Abstract
The need for fitting experimental data with a model containing nonlinear parameters occurs frequently. Estimates for the confidence intervals of the nonlinear parameters are occasionally stated, however the method of determination, if any, is rarely explained. Plots of the error surface near the minimum are popular but contain little information. Taylor's series expansion about a good estimate of the solution yields a convergent procedure for nonlinear fitting and a covariance matrix useful for estimation of the parameter confidence intervals. In order to make the advantages of this approach explicit, the data of Pindak and coworkers on the divergence of the cholesteric pitch in cholesteryl nonanoate near the smectic-A phase transition have been reanalyzed. Three-and five-parameter fits yield estimates of 0.674±0.005 and 0.702±0.035, respectively, for the critical exponentv. The results indicate that the validity of the published estimate ofv=0.675±0.025 depends on the independent measurement of two of the five model parameters originally proposed. The influence of weighting factors and highly correlated expansion functions is also demonstrated.
Similar content being viewed by others
References
The words “linear” and “nonlinear” are used throughout this paper to describe the occurrence of adjustable parameters within a proposed model. Knowledge of the use of linear combinations of expansion functions containing nonlinear combinations of independent variables is assumed
See, e.g., F. B. Hildebrand:Introduction to Numerical Analysis (McGraw-Hill, New York 1956) Chap. 7
See, e.g., J. J. White, H. I. Song, J. E. Rives, D. P. Landau: Phys. Rev. B4, 4605 (1971)
See, e.g., J. J. White, J. E. Rives: Phys. Rev. B6, 4352 (1972), and cited references
R. S. Pindak, C. C. Huang, J. T. Ho: Phys. Rev. Letters32, 43 (1974). The list of disquieting analyses of data on critical phenomena is extensive. This paper was selected for its timeliness and is not the oject of the caveat given in the introduction
Author information
Authors and Affiliations
Additional information
A preliminary account of this paper was presented to the Summer Meeting of the AAPT, Boone, N. C., June 20–22, 1974, AAPT Announcer4, No. 2, 30 (1974).
Rights and permissions
About this article
Cite this article
White, J.J. Estimation of confidence intervals for the nonlinear parameters of a least-squares fit. Application to cholesteryl nonanoate data. Appl. Phys. 5, 57–61 (1974). https://doi.org/10.1007/BF01193396
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF01193396