Adaptive Imperialist Competitive Algorithm(AICA) - Semantic Scholar
Adaptive Imperialist Competitive Algorithm (AICA) Marjan Abdechiri Elec., comp. & IT Department, Qazvin Azad University, Qazvin, Iran, [email protected]
Karim Faez Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran, [email protected]
Abstract— The novel Imperialist Competitive Algorithm (ICA) that was recently introduced has a good performance in some optimization problems. The ICA inspired by sociopolitical process of imperialistic competition of human being in the real world. In this paper, a new Adaptive Imperialist Competitive Algorithm (AICA) is proposed. In the proposed algorithm, for an effective search, the Absorption Policy changed dynamically to adapt the angle of colonies movement towards imperialist’s position. The ICA is easily stuck into a local optimum when solving high-dimensional multi-model numerical optimization problems. To overcome this shortcoming, we use probabilistic model that utilize the information of colonies positions to balance the exploration and exploitation abilities of the imperialistic competitive algorithm. Using this mechanism, ICA exploration capability will enhance. Some famous unconstraint benchmark functions used to test the AICA performance. Also, we use the AICA Algorithm to adjust the weights of a three-layered Perceptron neural network to predict the maximum worth of the stocks change in Tehran’s Bourse Market. Simulation results show this strategy can improve the performance of the ICA algorithm significantly.
gradient descent is a very popular optimization method, it plagued by slow convergence and susceptibility to local minima. Therefore, other approaches to improve NN training introduced. These methods include global optimization algorithms, such as Simulated Annealing [15], Genetic Algorithms [16,17], Particle Swarm Optimization algorithms [18,19,20] and other Evolutionary Algorithms. Recently, a new algorithm has been proposed by Atashpaz-Gargari and lucas [21], in 2007 that has inspired from a socio-human phenomenon. In this paper, we have proposed a new algorithm called Adaptive Imperialist Competitive Algorithm (AICA) that uses the probability density function to adapt the angle of colonies movement towards imperialist’s position during iterations dynamically. This mechanism, enhance the global search capability of the algorithm. This idea increases the performance of the ICA algorithm effectively in solving the optimization problems. We examined the proposed algorithm in several standard benchmark functions that usually tested in Evolutionary Algorithms. Also, we use the AICA Algorithm to adjust the weights of a three-layered Perceptron Neural Network to predict the maximum worth of the stocks change in Tehran’s Bourse Market[22]. The results of applying the proposed algorithm on benchmark functions and Neural Network to predict the maximum worth of the stocks change in Tehran’s Bourse Market indicated that the convergence speed and the quality of obtained solution in compare with ICA, PSO using a Sugeno function as inertia weight decline curve[23] and GA algorithm show a good performance. The rest of this paper organized as follows. Section two, provides an introduction the ICA algorithm. In section three, Adaptive Imperialistic Competitive Algorithm is proposed. Fourth section is devoted to the empirical results of proposed algorithm implementation and its compression with the results obtained by ICA, PSO and GA algorithms. The last section concludes the paper.
Keywords-Imperialist Competitive Algorithm; absorption policy; density probabilistic model.
I.
INTRODUCTION
The global optimization problem is applicable in every field of science, engineering and business. So far, many Evolutionary Algorithms (EA) [1,2], have been proposed for solving the global optimization problem. Inspired by the natural evolution, EA analogizes the evolution process of biological population, which can adapt the changing environments to the finding of the optimum of the optimization problem through evolving a population of candidate solutions. Some Evolutionary Algorithms for optimization problem are: the Genetic Algorithm (GA) [2,3,4,5,6,7], at first proposed by Holland, in 1962 [4], Particle Swarm Optimization algorithms (PSO) [8,9] that at first proposed by Kennedy and Eberhart [8], in 1995, Simulated Annealing (SA) [10,11,12], Cultural Evolutionary algorithms (CE) [13,14] at first was developed by Reynolds, in the early 1990s [14] and etc. The optimization methods are extensively used to adjust the weights of multi-layered Neural Networks. While Proc. 9th IEEE Int. Conf. on Cognitive Informatics (ICCI’10) F. Sun, Y. Wang, J. Lu, B. Zhang, W. Kinsner & L.A. Zadeh (Eds.) 978-1-4244-8040-1/10/$26.00 ©2010 IEEE
Helena Bahrami Elec., comp. & IT Department, Qazvin Azad University, Qazvin, Iran, [email protected]
II. INTRODUCTION OF IMPERIALIST COMPETITIVE ALGORITHMS (ICA) In this section, we introduce ICA algorithm and chaos theory.
W:? B*X@ YX ZXB? - \
A. Imperialist Competitive Algorithm (ICA)
^@Y PB*ZXYB'4'Y@ XY? U'''''''''''''''*_-
Imperialist Competitive Algorithm (ICA) is a new evolutionary algorithm in the Evolutionary Computation field based on the human's socio-political evolution. The algorithm starts with an initial random population called countries. Some of the best countries in the population selected to be the imperialists and the rest form the colonies of these imperialists. In an N dimensional optimization problem, a country is a array. This array defined as below
Where W:? is the total cost of the nth empire and ^ is a positive number which is considered to be less than one. `c*k" q O-'''''''''''''''''''''''''''''''''''''*zIn the absorption policy, the colony moves towards the imperialist by x unit. The direction of movement is the vector from colony to imperialist, as shown in Fig.1, in this figure, the distance between the imperialist and colony shown by d and x is a random variable with uniform distribution. Where q is greater than 1 and is near to 2. So, a proper choice can be q 7. In our implementation { is } * O- respectively. N
! " # " $ " % &'''''''''''''''''''''''''''''''*The cost of a country is found by evaluating the cost function f at the variables * ! " # " / " $ " % -. Then 3 4* 3 - 4* 3! " 3# " $ " 3% -'''''''''''*7-
`c*{" {The algorithm starts with N initial countries and the 389 best of them (countries with minimum cost) chosen as the imperialists. The remaining countries are colonies that each belong to an empire. The initial colonies belong to imperialists in convenience with their powers. To distribute the colonies among imperialists proportionally, the normalized cost of an imperialist is defined as follow
(8)
In ICA algorithm, to search different points around the imperialist, a random amount of deviation is added to the direction of colony movement towards the imperialist. In Fig. 1, this deflection angle is shown as , which is chosen randomly and with an uniform distribution. While moving toward the imperialist countries, a colony may reach to a better position, so the colony position changes according to position of the imperialist.
