Scientific Research

An Academic Publisher

A Gauss-Newton-Based Broyden’s Class Algorithm for Parameters of Regression Analysis

**Author(s)**Leave a comment

In this paper, a Gauss-Newton-based Broyden’s class method for parameters of regression problems is presented. The global convergence of this given method will be established under suitable conditions. Numerical results show that the proposed method is interesting.

KEYWORDS

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

X. Li and X. Zhao, "A Gauss-Newton-Based Broyden’s Class Algorithm for Parameters of Regression Analysis,"

*Applied Mathematics*, Vol. 2 No. 1, 2011, pp. 39-46. doi: 10.4236/am.2011.21005.

[1] | D. M. Bates and D. G. Watts, “Nonlinear Regression Analysis and Its Applications,” John Wiley & Sons, New York, 1988. doi:10.1002/9780470316757 |

[2] | S. Chatterjee and M. Machler, “Robust Regression: A Weighted Least Squares Approach, Communications in Statistics,” Theorey and Methods, Vol. 26, No. 6, 1997, pp. 1381-1394. doi:10.1080/03610929708831988 |

[3] | R. Christensen, “Analysis of Variance, Design and Regression: Applied Statistical Methods,” Chapman and Hall, New York, 1996. |

[4] | N. R. Draper and H. Smith, “Applied Regression Analysis,” 3rd Edtion, John Wiley & Sons, New York, 1998. |

[5] | F. A. Graybill and H. K. Iyer, “Regression Analysis: Concepts and Applications,” Duxbury Press, Belmont, 1994. |

[6] | R. F. Gunst and R. L. Mason, “Regression Analysis and Its Application: A Data-Oriented Approach,” Marcel Dekker, New York, 1980. |

[7] | R. H. Myers, “Classical and Modern Regression with Applications,” 2nd Edtion, PWS-Kent Publishing Company, Boston, 1990. |

[8] | R. C. Rao, “Linear Statistical Inference and Its Applications,” John Wiley & Sons, New York, 1973. doi:10.1002/9780470316436 |

[9] | D. A. Ratkowsky, “Nonlinear Regression Modeling: A Unified Practical Approach,” Marcel Dekker, New York, 1983. |

[10] | D. A. Ratkowsky, “Handbook of Nonlinear Regression Modeling,” Marcel Dekker, New York, 1990. |

[11] | A. C. Rencher, “Methods of Multivariate Analysis,” John Wiley & Sons, New York, 1995. |

[12] | G. A. F. Seber and C. J. Wild, “Nonlinear Regression,” John Wiley & Sons, New York, 1989. doi:10.1002/ 0471725315 |

[13] | A. Sen and M. Srivastava, “Regression Analysis: Theory, Methods, and Applications,” Springer-Verlag, New York, 1990. |

[14] | J. Fox, “Linear Statistical Models and Related Methods,” John Wiley & Sons, New York, 1984. |

[15] | S. Haberman and A. E. Renshaw, “Generalized Linear Models and Actuarial Science,” The Statistician, Vol. 45, No. 4, 1996, pp. 407-436. doi:10.2307/2988543 |

[16] | S. Haberman and A. E. Renshaw, “Generalized Linear Models and Excess Mortality from Peptic Ulcers,” Insurance: Mathematics and Economics, Vol. 9, No. 1, 1990, pp. 147-154. doi:10.1016/0167-6687(90)90012-3 |

[17] | R. R. Hocking, “The Analysis and Selection of Variables in Linear Regression,” Biometrics, Vol. 32, No. 1, 1976, pp. 1-49. doi:10.2307/2529336 |

[18] | P. McCullagh and J. A. Nelder, “Generalized Linear Models,” Chapman and Hall, London, 1989. |

[19] | J. A. Nelder and R. J. Verral, “Credibility Theory and Generalized Linear Models.” ASTIN Bulletin, Vol. 27, No. 1, 1997, pp. 71-82. doi:10.2143/AST.27.1.563206 |

[20] | M. Raydan, “The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem,” SIAM Journal on Optimization, Vol. 7, No. 1, 1997, pp. 26-33. doi:10.1137/S1052623494266365 |

[21] | J. Schropp, “A Note on Minimization Problems and Multistep Methods,” Numerical Mathematics, Vol. 78, 1997, pp. 87-101. doi:10.1007/s002110050305 |

[22] | J. Schropp, “One-Step and Multistep Procedures for Constrained Minimization Problems,” IMA Journal of Numerical Analysis, Vol. 20, No. 1, 2000, pp. 135-152. doi:10.1093/imanum/20.1.135 |

[23] | D. J. Van Wyk, “Differential Optimization Techniques,” Applied Mathematical Modelling, Vol. 8, 1984, pp. 419-424. doi:10.1016/0307-904X(84)90048-9 |

[24] | M. N. Vrahatis, G. S. Androulakis, J. N. Lambrinos and G. D. Magolas, “A Class of Gradient Unconstrained Minimization Algorithms with Adaptive Stepsize,” Journal of Computational and Applied Mathematics, Vol. 114, No. 2, 2000, pp. 367-386. doi:10.1016/S0377-0427(99) 00276-9 |

[25] | G. Yuan and X. Lu, “A New Line Search Method with Trust Region for Unconstrained Optimization,” Communications on Applied Nonlinear Analysis, Vol. 15, 2008, No. 2, pp. 35-49. |

[26] | G. Yuan and Z. Wei, “New Line Search Methods for Unconstrained Optimization,” Journal of the Korean Statistical Society, Vol. 38, No. 1, 2009, pp. 29-39. doi: 10.1016/j.jkss.2008.05.004 |

