Performance of Differential Evolution Method in Least Squares Fitting of Some Typical Nonlinear Curves

Mishra, SK (2007): Performance of Differential Evolution Method in Least Squares Fitting of Some Typical Nonlinear Curves.

Abstract

No foolproof method exists to fit nonlinear curves to data or estimate the parameters of an intrinsically nonlinear function. Some methods succeed at solving a set of problems but fail at the others. The Differential Evolution (DE) method of global optimization is an upcoming method that has shown its power to solve difficult nonlinear optimization problems. In this study we use the DE to solve some nonlinear least squares problems given by the National Institute of Standards and Technology (NIST), US Department of Commerce, USA and some other challenge problems posed by the CPC-X Software (the makers of the AUTO2FIT software). The DE solves the test problems given by the NIST and most of the challenge problems posed by the CPC-X, doing marginally better than the AUTO2FIT software in a few cases.

• Item Type: MPRA Paper
• Institution: North-Eastern Hill University, Shillong (India)
• Language: English
• Keywords: Nonlinear least squares; curve fitting; Differential Evolution; global optimization; AUTO2FIT; CPC-X Software; NIST; National Institute of Standards and Technology; test problems
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: 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 > C2 - Single Equation Models ; Single Variables > C20 - General

References

· 7d-soft High Technology Inc (--) AUTO2FIT Software The New Website of CPC-X Software http://www.geocities.com/neuralpower now at www.7d-soft.com · Bates, D and Watts, D (1988) Nonlinear Regression Analysis and Its Applications, John Wiley and Sons, New York. · Bennett, L, Swartzendruber, L and Brown, H (1994) Superconductivity Magnetization Modeling, National Institute of Standards and Technology (NIST), US Department of Commerce, USA. · Box, GEP (1957) “Evolutionary Operation: A Method for Increasing Industrial Productivity”, Applied Statistics, 6 , pp. 81-101. · Box, GP, Hunter, WG and Hunter, JS (1978) Statistics for Experimenters. Wiley, New York, pp. 483-487. · Box, MJ (1965) “A New Method of Constrained Optimization and a Comparison with Other Methods”. Comp. J. 8, pp. 42-52. · Cerny, V (1985) "Thermodynamical Approach to the Traveling Salesman Problem: An Efficient Calculations by Fast Computing Machines", J. Chem. Phys.,21, 6, 1087-1092. · CPC-X Software (--) At http://www.geocities.com/neuralpower/Regression_Test.htm#3. %20One%20More%20Test%20Data of http://www.geocities.com/neuralpower · Eberhart RC and Kennedy J (1995) “A New Optimizer using Particle Swarm Theory”, Proceedings Sixth Symposium on Micro Machine and Human Science, pp. 39–43. IEEE Service Center, Piscataway, NJ. · Eckerle, K (197?) Circular Interference Transmittance Study, National Institute of Standards and Technology (NIST), US Department of Commerce, USA. · Glover F (1986) " Future Paths for Integer Programming and Links to Artificial Intelligence", Computers and Operations Research, 5:533-549. · Goffe, Ferrier and Rogers (1994) "Global Optimization of Statistical Functions with Simulated Annealing," Journal of Econometrics, 60 (1/2), pp. 65-100. · Hahn, T (197?) Copper Thermal Expansion Study. National Institute of Standards and Technology (NIST), US Department of Commerce, USA. · Holland, J (1975) Adaptation in Natural and Artificial Systems, Univ. of Michigan Press, Ann Arbor. · Judge, GG, Griffith, WE, Hill, RC, Lee, CH and Lotkepohl, H (1990) The Theory and Practice of Econometrics, John Wiley, New York. · Kahaner, D, Moler, C and Nash, S (1989) Numerical Methods and Software. Prentice Hall, Englewood Cliffs, NJ: pp. 441-445. · Kirby, R. (1979) Scanning Electron Microscope Line Width Standards. National Institute of Standards and Technology (NIST), US Department of Commerce, USA. · Kirkpatrick, S, Gelatt, CD Jr., and Vecchi, MP (1983) "Optimization by Simulated Annealing", Science, 220, 4598, 671-680. · Kowalik, JS and Osborne, MR (1978) Methods for Unconstrained Optimization Problems. Elsevier North-Holland, New York. · Lanczos, C. (1956). Applied Analysis. Prentice Hall, Englewood Cliffs, NJ, pp. 272-280. · Mathworks.com (.) Statistical Toolbox - Example: Nonlinear Modeling Hougen-Watson Model http://www.mathworks.com/access/helpdesk_r13/help/toolbox/stats/nonlin_3.html · Metropolis, N (1987) The Beginning of the Monte Carlo Method. Los Alamos Science, No. 15, Special Issue, pp. 125-130. · Metropolis, N, Rosenbluth, A, Rosenbluth, M, Teller, A, and Teller, E (1953) "Equation of State Simulation Algorithm", J. Opt. Theory Appl., 45, 1, 41-51. · Meyer, RR (1970) Theoretical and Computational Aspects of Nonlinear Regression. in Nonlinear Programming, Rosen, JB, Mangasarian, OL and Ritter, K (Eds). Academic Press, New York, pp. 465-486. · Mishra, SK (2006) “Fitting a Logarithmic Spiral to Empirical Data with Displaced Origin”, SSRN http://ssrn.com/abstract=897863 · Misra, D (1978) Dental Research Monomolecular Adsorption Study, National Institute of Standards and Technology (NIST), US Department of Commerce, USA. · More, JJ, Garbow, BS, and Hillstrom, KE (1981) Testing unconstrained optimization software. ACM Transactions on Mathematical Software. 7(1), pp. 17-41. · Nelder, JA and Mead, R (1964) “A Simplex Method for Function Minimization” Computer Journal, 7: pp. 308-313. · Nelson, W (1981) Analysis of Performance-Degradation Data, IEEE Transactions on Reliability, 2-R-30(2), pp. 149-155. · NIST (--) Nonlinear Regression http://www.itl.nist.gov/div898/strd/nls/nls_main.shtml · Osborne, MR (1972) Some Aspects of Nonlinear Least Squares Calculations. in Numerical Methods for Nonlinear Optimization, Lootsma (Ed). Academic Press, New York, pp. 171-189. · Rao, CR and Mitra, SK (1971) Generalized Inverse of Matrices and its Applications, Wiley, New York. · Ratkowsky, DA (1983) Nonlinear Regression Modeling. Marcel Dekker, New York. · Roszman, L (19??) Quantum Defects for Sulfur I Atom, National Institute of Standards and Technology (NIST), US Department of Commerce, USA. · Storn, R and Price, K (1995) "Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces": Technical Report, International Computer Science Institute, Berkley. · Thurber, R (197?) Semiconductor electron mobility modeling. National Institute of Standards and Technology (NIST), US Department of Commerce, USA. · Törn, AA (1978) “A search Clustering Approach to Global Optimization” , in Dixon, LCW and Szegö, G.P. (Eds) Towards Global Optimization – 2, North Holland, Amsterdam. · Törn, AA and Viitanen, S (1994) “Topographical Global Optimization using Presampled Points”, J. of Global Optimization, 5, pp. 267-276. · Wild, J (2001) “Simann.f - Bug in Description of Judge's Function” letter to [e-mail] and [e-mail], in the Simulated Annealing based Fortran Computer program for nonlinear optimization Simann.f available at http://netlib2.cs.utk.edu/opt/simann.f