:? @ 3 3 ? ''''''''''''''''''''''''''''''''*AWhere, B? is the cost of nth imperialist and :? is its normalized cost. Each imperialist that has more cost value, will have less normalized cost value. Having the normalized cost, the power of each imperialist is calculated as below and based on that the colonies distributed among the imperialist countries. ? C
DE HGIJ
FGKM
C'''''''''''''''''''''''''''''''''*NDG
Figure1. Moving colonies toward their imperialist
On the other hand, the normalized power of an imperialist is assessed by its colonies. Then, the initial number of colonies of an empire will be
In this algorithm, the imperialistic competition has an important role. During the imperialistic competition, the weak empires will lose their power and their colonies. To model this competition, firstly we calculate the probability of possessing all the colonies by each empire considering the total cost of empire.
:? OP ? Q *RST -U''''''''''''''''''''''*VWhere, :? is initial number of colonies of nth empire and RST is the number of all colonies. To distribute the colonies among imperialist, :? of the colonies is selected randomly and assigned to their imperialist. The imperialist countries absorb the colonies towards themselves using the absorption policy. The absorption policy shown in Fig.1, makes the main core of this algorithm and causes the countries move towards to their minimum optima. The imperialists absorb these colonies towards themselves with respect to their power that described in (6). The total power of each imperialist is determined by the power of its both parts, the empire power plus percents of its average colonies power.
W:? @ 3 PW:3 U W:? '''''''''''''''''''''''''*Where, W:? is the total cost of nth empire and W:? is the normalized total cost of nth empire. Having the normalized total cost, the possession probability of each empire is calculated as below 9E C
%DE HGIJ FGKM %DG
C''''''''''''''''''''''''''''''''*k-
In each iteration, the country densities compute using the probabilistic model in Eq(11). If the countries density in the current iteration is more than the previous iteration, then with 85% the previous angle of the movement of the countries towards their empires will be shrunk and with 15% the mentioned angle will be expanded.
after a while all the empires except the most powerful one will collapse and all the colonies will be under the control of this unique empire. III.
THE PROPOSED IMPERIALIST COMPETITIVE ALGORITHM
3 kQ V*3¡! \ ¢- \ kQV*3¡! ¢-'''*V-
The ICA algorithm like many Evolutionary Algorithms suffers the lack of ability to global search properly in the problem space. During the search process, the algorithm may trap into local optima and it is possible to get far from the global optima. This causes the premature convergence. In this paper, a new method suggested that balance the exploration and exploitation abilities of the proposed algorithm, using colonies positions information. In the ICA algorithm absorption policy that mentioned in the previous section, the colonies move towards imperialists with an angle, which is a random variable. The colonies movement because of the constant parameter has a monotonic nature, so the colonies movement could not be adapted with the search process. Therefore, if the algorithm traps in the local optima, it cannot leave it and move towards the global optima. For solving this problem, and make balance between the explorative and exploitative search, we define the parameter adaptively, and dynamically adjust the movement of colonies to the imperialists during the search process.
3 , is the current angle of movement. 3¡! , is the previous angle and £ is the step size of shrinking and expanding the angle of movement. The value of this step size is varying between 0.0001 and 0.1. Otherwise, if the countries density in the current iteration is less than the previous iteration, then with 85% the previous angle of the movement of the countries towards their empires will be expanded and with 15% the mentioned angle will be shrunk. 3 kQ7V*3¡! \ ¢- \ kQzV*3¡! ¢-'''*_If the countries density in the current iteration is more than the previous iteration, it means that may be the countries are converging to an optimum point. So, in Eq. (15), depending on the density of the countries distribution, we set the angle of movement so that each country can escape from the dense area with 15% and with 85% the country will move towards its empire with a shrinking angle. In the cases that the countries converge to a local optima, this method will help to escape from falling into the local optima’s trap with possibility of 15% . In this way, we add explorative search ability to the ICA algorithm. In Eq. (16), if the countries density in the current iteration is less than the previous iteration, each country with possibility of 15% will move towards its empire with a shrinking angle and with 85% the country will move towards its empire with an expanding angle. This way, provides a more efficient search in all over the search space of the problem. The results show that the quality of solutions and the speed of convergence of imperialist competitive algorithm with adaptive absorption policy is better than to ICA, PSO using a Sugeno function as inertia weight and GA algorithms. This is observable in analysis and conclusion section.
A. the definition of adaptive movement angle in the absorption policy As mentioned before in ICA algorithm the colonies move towards the imperialist by a random amount of deviation. The parameter is this deviation. In this paper, we extract the statistical information about the search space from the current population of solutions to provide an adaptive movement angle. We proposed a probabilistic model, to modify the ICA global search capability. The probabilistic model P(x) that we use here is a Gaussian distribution model [24,25,26,27]. The joint probability distribution of all the countries, is given by the product of the marginal probabilities of the countries: *: - ' ?3 ! *: 3 3 " 3 -''''''''''*Where *: 3 3 " 3 -
! #G
Y
M EG G # * G
''''''''''''*7-
(1) Initialize the empires and their colonies positions randomly. (2) Compute the adaptive ¤ (colonies movement angle towards the imperialist’s position) using the probabilistic model. (3) Compute the total cost of all empires (Related to the power of both the imperialist and its colonies). (4) Pick the weakest colony (colonies) from the weakest empire and give it (them) to the empire that has the most likelihood to possess it (Imperialistic competition). (5) Eliminate the powerless empires. (6) If there is just one empire, then stop else continue. (7) Check the termination conditions.
The average, μ, and the standard deviation, , for the colony countries of each empire is approximated as below:
' ' : "3 '''''''''''''''''*A- ' : !
! # ' F !*': "3 : '- '' '''''''''''''*N
Figure2. The AICA algorithm.
IV.
testing. The neural network trained by CICA, ICA, PSO and GA algorithms and the results compared with each other. The results of these experiments presented in Table 2 and 3. In the Fig.3, which belongs to Sphere it is observable that the quality of global optima solution and the convergence velocity towards the optima point has improved in compare with the other three algorithms. In the log plot of the Sphere function, at the first 20 iterations, PSO algorithm has better convergence speed than the ICA and AICA algorithms but after that iteration the AICA won the competition.
