behaviors of the bee swarm.In a natural bee swarm,there are three kinds of bees to search food,the employed bees,the onlookers,and the scouts.The employed bees search food around the food source in their memories,then they share the food information with the onlookers through waggle dancing.Each onlookerbee chooses a food source found by the employed bees,and then further searches around the selected food source.The food source with more nectarhas largerchance to be selected by the onlooker bees than the one with less nectar.The scouts are a few employed bees which abandon their original food sources and randomly search for new ones.

For an optimization problem in a D-dimensionalspace, the position of a food source represents a potential solution,and the nectar amount of a food source represents the fi tness value of the solution.The number of employed bees or the onlooker bees is equal to the number of food sources,which means one food source is exploited by one employed bee.

Xi=(xi1,xi2,...,xiD)denotes the i th food source in the population,where D is the dimension of the problem. The exploitation mechanism used by both the employed bees and the onlookers is given as follows:

where Vi=(vi1,vi2,...,viD)is the new candidate solution generating from the neighborhood of current solution Xi(i=1,2,...,N),N is the population size,Xkis a randomly selected solution in the population(k= 1,2,...,N and k/=i),j=1,2,...,D is a random index, andφijis a uniform random number in the range[-1,1]. Then the greedy selection is operated between Viand Xito retain the better solution.

When all the employed bees fi nish their neighborhood search according to(1),they share allthe food information with allthe onlookers.Each onlookerselects a food source to do furthersearch according to the probability calculated by(2).

where f itiis the fi tness value of the i th solution in the population,and piis called the following probability.As shown in(2),piis proportionalto the fi tness value,and the solution with a larger fi tness value has a higher chance to be selected.

Ifone food source is notupdated overa prede fi ned numberofcycles,which means there is no betterfood source in its neighborhood,this food source is abandoned by the employed bee.The prede fi ned number of cycles is a control parameter called“l(fā)imit”.When the employed bee abandons its food source,it becomes a scout and searches a new food source randomly in the whole searching space according to(3).

3.ACEDA

EDA is an evolutionary algorithm derived from GA[16], which combines intelligence computation and the knowl-edge of statistics.EDA retains the selection operator in GA,but replaces the crossover and the mutation operator with the statisticalmodeland the sampling theory.One of the most essentialparts in EDA is to build a proper statisticalmodel.

The copula theory is a new branch in statistics,which constructs a multivariate joint distribution function with a given marginal distribution function and correlations among all variables.Basic de fi nitions and theorems are given in[16].One essential theorem in the copula theory is the Sklar theorem[16].This theorem expounds the construction method of multivariate jointdistribution by using the copula function and marginaldistribution functions.In the copula theory,the multivariate joint distribution function is constructed based on the essential Sklar theorem.

Copula functions mainly includes elliptic functions[17] and Archimedean copula functions[18].De fi nitions of the Archimedean copula function are given in[19].Two types of Archimedean copula functions Clayton and Gumbelare mainly used in this paper,as is shown in Table 1.

Table 1 Clayton and Gumbelfunction

ACEDA is based on EDA and the copula theory,which constructs the statistical model with the Archimedean copula function and the marginaldistribution function based on the Sklar theorem.In this algorithm,the estimation of the statistical model includes copula function estimation and marginal function estimation,then new population is generated by sampling from the copula function. The framework of ACEDA is shown in Fig.1.

Fig.1 Framework of ACEDA

In the following,the estimation of the marginal distribution function and the sampling method of the ndimensional Archimedean copula function are fi rst introduced,then the speci fi c processofthe Archimedean copula EDA is listed.