[27] | G. Yuan and X. Lu, “A Modified PRP Conjugate Gradient Method,” Annals of Operations Research, Vol. 166, No. 1, 2009, pp. 73-90. doi:10.1007/s10479-008-0420-4 |

[28] | G. Yuan, “Modified Nonlinear Conjugate Gradient Methods with Sufficient Descent Property for Large-Scale Optimization Problems,” Optimization Letters, Vol. 3, No. 1, 2009, pp.11-21. doi:10.1007/s11590-008-0086-5 |

[29] | G. Yuan, X. Lu and Z. Wei, “A Conjugate Gradient Method with Descent Direction for Unconstrained Optimization,” Journal of Computational and Applied Mathematics, Vol. 233, No. 2, 2009, pp. 519-530. doi:10.1016/ j.cam.2009.08.001 |

[30] | G. Yuan, “A Conjugate Gradient Method for Unconstrained Optimization Problems,” International Journal of Mathematics and Mathematical Sciences, Vol. 2009, 2009, pp. 1-14. doi:10.1155/2009/329623 |

[31] | G. Yuan and Z. Wei, “A Nonmonotone Line Search Method for Regression Analysis,” Journal of Service Science and Management, Vol. 2, No. 1, 2009, pp. 36-42. doi: 10.4236/jssm.2009.21005 |

[32] | A. Griewank, “The ‘Global’ Convergence of Broyden-Like Methods with a Suitable Line Search,” Journal of the Australian Mathematical Society. Series B, Vol. 28, No. 1, 1986, pp. 75-92. doi:10.1017/S0334270000005208 |

[33] | D. T. Zhu, “Nonmonotone Backtracking Inexact Quasi-Newton Algorithms for Solving Smooth Nonlinear Equations,” Applied Mathematics and Computation, Vol. 161, No. 3, 2005, pp. 875-895. doi:10.1016/j.amc.2003.12.074 |

[34] | J. Y. Fan, “A Modified Levenberg-Marquardt Algorithm for Singular System of Nonlinear Equations,” Journal of Computational Mathematics, Vol. 21, No. 5, 2003, pp. 625-636. |

[35] | Y. Yuan, “Trust Region Algorithm for Nonlinear Equations,” Information, Vol. 1, 1998, pp. 7-21. |

[36] | D. Li and M. Fukushima, “A Global and Superlinear Convergent Gauss-Newton-Based BFGS Method for Symmetric Nonlinear Equations,” SIAM Journal on Numerical Analysis, Vol. 37, No. 1, 1999, pp. 152-172. doi: 10.1137/S0036142998335704 |

[37] | Z. Wei, G. Yuan and Z. Lian, “An Approximate Gauss-Newton-Based BFGS Method for Solving Symmetric Nonlinear Equations,” Guangxi Sciences, Vol. 11, No. 2, 2004, pp. 91-99. |

[38] | G. Yuan and X. Li, “An Approximate Gauss-Newton-Based BFGS Method with Descent Directions for Solving Symmetric Nonlinear Equations,” OR Transactions, Vol. 8, No. 4, 2004, pp. 10-26. |

[39] | G. Yuan and X. Lu, “A Nonmonotone Gauss-Newton-Based BFGS Method for Solving Symmetric Nonlinear Equations,” Journal of Lanzhou University, Vol. 41, 2005, pp. 851-855. |

[40] | G. Yuan abd X. Lu, “A New Backtracking Inexact BFGS Method for Symmetric Nonlinear Equations,” Computer and Mathematics with Application,” Vol. 55, No. 1, 2008, pp. 116-129. doi:10.1016/j.camwa.2006.12.081 |

[41] | G. Yuan, X. Lu and Z. Wei, “BFGS Trust-Region Method for Symmetric Nonlinear Equations,” Journal of Computational and Applied Mathematics, Vol. 230, No. 1, 2009, pp. 44-58. doi:10.1016/j.cam.2008.10.062 |

[42] | G. Yuan, Z. Wang and Z. Wei, “A Rank-One Fitting Method with Descent Direction for Solving Symmetric Nonlinear Equations,” International Journal of Communications, Network and System Sciences, Vol. 2, No. 6, 2009, pp. 555-561. doi:10.4236/ijcns.2009.26061 |

[43] | G. Yuan, S. Meng and Z. Wei, “A Trust-Region-Based BFGS Method with Line Search Technique for Symmetric Nonlinear Equations,” Advances in Operations Research, Vol. 2009, 2009, pp. 1-20. doi:10.1155/2009/ 909753 |

[44] | G. Yuan and X. Li, “A Rank-One Fitting Method for Solving Symmetric Nonlinear Equations,” Journal of Applied Functional Analysis, Vol. 5, No. 4, 2010, pp. 389-407. |

[45] | G. Yuan, Z. Wei and X. Lu, “A Nonmonotone Trust Region Method for Solving Symmetric Nonlinear Equations,” Chinese Quarterly Journal of Mathematics, Vol. 24, No. 4, 2009, pp. 574-584. |

[46] | R. Byrd and J. Nocedal, “A Tool for the Analysis of Quasi-Newton Methods with Application to Unconstrained Minimization,” SIAM Journal on Numerical Analysis, Vol. 26, No. 3, 1989, pp. 727-739. doi:10.1137/0726042 |

[47] | D. Xu, “Global Convergence of the Broyden’s Class of Quasi-Newton Methods with Nonomonotone Linesearch,” ACTA Mathematicae Applicatae Sinica, English Series, Vol. 19, No. 1, 2003, pp.19-24. doi:10.1007/ s10255-003-0076-4 |

[48] | S. Chatterjee, A. S. Hadi and B. Price, “Regression Analysis by Example,” 3rd Edition, John Wiley & Sons, New York, 2000. |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.