ANALYSIS AND CONSIDERATION OF EMPIRICAL RESULTS
In this paper, the proposed algorithm, that called Adaptive Imperialist Competitive Algorithm (AICA), applied to some well-known benchmark functions and a three-layered Perceptron Neural Network to update its weights, in order to verify the AICA algorithm performance and compared with ICA and PSO using a Sugeno function as inertia weight and GA algorithms. These benchmarks presented in Table1. TABLE I. BENCHMARKS FOR SIMULATION Mathematical representation
Range
comparative Result for Sphere
6
10
4*-
Sphere
' F¥3 ! 3#
AICA ICA GA PSO
(-100,100) 4
10
Rosenbrock Rastrigin
# # # 47*- F¥¡! 3 ! *kk' *3¦! 3 - \ ' *3 - -
(-100,100) 2
4A*- F¥3 !*3# k §¨©*7}3 - \
10
k-
(-10,10) 0
! ª«««
¬
G # ¥ F¥ 3 ! 3 3 ! §¨©'* - \
3
10
C ost
Griewank
4N*- '
(-600,600)
-2
10
!
Ackley
4V(x)=-20exp(-0.2 F?3 ! ®# -?
(-32,32)
!
-4
exp( F?¯ ! §¨© 7}¯ -+20+e
10
?
michalewicz
4_ (x)= -F?3 ! ©°±*3 - BX
3¬G #8 &
-6
(0,})
10
-8
10
We made simulations for considering the rate of convergence and the quality of the proposed algorithm optima solution, in comparison to ICA, PSO using a Sugeno function as inertia weight and GA algorithms that all the benchmarks tested by 30 dimensions separately. The average of optimum value for 20 trails obtained. In these experiments, all the simulations done during 1000 generations for Sphere and Rosenbrock uni-modal functions and Rastrigin, Griwank, Ackley and michalewicz multimodal functions. In these simulations for AICA and ICA algorithms, we set the parameters q 7,' £=0.001. The number of imperialists and the colonies are set respectively to 8 and 80. In PSO algorithm the parameters ! and # 'are fix to 1.5 and the number of the particle is 80. Determining this amount for c1 and c2 we have given equal chance to social and cognition components take part in search process. In GA the population size is 80, the mutation and crossover rate are respectively set to 0.01 and 0.5. We applied the trained neural network with AICA, ICA, PSO and GA algorithms on the data of TEHRAN's bourse market. The inputs of this network are the volume of changed stocks, the last price, the least price and the most prices in different times. The output of this network is the approximation of the most prices of the changed stocks in TEHRAN's bourse market. In these simulations, we used a three-layered Perseptron Neural Network containing an input layer with 7 nodes, a hidden layer with 5 nodes and an output layer with one node. The dataset include of 1155 instances. Using Holdout method (The holdout method splits the data into two mutually exclusive sets, sometimes referred to as the training and test sets) we apply 80% of instance data for training the Neural Network and the remaining 20 % for
0
100
200
300
400
500
600
700
800
900
1000
Generation Figure3. The cost of Sphere function
In Rosenbrock uni-modal function the speed of convergence of PSO algorithm is better than ICA, GA and AICA algorithm until the 200th iteration. After the 200th iteration, the velocity and quality of optima solution recovered in AICA algorithm. comparative Result for Rosenbrock
10
10
AICA ICA GA PSO
9
10
8
10
7
10
6
C ost
10
5
10
4
10
3
10
2
10
1
10
0
100
200
300
400
500
600
700
800
900
1000
Generation Figure4. The cost of Rosenbrock function.
As we can see in Fig.5, for Rasrigin multi-modal function the ICA algorithm has better performance rather than the PSO and GA algorithms. The proposed algorithm has shown a good performance in this function and has been able to escape from the local peaks and reach to global optima.
4
comparative Result for Ackley
comparative Result for Rastrigin
3
20
10
AICA ICA GA PSO
2
10
AICA ICA GA PSO
18 16 14
1
10
C ost
C ost
12 0
10
10 8
-1
10
6 4
-2
10
2 -3
10
0
100
200
300
400
500
600
700
800
900
0
1000
Generation Figure 5. The cost of Rastrigin function.
0
100
AICA ICA GA PSO
-8
-10
C ost
-12
-14
-16
TABEL II. Sphere Rosenbrock Rastrigin Michalewicz Griewank Ackley
-20
300
400
500
600
700
800
600
700
800
900
1000
900
1000
Generation Figure 6. The cost of Michalewicz function.
In Fig.7, Griewank multi-modal function the proposed algorithm has had remarkable improved in this function both in optima solution quality and in convergence speed rather than the ICA, PSO and GA algorithms.
Average optimum value for 20 trails for benchmarks. PSO
GA
ICA
AICA
9.0371
18.8398
28.7678
1.4052 k¡²
32.6318
1.7112 kª
3.6425 kª
25.6721
4.6395
19.2644
0.0994
0.0052
-16.8100
-18.2950
-20.9049
-21.4124
-0.3687
-0.3702
-0.8319
-2.3712
0.7863
2.6179
1.4384
6.1457 k¡ª
-18
200
500
In Fig.8, Ackley multi-modal function, the proposed algorithm has better performance in this function both in optima solution quality and in convergence speed rather than the ICA, PSO and GA algorithms reach to a better optima. In table 2, the average of optimum value for 20 trails, which is obtained from proposed algorithm, ICA, PSO and GA are shown. The benchmarks, were tested by 30 dimensions and the stop condition was 1000 generations. The numerical results show that the proposed algorithm has recovered the global optima solution remarkably.
comparative Result for Michalewicz
100
400
Figure8. The cost of Ackley function.
-6
0
300
Generation
In Fig.6, Michalewicz multi-modal function, the porposed algorithm has shown good performance.
-22
200
In Fig9, comparison of Mean Square Error (MSE) of Neural Network trained by AICA, ICA, PSO and GA indicated that the proposed algorithm trained very well rather than the other algorithms.
comparative Result for Griewank 16
AICA ICA GA PSO
14 12
Mean Square Error
-1
10
ICA AICA PSO GA
10
M SE
C ost
8 6
-2
10
4 2 0 -2
-3
10
-4
0
100
200
300
400
500
600
700
800
900
0
100
200
300
400
500
600
700
800
900
1000
Epoch # Figure 9. The comparison of Mean Square Errors (MSE).
1000
Generation Figure7. The cost of Griewank function.
Table 2, shows the result of AICA, ICA, PSO and GA training algorithms mean square errors. As it is observable, the
AICA algorithm has the least MSE in compare with the other algorithms.