In the n-dimensional ACEDA,there are mainly two marginaldistribution estimation methods.One is based on the empiricalfunction,and the other is based on the Gaussian probability model.In this paper,the Gaussian probability model is used to estimate the marginal distribution. After selecting the dominantpopulation which consists of S individuals,mean valueμjand standard deviationσjof the j th dimension variable are calculated according to(4) and(5)respectively.The j th Gaussian marginal distribution is denoted as

The sampling method of the n-dimensional Archimedean copula function is as follows.C denotes the Archimedean copula function,and?represents its generator,while(U1,U2,...,Un)is the random vector which obeys the jointdistribution C.According to the algorithm proposed by Marshall and Olkin in[20],if there is a distribution function F which yields that F(0)=0,and the Laplace transform of F is equalto the inverse function of generator?,thatis to say??1=L?1[F],then the samples (u1,u2,...,un)of(U1,U2,...,Un)can be generated as follows.

Algorithm 1The sampling method of the ndimensionalArchimedean copula function

Step 1Generate variable v which obeys the distribution F,v～F=L?1[??1],where L?1[??1]denotes the inverse Laplace transform of??1.

Step 2Generate independent variables xj～U[0,1], j=1,2,...,n;

Step 3uj=??1((-ln xj)/v),j=1,2,...,n,then the sample(u1,u2,...,un)which obeys the jointdistribution function C is generated.

The process of Clayton and Gumbel ACEDA is shown as follows.

Algorithm 2The Clayton and GumbelACEDA

Step 1Population initialization.Initialize the population size NP,iterations Cycle,selection ratio s,and mutation ratio c.Randomly generate population size solutions as the initialpopulation,and then calculate the fi tness value of each population.

Step 2Construct dominantpopulation.Sortthe population in descending order according to the fi tness values; select the top S individuals according to(6)to construct the dominantpopulation.

Step 3Estimate the marginal Gaussian distribution of each dimension N(μj,σ2j)(j=1,2,...,n),according to (4)and(5).

Step 4Perform sampling operation on the given copula function to generate L new individuals which obey the jointdistribution.

Step 4.1The inverse Laplace transform of??1for the Clayton and Gumbel function obeys Gamma distribution and Alpha-stable distribution respectively[21],as is shown in(7)and(8).Generate variable v which subjects to Gamma distribution or Alpha-stable distribution according to(7)or(8).Generate independentvariables vj(j=1,2,...,n),which is uniformly distributed in the range[0,1],and then uj(j=1,2,...,n),for the Clayton or Gumbelfunction which is obtained according to(9)or (10).

Step 4.2According to(11),generate the k th new individualbased on the combination of the one-dimension Gaussian distribution and the Clayton or Gumbelcopula function value.

Step 5Constructnew population.The new population consists of the top S individuals from the previous generation,L new individuals which subjectto the jointdistribution and NP-S-L mutated individuals which are randomly generated in the searching space.

Step 6Judge whetherthe algorithm satis fi es the terminal condition or not.If the terminal condition is reached, the program willstop.Otherwise,return to Step 2.

4.ACABC algorithm

In this paper,we propose an improved ABC algorithm by combining ABC with ACEDA to improve the convergent speed and searching precision.The novelhybrid algorithm is ACABC algorithm.

In the ABC algorithm,the onlookers obtain food information from all employed bees,choose a better food source and then do further search around it.The onlooker mechanism is one of the mostessentialmechanisms in the ABC algorithm.In the ACABC algorithm,we combine the ACEDA with onlookers and propose a modi fi ed onlooker mechanism.After obtaining all the food information,the modi fi ed onlookers select excellentfood sources with a certain ratio,analyze the distribution rule of excellent food sources by estimating the distribution of them. Then these onlookers take the following strategies to update the currentfood sources.Firstly,preserve the excellent food sources selected previously,then predict the location ofnew food sources according to the estimated distribution information,and fi nally replace those poor food sources with these predicted new ones.Thatis to say,the onlookers adjust the searching direction of the whole colony based on the distribution of promising food sources.This modifi ed onlooker mechanism has learning and analyzing ability and enhanced globalperformance,which is a more intelligentonlookermechanism.The implementation fl ow of this modi fi ed onlooker mechanism is shown as follows.

Algorithm 3Modi fi ed onlookermechanism

Step 1Sortthe solutions obtained by employed bees in descending orderbased on the fi tness values,and calculate the following probability ofeach solution according to(2).

