SENSOP: A derivative-free solver for nonlinear least squares with sensitivity scaling
Nonlinear least squares optimization is used most often in fitting a complex model to a set of data. An ordinary nonlinear least squares optimizer assumes a constant variance for all the data points. This paper presents SENSOP, a weighted nonlinear least squares optimizer, which is designed for fitting a model to a set of data where the variance may or may not be constant. It uses a variant of the Levenberg-Marquardt method to calculate the direction and the length of the step change in the parameter vector. The method for estimating appropriate weighting functions applies generally to 1-dimensional signals and can be used for higher dimensional signals. Sets of multiple tracer outflow dilution curves present special problems because the data encompass three to four orders of magnitude; a fractional power function provides appropriate weighting giving success in parameter estimation despite the wide range.
KeywordsOptimization Curve fitting Indicator dilution Probability density function Sensitivity function Model identifiability Weighting function
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- 4.Box, G.E.P.; Hill, W.J. Correcting inhomogeneity of variance with power transformation weighting. Technometrics 16:385–389; 1974.Google Scholar
- 6.Dennis, J.E.; Gay, D.M.; Welsch, R.E. Algorithm 573: NL2SOL—An adaptive nonlinear least-squares algorithm. ACM TOMS 7:369–383; 1981.Google Scholar
- 7.Donaldson, J.R.; Schnabel, R.B. Computational experience with confidence regions and confidence intervals for nonlinear least squares. Technometrics 29:67–82; 1987.Google Scholar
- 8.Duncan, G.T. An empirical study of jackknife-constructed confidence regions in nonlinear regression. Technometrics 20:123–129; 1978.Google Scholar
- 9.Hiebert, K.L. An evaluation of mathematical software that solves nonlinear least squares problems. ACM TOMS 7:1–16; 1981.Google Scholar
- 12.Lanczos, C. Applied analysis. Englewood Cliffs, NJ: Prentice-Hall; 1956.Google Scholar
- 13.Levenberg, K. A method for the solution of certain problems in least squares. Quart. Appl. Math. 2:164–168; 1944.Google Scholar
- 15.Moré, J.J.; Garbow, B.S.; Hillstrom, K.E. Testing unconstrained optimizer software ACM TOMS 7:17–41; 1981.Google Scholar
- 16.Press, S.J. Applied multivariate analysis. New York: Holt, Rinehart and Winston; 1972.Google Scholar
- 17.Ratkowsky, D.A. Nonlinear regression modeling: Aunified practical approach. New York: Marcel Dekker; 1983.Google Scholar