Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Methods of Global Optimization

Mishra, SK (2006): Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Methods of Global Optimization.

Abstract

In this paper an attempt has been made to fit the Gielis curves (modified by various functions) to simulated data. The estimation has been done by two methods - the Classical Simulated Annealing (CSA) and the Particle Swarm (PS) methods - of global optimization. The Repulsive Particle Swarm (RPS) optimization algorithm has been used. It has been found that both methods are quite successful in fitting the modified Gielis curves to the data. However, the lack of uniqueness of Gielis parameters to data (from which they are estimated) is corroborated.

From a technical viewpoint, this exercise may be considered as an application of CSA and RPS to extremely nonlinear least-squares curve-fitting to data that may exhibit a large number of local optima.

• Item Type: MPRA Paper
• Language: English
• Keywords: Gielis curves; superformula; nonlinear curve-fitting; Least squares; multi-modal; local optima; global optimization; simulated annealing; particle swarm; parameters estimation
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis

References

· Bhabhrawala, T. and Krovi, V.: “Shape Recovery from Medical Image Data using Extended Superquadrics”, ASME 2005 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, California USA, September 24-28, 2005. · Cerny, V., "Thermodynamical Approach to the Traveling Salesman Problem: An Efficient Simulation Algorithm", J. Opt. Theory Appl., 45, 1, 41-51, 1985 · Eberhart R.C. and Kennedy J.: “A New Optimizer using Particle Swarm Theory”, Proceedings Sixth Symposium on Micro Machine and Human Science, pp. 39–43. IEEE Service Center, Piscataway, NJ, 1995. · Gielis, J.: “A Generic Geometric Transformation that unifies a Wide Range of Natural and Abstract Shapes”, American Journal of Botany, 90(3): pp. 333–338, 2003. · Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley, Reading, Mass, 1989. · Holland, J.: Adaptation in Natural and Artificial Systems, Univ. of Michigan Press, Ann Arbor, 1975. · Huang, V.L., Suganthan, P.N. and Liang, J.J. “Comprehensive Learning Particle Swarm Optimizer for Solving Multiobjective Optimization Problems”, International Journal of Intelligent Systems, 21, pp.209–226 (Wiley Periodicals, Inc. Published online in Wiley InterScience www.interscience.wiley.com) , 2006 · Kirkpatrick, S., Gelatt, C.D. Jr., and Vecchi, M.P.: "Optimization by Simulated Annealing", Science, 220, 4598, 671-680, 1983. · Liang, J.J. and Suganthan, P.N. “Dynamic Multi-Swarm Particle Swarm Optimizer”, International Swarm Intelligence Symposium, IEEE # 0-7803-8916-6/05/$20.00. pp. 124-129, 2005. (obtained through personal request made by the author to [e-mail]). · Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E.: "Equation of State Calculations by Fast Computing Machines", J. Chem. Phys.,21, 6, 1087-1092, 1953. · Mishra, S.K.: "On Estimation of the Parameters of Gielis Superformula from Empirical Data" Social Science Research Network (SSRN): http://ssrn.com/abstract=905051, Working Paper Series, 2006. · Mishra, S.K.: "Experiments on Estimation of the Parameters of Gielis Super-Formula by Simulated Annealing Method of Optimization" Social Science Research Network (SSRN): http://ssrn.com/abstract=910800 , 2006. · Parsopoulos, K.E. and Vrahatis, M.N., “Recent Approaches to Global Optimization Problems Through Particle Swarm Optimization”, Natural Computing, 1 (2-3), pp. 235-306, 2002. · Wright, A.H.: “Genetic Algorithms for Real Parameter Optimization”, in GJE Rawlings (ed) Foundations of Genetic Algorithms, Morgan Kauffmann Publishers, San Mateo, CA, pp. 205- 218, 1991.