Step 2Set the selection ratio s=0.3,and select the top S solutions according to(6)to constructthe dominant population.

Step 3Estimate the marginal Gaussian distribution of each variable according to(4)and(5).

Step 4After choosing one solution according to the following probability,the onlooker decides the sort order of this solution.If the sort order is less than S,then the neighborhood search mechanism of ABC is operated to generate new candidate solution;otherwise,the new candidate solution is generated based on ACEDA.

Step 5Calculate the fi tness value of new solutions, choose a better solution between the new candidate solution and the old one based on the greedy criterion to constructthe new population.

On the basis of the modi fi ed onlooker mechanism,two more improved strategies are proposed to enhance the adaptability and the exploitation.

4.1 Dynamicaladjustment of the searching strategyThe modi fi ed onlooker mechanism shown above includes two searching strategies.According to the selection ratio s=0.3,the top 30%individuals are updated by the neighborhood search strategy while the rest 70%are updated based on ACEDA.However,as the iteration goes on,an increasing number of individuals are approaching the global best solution and the ratio of excellent individuals is getting larger.As a result,a growing number of individuals can notbe updated by ACEDA,which leads to the searching inef fi ciency.To improve this situation,an adaptive selection ratio is introduced,as is shown in(12).

where iter denotes the currentiteration number,and Cycle is the maximaliteration number.The initialvalue is set as 0.3,and the selection ratio grows linearly as the iteration increases.The numberof selected individuals in each iteration is calculated according to(13),which indicates that more individuals are updated by the neighborhood searching mechanism while fewer of them are updated by copula EDA.In a word,the searching strategies are adjusted dynamically by introducing the adaptive parameter to improve the search ef fi ciency.

4.2 Improved gbest-guided neighborhood search mechanism

Inspired by the gbest-guided mechanism in[6],the neighborhood search equation is modi fi ed to improve the exploitation of the algorithm,as is shown in(14).

where the third term is a new added term,and yjis the j th element of the current global best solution.As it is in(15),a new adaptive parameter w is introduced to adjust the neighborhood search strategy dynamically.w decreases linearly as the iteration goes on,which means that the generation ofthe new candidate solution is increasingly dependenton the globalbestvalue,while the in fl uence ofthe current solution xijis weakened progressively.The new candidate solution is driven towards the global best solution in this way to improve the exploitation.

The process of the ACABC algorithm is shown as follows.

Algorithm 4Archimedean copula estimation of distribution by the ABC algorithm

Step 1Parameterinitialization.Initialize the population size NP,iterations Cycle,trial,limit,selection ratio s, and the parameterθused in ACEDA.

Step 2Population initialization.Randomly generate NP solution X={Xi|i=1,2,...,NP}as the initial population,calculate the fi tness value of each population, set triali=0 for each solution Xiand then initialize the gbestsolution.

Step 3Employed bee stage.Each employed bee conducts the gbest-guided neighborhood search according to (14)to generate the new candidate solution,calculates the fi tness value,and updates the currentsolution based on the greedy criterion.If Xiis updated as the new candidate solution,set triali=0;otherwise,triali=triali+1.

Step 4Calculate the following probability according to (2),sort allsolutions in descending order according to fi tness values,compute the numberofexcellentindividuals S according to(12)and(13),and selectthe top S individuals to constructthe dominantpopulation.

Step 5The onlooker stage.Carry out Step 3-Step 5 in Algorithm 3.

Step 6Update the globalbestsolution.Selectthe solution with the largest fi tness value as the currentgbestsolution.

Step 7The scouts stage.Give up the solution Xiwith tiraliexceeding the prede fi ned value limit,randomly generate a new solution in the search space,calculate the fi tness value,and set tirali=0.

Step 8Judge whetherthe algorithm satis fi es the terminal condition or not.If the terminal condition is reached, outputthe globalbestsolution as the fi naloptimization result.Otherwise,return to Step 3.

