Mishra, SK (2006): Least Squares Fitting of ChacónGielis Curves by the Particle Swarm Method of Optimization.

PDF
MPRA_paper_466.pdf Download (215kB)  Preview 
Abstract
Ricardo Chacón generalized Johan Gielis's superformula by introducing elliptic functions in place of trigonometric functions. In this paper an attempt has been made to fit the ChacónGielis curves (modified by various functions) to simulated data by the least squares principle. Estimation has been done by the Particle Swarm (PS) methods of global optimization. The Repulsive Particle Swarm optimization algorithm has been used. It has been found that although the curvefitting exercise may be satisfactory, a lack of uniqueness of ChacónGielis parameters to data (from which they are estimated) poses an insurmountable difficulty to interpretation of findings.
Item Type:  MPRA Paper 

Original Title:  Least Squares Fitting of ChacónGielis Curves by the Particle Swarm Method of Optimization 
Language:  English 
Keywords:  Least squares multimodal nonlinear curvefitting; Ricardo Chacón; Jacobian Elliptic functions; Weierstrass ; Gielis superformula; supershapes; Particle Swarm method; Repulsive Particle Swarm method of Global optimization; nonlinear programming; multiple suboptima; global; local optima; fit; empirical; estimation; cellular automata; fractals 
Subjects:  C  Mathematical and Quantitative Methods > C4  Econometric and Statistical Methods: Special Topics > C44  Operations Research; Statistical Decision Theory 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 
Item ID:  466 
Depositing User:  Sudhanshu Kumar Mishra 
Date Deposited:  15. Oct 2006 
Last Modified:  08. Jan 2014 11:27 
References:  · Abramowitz, M. and Stegun, I.A.: Handbook of Mathematical Functions, Dover Publications, New York, 1964. · Barnsley, M.: Fractals Everywhere, Academic Press, Boston, MA.1993. · Chacón, R.: “A Mathematical Description of Natural Shapes in our Nonlinear World”, Working paper, arXive:nlin. AO/0405057 v1 25 May, 2004 · 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. · 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 · Liang, J.J. and Suganthan, P.N. “Dynamic MultiSwarm Particle Swarm Optimizer”, International Swarm Intelligence Symposium, IEEE # 0780389166/05/$20.00. pp. 124129, 2005. (obtained through personal request made by the author to epnsugan@ntu.edu.sg). · 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 Papers Series, 2006 (a). · Mishra, S.K.: "Experiments on Estimation of the Parameters of Gielis SuperFormula by Simulated Annealing Method of Optimization" Social Science Research Network (SSRN): http://ssrn.com/abstract=910800 , Working Papers Series, 2006 (b). · Mishra, S.K.: “A Comparative Study on Fitting of Gielis Curves by Classical Versus Generalized Simulated Annealing Methods”, Social Science Research Network (SSRN): http://ssrn.com/abstract=912336 , Working Papers Series, 2006 (c). · Mishra, S.K.: “Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Methods of Global Optimization”, Social Science Research Network (SSRN): http://ssrn.com/abstract=913667, Working Papers Series, 2006 (d). · Qian, F., Zhao, Y. and Hirata, H.: “Learning Cellular Automata for Function Optimization Problems”, T.IEE. Japan, 121C(1), pp. 261268, 2001. · Parsopoulos, K.E. and Vrahatis, M.N., “Recent Approaches to Global Optimization Problems Through Particle Swarm Optimization”, Natural Computing, 1 (23), pp. 235 306, 2002. · Peitgen, H.O., Jürgens, H.O. and Saupe, D.: Chaos and Fractals: New Frontiers of Science, SpringerVerlag, New York. 1992. · Von Neumann, J.: The Theory of Selfproducing Automata, (Edited and completed by A.W. Burks), Univ. of Illinois Press, Urbana, 1966. · Whittaker, E.T. and Watson, G.N.: A Course of Modern Analysis, Cambridge Univ. Press, 1996. · Wolfram, S.: A New Kind of Science, Wolfram Media, 2002. (The online version of the book is available on http://www.wolframscience.com/nksonline/toc.html ). · Yatapanage, N.: Cellular Automata as a Model for Dynamic Leaf Structure, BE Dissertation in Software Engineering, School of Information Technology and Electrical Engineering, University of Queensland, 2003. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/466 