[8]J. Kennedy and R.C. Eberhart, "Particle swarm optimization", in: Proceedings of IEEE International Conference on Neural Networks, Piscataway: IEEE, pp. 1942–1948, 1995. [9]X. Yang, J. Yuan, J. Yuan and H. Mao," A modified particle swarm optimizer with dynamic adaptation", Applied Mathematics and Computation, Volume 189, Issue 2, pp. 1205-1213, 2007. [10]B.E. Rosen and J.M. Goodwin, "Optimizing Neural Networks Using Very Fast Simulated Annealing. Neural", Parallel & Scientific Computations, pp.383–392, 1997. [11]L.A. Ingber, "Simulated annealing: practice versus theory", J. Math. Comput. Modell. 18 (11), pp.29–57, 1993. [12]M.F. Cardoso, R.L. Salcedo, S.F. Azevedo, D. Barbosa, "A simulated annealing approach to the solution of minlp problems", Comput. Chem. Eng. 21 (12) ,pp.1349–1364, 1997. [13]B. Franklin and M. Bergerman, "Cultural Algorithms: Concepts and Experiments", In Proceedings of the IEEE Congress on Evolutionary Computation, volume 2, pp. 1245–1251, 2000. [14] X. Jin and R.G. Reynolds, "Using Knowledge-Based Evolutionary Computation to Solve Nonlinear Constraint Optimization Problems: A Cultural Algorithm Approach", In Proceedings of the IEEE Congress on Evolutionary Computation, volume 3, pp. 1672–1678, 1999. [15]B.E. Rosen and J.M. Goodwin, "Optimizing Neural Networks Using Very Fast Simulated Annealing", Neural, Parallel & Scientific Computations, pp.383–392, 1997. [16]C.L. Wu, K.W. Chau, "A flood forecasting neural network model with genetic algorithm", International Journal of Environment and Pollution 28(3–4) pp. 261–273 ,(2006). [17]N. Muttil, K.W. Chau, "Neural network and genetic programming for modelling coastal algal blooms", International Journal of Environment and Pollution 28 (3–4) pp. 223–238, 2006. [18] J. Kennedy and R.C. Eberhart, "Particle swarm optimization", in: Proceedings of IEEE International Conference on Neural Networks, Piscataway: IEEE, pp. 1942–1948, 1995. [19]K. Lei, Y. Qiu and Y. He, "A New Adaptive Well-Chosen Inertia Weight Strategy to Automatically Harmonize Global and Local Search Ability in Particle Swarm Optimization", ISScAA, 2006. [20]Y. Da, X.R. Ge, "An improved PSO-based ANN with simulated annealing technique", Neurocomput. Lett. 63 pp. 527–533, 2005. [21]E. Atashpaz-Gargari and C. Lucas, "Imperialist Competitive Algorithm: An Algorithm for Optimization Inspired by Imperialistic Competition", IEEE Congress on Evolutionary Computation (CEC 2007).pp 4661-4667, 2007. [22]http://www.IRBourse.com, The dataset for training the Neural Network. [23]S. Kirtrick and C. D. Gelatt and M. P. Vecchi, “Optimization by Simulated Annealing”, Science, Vol 220, Number 4598, pp. 671-680, 1983. [24]A. Papoulis, ”Probability, Random Variables and Stochastic Processes”, McGraw-Hill, 1965. [25]Randall C. Smith, Peter Cheeseman, ”On the Representation and Estimation of Spatial Uncertainty,” the International Journal of Robotics Research,Vol.5, No.4, Winter 1986. [26]T.K. Paul and H. Iba, “Linear and Combinatorial Optimizations by Estimation of Distribution Algorithms,” 9th MPS Symposium on Evolutionary Computation, IPSJ, Japan, 2002. [27]Yaakov Bar-Shalom, X. Rong Li, Thiagalingam Kirubarajan, ”Estimation with Applications to Tracking and Navigation,” John Wiley & Sons, 2001.
TABLE II. COMPARE RESULTS Train Error GA PSO ICA
0.0031 0.0016 0.0021
AICA
0.0011
Test Error 0.0063 0.0016 0.0014 6.2798 ³´¡µ
V.
Train correlation
Test correlation
0.9909 0.9949 0.9936
0.9784 0.9943 0.9952
Time of training (second) 511.880180 1363.970089 1165.571148
0.9964
0.9979
1178.611372
CONCLUSION
In this paper, an improved imperialist algorithm called Adaptive Imperialist Competitive Algorithm (AICA) introduced. The proposed algorithm uses the probability density function to adapt the angle of colonies movement towards imperialist’s position during iterations dynamically. This mechanism, enhance the global search capability of the algorithm. This idea balances the exploration and exploitation abilities of the proposed algorithm, using colonies positions information. We examined the proposed algorithm in several standard benchmark functions that usually tested in Evolutionary Algorithms. Also, we use the AICA Algorithm to adjust the weights of a three-layered Perceptron Neural Network to predict the maximum worth of the stocks change in Tehran’s Bourse Market. Experimental results show that the proposed algorithm is a promising method with good global convergence performance than the ICA, GA and PSO algorithms. In the future, we will work on the affect of the different probability models on the performance of the proposed algorithm. REFERENCES [1]H. Sarimveis and A. Nikolakopoulos, "A Line Up Evolutionary Algorithm for Solving Nonlinear Constrained Optimization Problems", Computers & Operations Research, 32(6):pp.1499–1514, 2005. [2]H. M¨uhlenbein, M. Schomisch, J.Born, "The Parallel Genetic Algorithm as Function Optimizer", Proceedings of The Fourth International Conference on Genetic Algorithms, University of California, San diego, pp. 270-278,1991. [3]C. Bing-rui and F. Xia-ting, "Self-adapting Chaos-genetic Hybrid Algorithm with Mixed congruential Method", Forth International Conference, pp. 674-677,2008. [4]J.H. Holland. "ECHO: Explorations of Evolution in a Miniature World", In J.D. Farmer and J. Doyne, editors, Proceedings of the Second Conference on Artificial Life, 1990. [5] M. Gao, J. Xu, J. Tian and H. Wu, "Path Planning for Mobile Robot based on Chaos Genetic Algorithm", Forth International Conference, pp. 409-413,2008. [6]M. Melanie, "An Introduction to Genetic Algorithms", Massachusett"s: MIT Press, 1999. [7]May RM, "Simple mathematical models with very complicated dynamics,. Nature 1976;261:459.
