Journal of Institute of Control, Robotics and Systems (2014) 20(8): ISSN: eissn:
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1 Journal of Institute of Control, Robotics and Systems (0) 0(8): ISSN:97- eissn:- On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution *,, (ByungSeok Lee,*, Joon Hwa Lee, and Moon-Beom Heo ), School of Electrical and Computer Engineering, University of Seoul, Satellite Navigation Team, Korea Aerospace Research Institute Abstract: This paper deals with the performance comparison of a PSO algorithm inspired in the process of simulating the behavior pattern of the organisms. The PSO algorithm finds the optimal solution (fitness value) of the objective function based on a stochastic process. Generally, the stochastic process, a random function, is used with the expression related to the velocity included in the PSO algorithm. In this case, the random function of the normal distribution (Gaussian) or uniform distribution are mainly used as the random function in a PSO algorithm. However, in this paper, because the probability distribution which is various with shape parameters can be expressed, the performance comparison of a PSO algorithm using the beta probability distribution function, that is a random function which has a high degree of freedom, is introduced. For performance comparison, functions (Rastrigin, Rosenbrock, Schwefel) were selected among the benchmark Set. And the convergence property was compared and analyzed using PSO-FIW to find the optimal solution. Keywords: PSO (Particle Swarm Optimization), PSO-FIW (PSO with Fixed Inertia Weight), beta probability distribution, benchmark set I. 서론 99 Kennedy Eberhart (PSO: Particle Swarm Optimization) PSO [-7]. PSO (AL: Artificial Life) (Swarm theory) (EC: Evolutionary Com- putation)., (GA: Genetic Algorithms) (EP: Evolutionary Programming) [,]. AL PSO (social behavior), (herd), (bird flock), (fish school) []. GA EC PSO (SI: Swarm Intelligence) * Corresponding Author Manuscript received December, 0 / revised April, 0 / accepted June, 0 : (byungseok@uos.ac.kr) : (joonhwa@uos.ac.kr) : (hmb@kari.re.kr). GA agent individual point (particle) [,]., particle solution. GA population PSO Swarm. PSO. PSO (convergence speed) (local search) (global search), PSO., PSO (normal distribution) (uniform distribution) (quasi-random or pseudo random) (low-discrepancy sequences) Sobol, Faure, Halton, Van der Corput [8,9], [0] Gamma. [0], (shape parameter), Copyright ICROS 0
2 On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution 8 8, PSO Normal, Uniform., 00 ( 00 ). II PSO. III PSO (Inertia Weight) PSO (, PSO-IW) PSO-IW (convergence rate) (Constriction Coefficient) PSO (, PSO-CC). IV,. V, VI., PSO. II. PSO 알고리즘. PSO 알고리즘연산식 PSO, [,]. PSO (solution) (particle) (position) (velocity). () () PSO. () () (), PSO (). ().,. (acceleration constant) (weighting value). ()., Cognitive Component Nostalgia Swarm ( ) ( ), Social Component ( ). PSO. Fig.. Pseudo code of PSO algorithm. ( ).,, [-].. PSO 알고리즘의사코드 PSO () (). (fitness value) (objective function)., PSO best position Swarm (personal best position, ).,, personal best position (global best position, )., personal best position pbest lbest, global best position gbest. (pseudo code). III. 계수및제한성이개선된 PSO 알고리즘. 관성하중계수를갖는 PSO 알고리즘 PSO () exploitation exploration. Reynolds (bird flocks) (Collision Avoidance, Velocity Matching, Flock Centering) [7] ( ) (neighbor) swarm.
3 8 이병석, 이준화, 허문범 exploration exploitation () inertia weight PSO PSO-IW Shi Eberhart []. (), (diversity),. []. ( 遞增 ). 랜덤조절방법 (random adjustments). () 0.7 Normal(Gaussian) distribution. () (), () () cognitive social component [,].,.. 선형감소방법 () (linear decreasing). 0.9, 0.. () [,]. (). 비선형감소방법 (nonlinear decreasing) (7)-(9) [,]. (7) (8) (9) (7)., (8) (relative improvement).. 퍼지적응방법 (fuzzy adaptive)..,. LOW, MEDIUM, HIGH []. 관성체증방법 (increasing inertia) [].. 제한계수를갖는 PSO 알고리즘 () (0). (0) () () []. () (),. ().. 베타확률분포 IV. 베타확률분포와벤치마크함수 ( ),. ( ) (RV: Random Variable).,., () PDF (shape parameter)., (beta function). ()
![(). 0, (). () -.,.. 벤치마크함수... (dimensionality) (non-linearity) (multi-modal),, [8-0]., domain MATLAB surf. Min. Value.](/docs-images/113/207085288/images/4-2.jpg "(a) Rastrigin function (b) Rosenbrock function (c) Schwefel function..... Fig.. Shape of the benchmark functions.")
