Algorithm 676: ODRPACK: software for weighted orthogonal distance regression

ABSTRACT

In this paper, we describe ODRPACK, a software package for the weighted orthogonal distance regression problem. This software is an implementation of the algorithm described in [2] for finding the parameters that minimize the sum of the squared weighted orthogonal distances from a set of observations to a curve or surface determined by the parameters. It can also be used to solve the ordinary nonlinear least squares problem. The weighted orthogonal distance regression procedure application to curve and surface fitting and to measurement error models in statistics. The algorithm implemented is an efficient and stable trust region (Levenberg-Marquardt) procedure that exploits the structure of the problem so that the computational cost per iteration is equal to that for the same type of algorithm applied to the ordinary nonlinear least squares problem. The package allows a general weighting scheme, provides for finite difference derivatives, and contains extensive error checking and report generating facilities.

REFERENCES

• 1: BOGGS, P. T., BYRD, R. H., DONALDSON, J. R., AND SCHNABEL, R.B. User's reference guide for ODRPACK--Software for weighted orthogonal distance regression. National Institute of Standards and Technology Internal Rep. 89-4103.
• 2: Paul T. Boggs , Richard H. Byrd , Robert B. Schnabel, A stable and efficient algorithm for nonlinear orthogonal distance regression, SIAM Journal on Scientific and Statistical Computing, v.8 n.6, p.1052-1078, November 1, 1987 [doi>10.1137/0908085]
• 3: BOGGS, P. T., AND DONALDSON, J.R. The computation and use of the asymptotic covariance matrix for measurement error models. National Institute of Standards and Technology Internal Rep. 89-4102.
• 4: BOGGS, P. T., DONALDSON, J. R., SCHNABEL, R. B., AND SPIEGELMAN, C.H. A computational examination of orthogonal distance regression. J. Econometrics 38, 1/2 (1988), 169-201.
• 5: COWELL, W. R., HAGUE, S. J., AND ILES, R. M.J. Toolpack/1 Introductory Guide. Issued jointly as Publication NP1277, Numerical Algorithms Group, Oxford, England, and Tech. Rep. ANL- 86-43, Argonne National Laboratory, Argonne, Ill., 1986.
• 6: J. E. Dennis, Jr. , Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16), Soc for Industrial & Applied Math, 1996
• 7: Janet R. Donaldson , Robert B. Schnabel, Computational experience with confidence intervals for nonlinear least squares, Technometrics, v.29 n.1, p.67-82, February 1, 1987 [doi>10.2307/1269884]
• 8: DONGARRA, J. J., MOLER, C. B., BUNCH, J. R., AND STEWART, G.W. LINPACK Users' Guide. SIAM, Philadelphia, Pa., 1979.
• 9: DRAPER, N. R., AND SMITH, H. Applied Regression Analysis, 2d Ed. Wiley, New York, 1981.
• 10: EFRON, B. The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia, Pa., 1982.
• 11: FONG, K. W., JEFFERSON, T. H., AND SUYEH1RO, T. SLATEC common mathematical library source file format. Lawrence Livermore Laboratory UCRL-53313, Livermore, Calif., 1982.
• 12: Wayne A Fuller, Measurement error models, John Wiley & Sons, Inc., New York, NY, 1986
• 13: HIMMELBLAU, D.M. Process Analysis by Statistical Methods. Wiley, New York, 1970.
• 14: C. L. Lawson , R. J. Hanson , D. R. Kincaid , F. T. Krogh, Basic Linear Algebra Subprograms for Fortran Usage, ACM Transactions on Mathematical Software (TOMS), v.5 n.3, p.308-323, Sept. 1979 [doi>10.1145/355841.355847]
• 15: MoR~, J.J. The Levenberg-Marquardt algorithm: Implementation and theory. In Numerical Ana/ys/s, G. A. Watson, Ed. Lecture Notes in Mathematics 630, Springer Verlag, Berlin 1977, 105-116.
• 16: Jorge J. Moré , Burton S. Garbow , Kenneth E. Hillstrom, Testing Unconstrained Optimization Software, ACM Transactions on Mathematical Software (TOMS), v.7 n.1, p.17-41, March 1981 [doi>10.1145/355934.355936]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Statistical computing

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.6 Optimization
          Subjects: Least squares methods

General Terms:
Algorithms