C
Karim Faez Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran, [email protected]
Abstract— The novel Imperialist Competitive Algorithm (ICA) that was recently introduced has a good performance in some optimization problems. The ICA inspired by sociopolitical process of imperialistic competition of human being in the real world. In this paper, a new Adaptive Imperialist Competitive Algorithm (AICA) is proposed. In the proposed algorithm, for an effective search, the Absorption Policy changed dynamically to adapt the angle of colonies movement towards imperialist’s position. The ICA is easily stuck into a local optimum when solving high-dimensional multi-model numerical optimization problems. To overcome this shortcoming, we use probabilistic model that utilize the information of colonies positions to balance the exploration and exploitation abilities of the imperialistic competitive algorithm. Using this mechanism, ICA exploration capability will enhance. Some famous unconstraint benchmark functions used to test the AICA performance. Also, we use the AICA Algorithm to adjust the weights of a three-layered Perceptron neural network to predict the maximum worth of the stocks change in Tehran’s Bourse Market. Simulation results show this strategy can improve the performance of the ICA algorithm significantly.
gradient descent is a very popular optimization method, it plagued by slow convergence and susceptibility to local minima. Therefore, other approaches to improve NN training introduced. These methods include global optimization algorithms, such as Simulated Annealing [15], Genetic Algorithms [16,17], Particle Swarm Optimization algorithms [18,19,20] and other Evolutionary Algorithms. Recently, a new algorithm has been proposed by Atashpaz-Gargari and lucas [21], in 2007 that has inspired from a socio-human phenomenon. In this paper, we have proposed a new algorithm called Adaptive Imperialist Competitive Algorithm (AICA) that uses the probability density function to adapt the angle of colonies movement towards imperialist’s position during iterations dynamically. This mechanism, enhance the global search capability of the algorithm. This idea increases the performance of the ICA algorithm effectively in solving the optimization problems. We examined the proposed algorithm in several standard benchmark functions that usually tested in Evolutionary Algorithms. Also, we use the AICA Algorithm to adjust the weights of a three-layered Perceptron Neural Network to predict the maximum worth of the stocks change in Tehran’s Bourse Market[22]. The results of applying the proposed algorithm on benchmark functions and Neural Network to predict the maximum worth of the stocks change in Tehran’s Bourse Market indicated that the convergence speed and the quality of obtained solution in compare with ICA, PSO using a Sugeno function as inertia weight decline curve[23] and GA algorithm show a good performance. The rest of this paper organized as follows. Section two, provides an introduction the ICA algorithm. In section three, Adaptive Imperialistic Competitive Algorithm is proposed. Fourth section is devoted to the empirical results of proposed algorithm implementation and its compression with the results obtained by ICA, PSO and GA algorithms. The last section concludes the paper.
Keywords-Imperialist Competitive Algorithm; absorption policy; density probabilistic model.
I.
INTRODUCTION
The global optimization problem is applicable in every field of science, engineering and business. So far, many Evolutionary Algorithms (EA) [1,2], have been proposed for solving the global optimization problem. Inspired by the natural evolution, EA analogizes the evolution process of biological population, which can adapt the changing environments to the finding of the optimum of the optimization problem through evolving a population of candidate solutions. Some Evolutionary Algorithms for optimization problem are: the Genetic Algorithm (GA) [2,3,4,5,6,7], at first proposed by Holland, in 1962 [4], Particle Swarm Optimization algorithms (PSO) [8,9] that at first proposed by Kennedy and Eberhart [8], in 1995, Simulated Annealing (SA) [10,11,12], Cultural Evolutionary algorithms (CE) [13,14] at first was developed by Reynolds, in the early 1990s [14] and etc. The optimization methods are extensively used to adjust the weights of multi-layered Neural Networks. While Proc. 9th IEEE Int. Conf. on Cognitive Informatics (ICCI’10) F. Sun, Y. Wang, J. Lu, B. Zhang, W. Kinsner & L.A. Zadeh (Eds.) 978-1-4244-8040-1/10/$26.00 ©2010 IEEE
Helena Bahrami Elec., comp. & IT Department, Qazvin Azad University, Qazvin, Iran, [email protected]
II. INTRODUCTION OF IMPERIALIST COMPETITIVE ALGORITHMS (ICA) In this section, we introduce ICA algorithm and chaos theory.
W:? B*X@ YX ZXB? - \
A. Imperialist Competitive Algorithm (ICA)
^@Y PB*ZXYB'4'Y@ XY? U'''''''''''''''*_-
Imperialist Competitive Algorithm (ICA) is a new evolutionary algorithm in the Evolutionary Computation field based on the human's socio-political evolution. The algorithm starts with an initial random population called countries. Some of the best countries in the population selected to be the imperialists and the rest form the colonies of these imperialists. In an N dimensional optimization problem, a country is a array. This array defined as below
Where W:? is the total cost of the nth empire and ^ is a positive number which is considered to be less than one. `c*k" q O-'''''''''''''''''''''''''''''''''''''*zIn the absorption policy, the colony moves towards the imperialist by x unit. The direction of movement is the vector from colony to imperialist, as shown in Fig.1, in this figure, the distance between the imperialist and colony shown by d and x is a random variable with uniform distribution. Where q is greater than 1 and is near to 2. So, a proper choice can be q 7. In our implementation { is } * O- respectively. N
! " # " $ " % &'''''''''''''''''''''''''''''''*The cost of a country is found by evaluating the cost function f at the variables * ! " # " / " $ " % -. Then 3 4* 3 - 4* 3! " 3# " $ " 3% -'''''''''''*7-
`c*{" {The algorithm starts with N initial countries and the 389 best of them (countries with minimum cost) chosen as the imperialists. The remaining countries are colonies that each belong to an empire. The initial colonies belong to imperialists in convenience with their powers. To distribute the colonies among imperialists proportionally, the normalized cost of an imperialist is defined as follow
(8)
In ICA algorithm, to search different points around the imperialist, a random amount of deviation is added to the direction of colony movement towards the imperialist. In Fig. 1, this deflection angle is shown as , which is chosen randomly and with an uniform distribution. While moving toward the imperialist countries, a colony may reach to a better position, so the colony position changes according to position of the imperialist.