4 베타확률분포를이용한입자떼최적화알고리즘의성능비교 87 Probability Density Function of Distribution a=b=. a=b=.8 a=b=. a=b=.0 a=b=0... Table. Selected benchmark functions. Func. Name & Formula Rastrigin function cos Min. Value R. V. f f Rosenbrock function Schwefel function sin 0 0. #. Fig.. Probability density function of beta distribution #... Probability Density Function of Distribution a=., b=. a=, b=0.8 a=.8, b=. a=.0, b=. a=, b= a=., b= R. V.. #. Fig.. Probability density function of beta distribution #. (). 0, (). () -.,.. 벤치마크함수... (dimensionality) (non-linearity) (multi-modal),, [8-0]., domain MATLAB surf. Min. Value. (a) Rastrigin function (b) Rosenbrock function (c) Schwefel function..... Fig.. Shape of the benchmark functions.
5 88 ByungSeok Lee, Joon Hwa Lee, and Moon-Beom Heo. PSO. Table. Principle setting value for PSO algorithm test using Func. benchmark functions. Search Space [-0, 0] f [-0, 0] f [-00, 00] V. 시뮬레이션결과 MATLAB m-file. particle, PSO particle Swarm. PSO PSO-IW., 0.7 (fixed) (, PSO-FIW), [ ]. PSO-FIW ()~(). ~ f 0 ()~()., particle. PSO. () () ()~(),, particle, domain, search space (upper bound) (lower bound). Dim Swarm Size Iteration # per Algorithm [ ],, PSO Gussian, Uniform,, MATLAB randn, rand, random. (, ), (, ), (.8,.), (.,.8), (0., 0.), (.,.), (.8,.8), (.,.). [ ] Gaussian Uniform., f, f 0 0 PSO-FIW 00 ( 00 ).,, 8., a b pdf, a b pdf.,, Gaussian PSO-FIW Uniform ( )., Uniform PSO-FIW,, f 0, 0 (.8,.), (0., 0.) Unifrom., f 0, (, ), (.8,.), (0., 0.), (.,.), (.,.) Uniform, f 0, (.8,.), (0., 0.) Uniform. VI. 결론 (EC) PSO. PSO (MSC: Monte Carlo Simulation), (SA: Simulated Annealing) Gaussian Uniform Gaussian Uniform PSO. 8 (, ), (.,.8), (.8,.8) Gaussian Uniform, f, f 0, 0,
6 On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution 89.. Table. Performance Comparison according to the Func. probability distribution and the Decrement compared with the Gaussian and Uniform probability distribution against the Optimum probability distribution. Dim. Distribution Compared with Gaussian Compared with Uniform 0 > > 9.9% 70.7% 0 > > % % f 0 > > % 99.7% 0 > > % 99.99% f 0 > >.0% 8.789% 0 > >.00% 8.7% : (, ), : (.,.8), : (.8,.8). ( ) a, b.,., MATLAB Gaussian Uniform,. 부록,, Min. Value Mean Standard Deviation 00 ( 00 ).. PSO-FIW #. Table. PSO-FIW algorithms performance comparison according to the probability distribution #.., Uniform, Gaussian, %., (.,.) 0 f Uniform 0., (, ), (.,.8). particle [8-0] (low-dicrepancy). 0.,, particle, Gaussian, Uniform particle particle particle,.,, Probability Distribution Gaussian Uniform Func. Dim. Min. Value Mean Min. Value Standard Deviation f f f f
7 80 이병석, 이준화, 허문범. PSO-FIW #. Table. PSO-FIW algorithms performance comparison accord- ing to the probability distribution #.. PSO-FIW #. Table. PSO-FIW algorithms performance comparison accord- ing to the probability distribution #. Probability Distribution Func. Dim. Min. Value Mean Min. Value Standard Deviation Probability Distribution Func. Dim. Min. Value Mean Min. Value Standard Deviation (a=, b=) f (a=0., b=0.) f f f (a=, b=) f (a=., b=.) 0. 0 f f f (a=.8, b=.) f (a=.8, b=.8) f f f (a=., b=.8) f (a=., b=.) f f f
8 베타확률분포를이용한입자떼최적화알고리즘의성능비교 8 x 0 80 Algo no. 0 8 Algo no Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , Gaussian, Dimension = 0 f, Gaussian, Dimension = 0 f, Gaussian, Dimension = 0 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , Uniform, Dimension = 0 f, Uniform, Dimension = 0 f, Uniform, Dimension = Algo no. 0 Algo no. 0 x 0 7 Algo no. 0 Algo no. 0 x 0. Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , Gaussian, Dimension = 0 f, Gaussian, Dimension = 0 f, Gaussian, Dimension = 0 00 x 0 7 x 0 00 Algo no. 0 Algo no. 0. Algo no. 0 Algo no. 0. Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , Uniform, Dimension = 0 f, Uniform, Dimension = 0 f, Uniform, Dimension = 0. PSO-FIW #( = 0, 0). Fig.. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0, 0). Gaussian Uniform, f, f 0 0 PSO-FIW Uniform Gaussian.