5.Experiments and results

In this paper,two types of ACABC algorithms,namely the Clayton ACABC algorithm and the Gumbel ACABC algorithm are proposed based on Archimedean Clayton and Gumbel functions.In this part,the performance of the Clayton ACABC algorithm and the Gumbel ACABC algorithm are compared with ABC,GABC,Clayton CEDA and Gumbel CEDA by optimizing benchmark functions.

5.1 Benchmark functions

The details of six benchmark functions used in this paper are given in Table 2.

Table 2 Benchmark functions

5.2 Optimalparameter selection

Population size NP and maximaliteration times Cycle are essentialparameters which have a signi fi cantimpacton the performance of all population-based algorithms.The experimental results in[14,15,21]showed that the ACEDA obtains the best performance with large population sizes and few iteration times,while the ABC algorithm requires small population sizes and more iteration times[4-9].As a result,experiments are designed in this partto select the best parameters for the novel hybrid algorithms Clayton ACABC and Gumbel ACABC.

In this paper,the fi tness evaluation times,namely the productof population and iteration times of differentalgorithms,are equalto each other to evaluate the performance of differentalgorithms fairly.

The numberof fi tness evaluation is setas 200 000 times, and fi ve different parameter combinations are selected to optimize six benchmark functions respectively.The other parameters used in the algorithms are as follows.The dimension of allfunctions is D=100,for Clayton ACABC the parameter withθ=0.1,while for Gumbel ACABC withθ=1.In differenttestenvironments,each algorithm runs 10 times and computes the mean value as the fi nalresults.Results for Clayton ACABC and Gumbel ACABC are shown in Table 3 and Table 4 respectively.

Table 3 Optimization results of Clayton ACABC with different population sizes and iterations

Table 4 Optimization results of Gumbel ACABC with different population sizes and iterations

As shown in Table 3,data in bold are the best optimization results ofeach function.For Griewank,Rastrigin, Ackley and Rosenbrock,the bestoptimization resultis obtained when the parameters are NP=40,Cycle=5 000, especially for Rastrigin,the optimization result is much betterthan the others.While for Sphere and Schwefel2.22, the best optimization results are obtained when NP= 100,Cycle=2 000.However,we can also observe that the optimization results when NP=40,Cycle=5 000 are nearly the same with the bestones.

According to the results shown in Table 4,for Griewank, the bestoptimization resultis obtained when NP=100, Cycle=2 000,and when NP=40,Cycle=5 000,the optimization result is very close to the best result.While for the other fi ve functions,the best optimization results are allobtained when NP=40,Cycle=5 000.

Based on the results shown in Table 3 and Table 4,we can draw the conclusion thatboth the Clayton ACABC algorithm and the Gumbel ACABC algorithm can get the best performance when NP=40,Cycle=5 000.Thus in the following experiments,the population size is 40 and the iteration times is 5 000 for the Clayton ACABC algorithm and the GumbelACABC algorithm.

5.3 Performance comparison of different algorithmsIn this part,six algorithms including ABC,GABC,Clayton CEDA,Gumbel CEDA,Clayton ACABC and Gumbel ACABC are used to optimize six benchmark functions respectively,and the performance on search accuracy and the convergentspeed of the six algorithms are compared.

The numberof fi tness evaluation is setas 200 000 times. In ABC,GABC,Clayton ACABC and Gumbel ACABC, NP=40 and Cycle=5 000,while for Clayton CEDA and Gumbel CEDA,NP=100 and Cycle=2 000.The other parameters are setas follows.D=100,limit=50, θ=0.1 in Clayton ACABC andθ=1 in Gumbel ACABC.Each algorithm run 30 times and the mean value is calculated as the fi nal results.Experimental results are shown in Table 5.