:? @ 3 3 ? ''''''''''''''''''''''''''''''''*AWhere, B? is the cost of nth imperialist and :? is its normalized cost. Each imperialist that has more cost value, will have less normalized cost value. Having the normalized cost, the power of each imperialist is calculated as below and based on that the colonies distributed among the imperialist countries. ? C
DE HGIJ
FGKM
C'''''''''''''''''''''''''''''''''*NDG
Figure1. Moving colonies toward their imperialist
On the other hand, the normalized power of an imperialist is assessed by its colonies. Then, the initial number of colonies of an empire will be
In this algorithm, the imperialistic competition has an important role. During the imperialistic competition, the weak empires will lose their power and their colonies. To model this competition, firstly we calculate the probability of possessing all the colonies by each empire considering the total cost of empire.
:? OP ? Q *RST -U''''''''''''''''''''''*VWhere, :? is initial number of colonies of nth empire and RST is the number of all colonies. To distribute the colonies among imperialist, :? of the colonies is selected randomly and assigned to their imperialist. The imperialist countries absorb the colonies towards themselves using the absorption policy. The absorption policy shown in Fig.1, makes the main core of this algorithm and causes the countries move towards to their minimum optima. The imperialists absorb these colonies towards themselves with respect to their power that described in (6). The total power of each imperialist is determined by the power of its both parts, the empire power plus percents of its average colonies power.
W:? @ 3 PW:3 U W:? '''''''''''''''''''''''''*Where, W:? is the total cost of nth empire and W:? is the normalized total cost of nth empire. Having the normalized total cost, the possession probability of each empire is calculated as below 9E C
%DE HGIJ FGKM %DG
C''''''''''''''''''''''''''''''''*k-
In each iteration, the country densities compute using the probabilistic model in Eq(11). If the countries density in the current iteration is more than the previous iteration, then with 85% the previous angle of the movement of the countries towards their empires will be shrunk and with 15% the mentioned angle will be expanded.
after a while all the empires except the most powerful one will collapse and all the colonies will be under the control of this unique empire. III.
THE PROPOSED IMPERIALIST COMPETITIVE ALGORITHM
3 kQ V*3¡! \ ¢- \ kQV*3¡! ¢-'''*V-
The ICA algorithm like many Evolutionary Algorithms suffers the lack of ability to global search properly in the problem space. During the search process, the algorithm may trap into local optima and it is possible to get far from the global optima. This causes the premature convergence. In this paper, a new method suggested that balance the exploration and exploitation abilities of the proposed algorithm, using colonies positions information. In the ICA algorithm absorption policy that mentioned in the previous section, the colonies move towards imperialists with an angle, which is a random variable. The colonies movement because of the constant parameter has a monotonic nature, so the colonies movement could not be adapted with the search process. Therefore, if the algorithm traps in the local optima, it cannot leave it and move towards the global optima. For solving this problem, and make balance between the explorative and exploitative search, we define the parameter adaptively, and dynamically adjust the movement of colonies to the imperialists during the search process.
3 , is the current angle of movement. 3¡! , is the previous angle and £ is the step size of shrinking and expanding the angle of movement. The value of this step size is varying between 0.0001 and 0.1. Otherwise, if the countries density in the current iteration is less than the previous iteration, then with 85% the previous angle of the movement of the countries towards their empires will be expanded and with 15% the mentioned angle will be shrunk. 3 kQ7V*3¡! \ ¢- \ kQzV*3¡! ¢-'''*_If the countries density in the current iteration is more than the previous iteration, it means that may be the countries are converging to an optimum point. So, in Eq. (15), depending on the density of the countries distribution, we set the angle of movement so that each country can escape from the dense area with 15% and with 85% the country will move towards its empire with a shrinking angle. In the cases that the countries converge to a local optima, this method will help to escape from falling into the local optima’s trap with possibility of 15% . In this way, we add explorative search ability to the ICA algorithm. In Eq. (16), if the countries density in the current iteration is less than the previous iteration, each country with possibility of 15% will move towards its empire with a shrinking angle and with 85% the country will move towards its empire with an expanding angle. This way, provides a more efficient search in all over the search space of the problem. The results show that the quality of solutions and the speed of convergence of imperialist competitive algorithm with adaptive absorption policy is better than to ICA, PSO using a Sugeno function as inertia weight and GA algorithms. This is observable in analysis and conclusion section.
A. the definition of adaptive movement angle in the absorption policy As mentioned before in ICA algorithm the colonies move towards the imperialist by a random amount of deviation. The parameter is this deviation. In this paper, we extract the statistical information about the search space from the current population of solutions to provide an adaptive movement angle. We proposed a probabilistic model, to modify the ICA global search capability. The probabilistic model P(x) that we use here is a Gaussian distribution model [24,25,26,27]. The joint probability distribution of all the countries, is given by the product of the marginal probabilities of the countries: *: - ' ?3 ! *: 3 3 " 3 -''''''''''*Where *: 3 3 " 3 -
! #G
Y
M EG G # * G
''''''''''''*7-
(1) Initialize the empires and their colonies positions randomly. (2) Compute the adaptive ¤ (colonies movement angle towards the imperialist’s position) using the probabilistic model. (3) Compute the total cost of all empires (Related to the power of both the imperialist and its colonies). (4) Pick the weakest colony (colonies) from the weakest empire and give it (them) to the empire that has the most likelihood to possess it (Imperialistic competition). (5) Eliminate the powerless empires. (6) If there is just one empire, then stop else continue. (7) Check the termination conditions.
The average, μ, and the standard deviation, , for the colony countries of each empire is approximated as below:
' ' : "3 '''''''''''''''''*A- ' : !
! # ' F !*': "3 : '- '' '''''''''''''*N
Figure2. The AICA algorithm.
IV.
testing. The neural network trained by CICA, ICA, PSO and GA algorithms and the results compared with each other. The results of these experiments presented in Table 2 and 3. In the Fig.3, which belongs to Sphere it is observable that the quality of global optima solution and the convergence velocity towards the optima point has improved in compare with the other three algorithms. In the log plot of the Sphere function, at the first 20 iterations, PSO algorithm has better convergence speed than the ICA and AICA algorithms but after that iteration the AICA won the competition.