9 8 ByungSeok Lee, Joon Hwa Lee, and Moon-Beom Heo x 0 Algo no. 0 Algo no Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 00 Algo no. 0 Algo no. 0 Algo no. 0 x 0. Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=.8, b=.), Dimension = 0 f, (a=.8, b=.), Dimension = 0 f, (a=.8, b=.), Dimension = x 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=., b=.8), Dimension = 0 f, (a=., b=.8), Dimension = 0 f, (a=., b=.8), Dimension = 0. PSO-FIW #( = 0). Fig.. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0). a, b,, f, f 0 0 PSO-FIW. (, ) (.,.8).
10 On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 07 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=, b=), Dimension = 0 f, (a=, b=), Dimension = 0 f, (a=, b=), Dimension = Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=.8, b=.), Dimension = 0 f, (a=.8, b=.), Dimension = 0 f, (a=.8, b=.), Dimension = 0 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=., b=.8), Dimension = 0 f, (a=., b=.8), Dimension = 0 f, (a=., b=.8), Dimension = 0 7. PSO-FIW #( = 0). Fig. 7. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0) ,, f, f PSO-FIW. (, ) (.,.8).
11 8 이병석, 이준화, 허문범 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=0., b=0.), Dimension = 0 f, (a=0., b=0.), Dimension = 0 f, (a=0., b=0.), Dimension = 0 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 x 0 00 Algo no. 0. Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=.8, b=.8), Dimension = 0 f, (a=.8, b=.8), Dimension = 0 f, (a=.8, b=.8), Dimension = 0 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 8. PSO-FIW #( = 0). Fig. 8. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0). 8 a, b.,, f, f 0 0 PSO-FIW. 8 (.,.) 0 f,., 0 f (0., 0.), (.,.), (.8,.8).
12 베타확률분포를이용한입자떼최적화알고리즘의성능비교 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0 7. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=0., b=0.), Dimension = 0 f, (a=0., b=0.), Dimension = 0 f, (a=0., b=0.), Dimension = Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x 0 7. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 x 0 7 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=.8, b=.8), Dimension = 0 f, (a=.8, b=.8), Dimension = 0 f, (a=.8, b=.8), Dimension = 0 x 0 x Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 00. Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no. 0 Algo no , (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 f, (a=., b=.), Dimension = 0 9. PSO-FIW #( = 0). Fig. 9. PSO-FIW algorithm performance comparison according to the probability distributions on the benchmark functions #(Dim. = 0) ,, f, f PSO-FIW. 9 (.,.) f. (0.) 0 f (.8,.8).
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14 On the Comparison of Particle Swarm Optimization Algorithm Performance using Probability Distribution 87 이준화 ~ ~99 California Institute of Technology(Caltech). 99 ~.. 허문범 Illinois Institute of Technology. 00 Illinois Institute of Technology ~. GNSS,,.
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Problem New Case RETRIEVE Learned Case Retrieved Cases New Case RETAIN Tested/ Repaired Case Case-Base REVISE Solved Case REUSE Aamodt, A. and Plaza, E. (1994). Case-based reasoning; Foundational
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The e-business Studies Volume 17, Number 6, December, 30, 2016:275~289 Received: 2016/12/02, Accepted: 2016/12/22 Revised: 2016/12/20, Published: 2016/12/30 [ABSTRACT] SNS is used in various fields. Although
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