Table 5 Simulation results achieved by different algorithms

As the results shown in Table 5,allof the six algorithms can optimize successfully on Sphere,Griewank,Ackley and Schwefel 2.22,and both Clayton ACABC and GumbelACABCcan obtain optimization results with higheraccuracy compared with the other four algorithms.The optimization results on Rastrigin indicate that only Clayton ACABC and Gumbel ACABC can optimize successfully with high precision,while ABC,GABC,Clayton CEDA and Gumbel CEDA cannot optimize Rastrigin successfully.However,the results on Rosenbrock show that all algorithms cannot achieve satisfying optimization results, and ABC and GABC outperform the CEDA and ACABC algorithms.

According to the optimization results,we can obtain the conclusion that both Clayton ACABC and Gumbel ACABC can optimize mostof the test functions with high precision and outperform ABC,GABC,Clayton CEDA, and Gumbel CEDA.

In orderto evaluate the convergentspeed ofthe proposed ACABC algorithms,the convergence curves of fi ve functions except Rosenbrock are drawn in Figs.2-6 respectively.It can be observed that Clayton ACABC and Gumbel ACABC accelerate the search speed and improve the convergentspeed signi fi cantly.

According to the experimental results in this part,we can draw the conclusion that Clayton ACABC and Gumbel ACABC can speed up the convergent speed with greater precision for mosttestfunctions.The novelhybridACABC algorithms outperform the ABC algorithms and ACEDA.

Fig.2 Convergence curve on Sphere(Sphere with dim=100)

Fig.3 Convergence curve on Griewank(Griewank with dim=100)

Fig.4 Convergence curve on Rastrigin(Rastrigin with dim=100)

Fig.5 Convergence curve on Ackley(Ackley with dim=100)

Fig.6 Convergence curve on Schwefel 2.22(Schwefel 2.22 with dim=100)

6.Conclusions

In this paper,we propose a novelhybrid algorithm called ACABC algorithm.The ABC algorithm is a biologicalinspired optimization algorithm with a random search process.In order to improve the search ef fi ciency,we introduce ACEDA into the onlooker stage and propose a more intelligent onlooker mechanism.In the modi fi ed mechanism,onlookers fi rst sort all solutions obtained by employed bees in descending order according to the fi tness values,select excellent individuals to construct dominant population,then constructa probability distribution model based on ACEDA,and sample based on the estimated distribution to generate new individuals.The updating strategy ensures that the new population obeys the distribution ofthe dominantpopulation.In addition,we modify the neighborhood search strategy with the gbestvalue by guiding the operatorand the adaptive parameter to improve the exploitation of the algorithm.Experimental results show thatthe ACABCalgorithm speeds up the convergentspeedwith greater precision for most test functions.In conclusion,the novelhybrid ACABC algorithms outperform the ABC algorithms and ACEDA.

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Biographies

Haidong Xu was born in 1990.She received her B.S.degree from Shandong University in June, 2012 in communication engineering and now she is an M.Sc.student in the same University.Her main research area includes arti fi cial intelligence and computing intelligence,arti fi cial neural network,swarm intelligence algorithms and mathematicalstatistics.

E-mail:xu-hai-dong1990@163.com

Mingyan Jiang was born in 1964.He received his M.S.degree from Shandong University in 1992 and his Ph.D.degree in 2005.He fi nished his postdoctoral research in Spain(CTTC)in communication signaland system in 2007.Now he is a fullprofessor and a doctoral supervisor in the School of Information Science and Engineering in Shandong University,China.His research interests include softcomputing,signal and image processing,computer network,arti fi cial intelligence and data mining.He has published more than 200 professional papers and 6 books.

E-mail:jiangmingyan@sdu.edu.cn

Kun Xu was born in 1988.He received his B.S. degree from Shandong Normal University in June, 2011 in electronic information engineering and now is an M.Sc.student in Shandong University.His main research interests include machine learning, parallel computation,optimization and arti fi cial intelligence.

E-mail:xukun sdu@163.com

10.1109/JSEE.2015.00045

Manuscriptreceived on March 19,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61201370),the Special Funding Project for Independent Innovation Achievement Transform of Shandong Province (2012CX30202)and the Natural Science Foundation of Shandong Province(ZR2014FM039).