ANALYSIS AND CONSIDERATION OF EMPIRICAL RESULTS
In this paper, the proposed algorithm, that called Adaptive Imperialist Competitive Algorithm (AICA), applied to some well-known benchmark functions and a three-layered Perceptron Neural Network to update its weights, in order to verify the AICA algorithm performance and compared with ICA and PSO using a Sugeno function as inertia weight and GA algorithms. These benchmarks presented in Table1. TABLE I. BENCHMARKS FOR SIMULATION Mathematical representation
Range
comparative Result for Sphere
6
10
4*-
Sphere
' F¥3 ! 3#
AICA ICA GA PSO
(-100,100) 4
10
Rosenbrock Rastrigin
# # # 47*- F¥¡! 3 ! *kk' *3¦! 3 - \ ' *3 - -
(-100,100) 2
4A*- F¥3 !*3# k §¨©*7}3 - \
10
k-
(-10,10) 0
! ª«««
¬
G # ¥ F¥ 3 ! 3 3 ! §¨©'* - \
3
10
C ost
Griewank
4N*- '
(-600,600)
-2
10
!
Ackley
4V(x)=-20exp(-0.2 F?3 ! ®# -?
(-32,32)
!
-4
exp( F?¯ ! §¨© 7}¯ -+20+e
10
?
michalewicz
4_ (x)= -F?3 ! ©°±*3 - BX
3¬G #8 &
-6
(0,})
10
-8
10
We made simulations for considering the rate of convergence and the quality of the proposed algorithm optima solution, in comparison to ICA, PSO using a Sugeno function as inertia weight and GA algorithms that all the benchmarks tested by 30 dimensions separately. The average of optimum value for 20 trails obtained. In these experiments, all the simulations done during 1000 generations for Sphere and Rosenbrock uni-modal functions and Rastrigin, Griwank, Ackley and michalewicz multimodal functions. In these simulations for AICA and ICA algorithms, we set the parameters q 7,' £=0.001. The number of imperialists and the colonies are set respectively to 8 and 80. In PSO algorithm the parameters ! and # 'are fix to 1.5 and the number of the particle is 80. Determining this amount for c1 and c2 we have given equal chance to social and cognition components take part in search process. In GA the population size is 80, the mutation and crossover rate are respectively set to 0.01 and 0.5. We applied the trained neural network with AICA, ICA, PSO and GA algorithms on the data of TEHRAN's bourse market. The inputs of this network are the volume of changed stocks, the last price, the least price and the most prices in different times. The output of this network is the approximation of the most prices of the changed stocks in TEHRAN's bourse market. In these simulations, we used a three-layered Perseptron Neural Network containing an input layer with 7 nodes, a hidden layer with 5 nodes and an output layer with one node. The dataset include of 1155 instances. Using Holdout method (The holdout method splits the data into two mutually exclusive sets, sometimes referred to as the training and test sets) we apply 80% of instance data for training the Neural Network and the remaining 20 % for
0
100
200
300
400
500
600
700
800
900
1000
Generation Figure3. The cost of Sphere function
In Rosenbrock uni-modal function the speed of convergence of PSO algorithm is better than ICA, GA and AICA algorithm until the 200th iteration. After the 200th iteration, the velocity and quality of optima solution recovered in AICA algorithm. comparative Result for Rosenbrock
10
10
AICA ICA GA PSO
9
10
8
10
7
10
6
C ost
10
5
10
4
10
3
10
2
10
1
10
0
100
200
300
400
500
600
700
800
900
1000
Generation Figure4. The cost of Rosenbrock function.
As we can see in Fig.5, for Rasrigin multi-modal function the ICA algorithm has better performance rather than the PSO and GA algorithms. The proposed algorithm has shown a good performance in this function and has been able to escape from the local peaks and reach to global optima.
4
comparative Result for Ackley
comparative Result for Rastrigin
3
20
10
AICA ICA GA PSO
2
10
AICA ICA GA PSO
18 16 14
1
10
C ost
C ost
12 0
10
10 8
-1
10
6 4
-2
10
2 -3
10
0
100
200
300
400
500
600
700
800
900
0
1000
Generation Figure 5. The cost of Rastrigin function.
0
100
AICA ICA GA PSO
-8
-10
C ost
-12
-14
-16
TABEL II. Sphere Rosenbrock Rastrigin Michalewicz Griewank Ackley
-20
300
400
500
600
700
800
600
700
800
900
1000
900
1000
Generation Figure 6. The cost of Michalewicz function.
In Fig.7, Griewank multi-modal function the proposed algorithm has had remarkable improved in this function both in optima solution quality and in convergence speed rather than the ICA, PSO and GA algorithms.
Average optimum value for 20 trails for benchmarks. PSO
GA
ICA
AICA
9.0371
18.8398
28.7678
1.4052 k¡²
32.6318
1.7112 kª
3.6425 kª
25.6721
4.6395
19.2644
0.0994
0.0052
-16.8100
-18.2950
-20.9049
-21.4124
-0.3687
-0.3702
-0.8319
-2.3712
0.7863
2.6179
1.4384
6.1457 k¡ª
-18
200
500
In Fig.8, Ackley multi-modal function, the proposed algorithm has better performance in this function both in optima solution quality and in convergence speed rather than the ICA, PSO and GA algorithms reach to a better optima. In table 2, the average of optimum value for 20 trails, which is obtained from proposed algorithm, ICA, PSO and GA are shown. The benchmarks, were tested by 30 dimensions and the stop condition was 1000 generations. The numerical results show that the proposed algorithm has recovered the global optima solution remarkably.
comparative Result for Michalewicz
100
400
Figure8. The cost of Ackley function.
-6
0
300
Generation
In Fig.6, Michalewicz multi-modal function, the porposed algorithm has shown good performance.
-22
200
In Fig9, comparison of Mean Square Error (MSE) of Neural Network trained by AICA, ICA, PSO and GA indicated that the proposed algorithm trained very well rather than the other algorithms.
comparative Result for Griewank 16
AICA ICA GA PSO
14 12
Mean Square Error
-1
10
ICA AICA PSO GA
10
M SE
C ost
8 6
-2
10
4 2 0 -2
-3
10
-4
0
100
200
300
400
500
600
700
800
900
0
100
200
300
400
500
600
700
800
900
1000
Epoch # Figure 9. The comparison of Mean Square Errors (MSE).
1000
Generation Figure7. The cost of Griewank function.
Table 2, shows the result of AICA, ICA, PSO and GA training algorithms mean square errors. As it is observable, the
AICA algorithm has the least MSE in compare with the other algorithms.
[8]J. Kennedy and R.C. Eberhart, "Particle swarm optimization", in: Proceedings of IEEE International Conference on Neural Networks, Piscataway: IEEE, pp. 1942–1948, 1995. [9]X. Yang, J. Yuan, J. Yuan and H. Mao," A modified particle swarm optimizer with dynamic adaptation", Applied Mathematics and Computation, Volume 189, Issue 2, pp. 1205-1213, 2007. [10]B.E. Rosen and J.M. Goodwin, "Optimizing Neural Networks Using Very Fast Simulated Annealing. Neural", Parallel & Scientific Computations, pp.383–392, 1997. [11]L.A. Ingber, "Simulated annealing: practice versus theory", J. Math. Comput. Modell. 18 (11), pp.29–57, 1993. [12]M.F. Cardoso, R.L. Salcedo, S.F. Azevedo, D. Barbosa, "A simulated annealing approach to the solution of minlp problems", Comput. Chem. Eng. 21 (12) ,pp.1349–1364, 1997. [13]B. Franklin and M. Bergerman, "Cultural Algorithms: Concepts and Experiments", In Proceedings of the IEEE Congress on Evolutionary Computation, volume 2, pp. 1245–1251, 2000. [14] X. Jin and R.G. Reynolds, "Using Knowledge-Based Evolutionary Computation to Solve Nonlinear Constraint Optimization Problems: A Cultural Algorithm Approach", In Proceedings of the IEEE Congress on Evolutionary Computation, volume 3, pp. 1672–1678, 1999. [15]B.E. Rosen and J.M. Goodwin, "Optimizing Neural Networks Using Very Fast Simulated Annealing", Neural, Parallel & Scientific Computations, pp.383–392, 1997. [16]C.L. Wu, K.W. Chau, "A flood forecasting neural network model with genetic algorithm", International Journal of Environment and Pollution 28(3–4) pp. 261–273 ,(2006). [17]N. Muttil, K.W. Chau, "Neural network and genetic programming for modelling coastal algal blooms", International Journal of Environment and Pollution 28 (3–4) pp. 223–238, 2006. [18] J. Kennedy and R.C. Eberhart, "Particle swarm optimization", in: Proceedings of IEEE International Conference on Neural Networks, Piscataway: IEEE, pp. 1942–1948, 1995. [19]K. Lei, Y. Qiu and Y. He, "A New Adaptive Well-Chosen Inertia Weight Strategy to Automatically Harmonize Global and Local Search Ability in Particle Swarm Optimization", ISScAA, 2006. [20]Y. Da, X.R. Ge, "An improved PSO-based ANN with simulated annealing technique", Neurocomput. Lett. 63 pp. 527–533, 2005. [21]E. Atashpaz-Gargari and C. Lucas, "Imperialist Competitive Algorithm: An Algorithm for Optimization Inspired by Imperialistic Competition", IEEE Congress on Evolutionary Computation (CEC 2007).pp 4661-4667, 2007. [22]http://www.IRBourse.com, The dataset for training the Neural Network. [23]S. Kirtrick and C. D. Gelatt and M. P. Vecchi, “Optimization by Simulated Annealing”, Science, Vol 220, Number 4598, pp. 671-680, 1983. [24]A. Papoulis, ”Probability, Random Variables and Stochastic Processes”, McGraw-Hill, 1965. [25]Randall C. Smith, Peter Cheeseman, ”On the Representation and Estimation of Spatial Uncertainty,” the International Journal of Robotics Research,Vol.5, No.4, Winter 1986. [26]T.K. Paul and H. Iba, “Linear and Combinatorial Optimizations by Estimation of Distribution Algorithms,” 9th MPS Symposium on Evolutionary Computation, IPSJ, Japan, 2002. [27]Yaakov Bar-Shalom, X. Rong Li, Thiagalingam Kirubarajan, ”Estimation with Applications to Tracking and Navigation,” John Wiley & Sons, 2001.
TABLE II. COMPARE RESULTS Train Error GA PSO ICA
0.0031 0.0016 0.0021
AICA
0.0011
Test Error 0.0063 0.0016 0.0014 6.2798 ³´¡µ
V.
Train correlation
Test correlation
0.9909 0.9949 0.9936
0.9784 0.9943 0.9952
Time of training (second) 511.880180 1363.970089 1165.571148
0.9964
0.9979
1178.611372
CONCLUSION
In this paper, an improved imperialist algorithm called Adaptive Imperialist Competitive Algorithm (AICA) introduced. The proposed algorithm uses the probability density function to adapt the angle of colonies movement towards imperialist’s position during iterations dynamically. This mechanism, enhance the global search capability of the algorithm. This idea balances the exploration and exploitation abilities of the proposed algorithm, using colonies positions information. We examined the proposed algorithm in several standard benchmark functions that usually tested in Evolutionary Algorithms. Also, we use the AICA Algorithm to adjust the weights of a three-layered Perceptron Neural Network to predict the maximum worth of the stocks change in Tehran’s Bourse Market. Experimental results show that the proposed algorithm is a promising method with good global convergence performance than the ICA, GA and PSO algorithms. In the future, we will work on the affect of the different probability models on the performance of the proposed algorithm. REFERENCES [1]H. Sarimveis and A. Nikolakopoulos, "A Line Up Evolutionary Algorithm for Solving Nonlinear Constrained Optimization Problems", Computers & Operations Research, 32(6):pp.1499–1514, 2005. [2]H. M¨uhlenbein, M. Schomisch, J.Born, "The Parallel Genetic Algorithm as Function Optimizer", Proceedings of The Fourth International Conference on Genetic Algorithms, University of California, San diego, pp. 270-278,1991. [3]C. Bing-rui and F. Xia-ting, "Self-adapting Chaos-genetic Hybrid Algorithm with Mixed congruential Method", Forth International Conference, pp. 674-677,2008. [4]J.H. Holland. "ECHO: Explorations of Evolution in a Miniature World", In J.D. Farmer and J. Doyne, editors, Proceedings of the Second Conference on Artificial Life, 1990. [5] M. Gao, J. Xu, J. Tian and H. Wu, "Path Planning for Mobile Robot based on Chaos Genetic Algorithm", Forth International Conference, pp. 409-413,2008. [6]M. Melanie, "An Introduction to Genetic Algorithms", Massachusett"s: MIT Press, 1999. [7]May RM, "Simple mathematical models with very complicated dynamics,. Nature 1976;261:459.
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