Robust Nonlinear Least Squares Approaches for Evaluating OVA-Mediated Bleaching Reactions: An Experimental Comparative Study

Abstract

Nonlinear models are usually encountered in various areas including experimental studies such as physics, chemistry, biology etc. Ordinary least squares is one of the most widely used methods for parameter estimation in different types of nonlinear models. However, there are some regression assumptions need to be satisfied for obtaining efficient parameter estimates. In this paper, the parameter estimation process is evaluated carefully for some bleaching reactions by using chicken egg albumin (OVA) and some precautions are taken in the presence of violations of the assumptions (heteroscedasticity, autocorrelation, the presence of outliers). In this way, robust logged nonlinear least squares approaches are examined and compared under different conditions of reactions.

Keywords

• Robust nonlinear least squares,heteroscedasticity,autocorrelation,outlier,chicken egg albumin

References

1. Kittrell, J.R., Hunter, W.G. and Watson, C.C., “Nonlinear least squares analysis of catalytic rate models”, AIChE Journal, 11(6): 1051-1057, (1965).
2. Rieger, P.H., “Least squares analysis of ESR powder patterns with noncoincident principal axes of the g and hyperfine tensors”, Journal of Magnetic Resonance, 50(3): 485-489, (1982).
3. Johnson, M.L. and Frasier, S.G., “Nonlinear least squares analysis”, Methods in Enzymology, 117: 301-342, (1985).
4. Boukamp, B.A., “A nonlinear least squares fit procedure for analysis of immittance data of electrochemical systems”, Solid State Ionics, 20(1): 31-44, (1986).
5. Small, J.R., Libertini, L.J. and Small, E.W., “Analysis of photoacoustic waveforms using the nonlinear least squares method”, Biophysical Chemistry, 42(1): 29-48, (1992).
6. Oshannessy, D.J., Brighamburke, M., Soneson, K.K., Hensley, P. and Brooks, I., “Determination of rate and equilibrium binding constants for macromolecular interactions using surface plasmon resonance: use of nonlinear least squares analysis methods”, Analytical Biochemistry, 212(2): 457-468, (1993).
7. Asari, K.V., Kumar, S. and Radhakrishnan, D., “A new approach for nonlinear distortion correction in endoscopic images based on least squares estimation”, IEEE Transactions on Medical Imaging, 18(4): 345-354, (1999).
8. Özer, İ. and Çağlar, A., “Protein-mediated nonphotochemical bleaching of malachite green in aqueous solution”, Dyes and Pigments, 54: 11-16, (2002).

9. Küçükkılınç, T. and Özer, İ., “Multi-site inhibition of human plasma cholinesterase by cationic phenoxazine and phenothiazine dyes”, Archives of Biochemistry and Biophysics, 461(2): 294-298, (2007).
10. Carroll, R.J. and Ruppert, D., “Robust estimation in heteroscedastic linear models”, The Annals of Statistics, 10(2): 429-441, (1982).
11. Carroll, R.J. and Ruppert, D., Transformation and Weighting in Regression, Chapman and Hall, New York, (1988).
12. Çelik, R., “Stabilizing heteroscedasticity for butterfly-distributed residuals by the weighting absolute centered external variable”, Journal of Applied Statistics, 42(4): 705-721, (2015).
13. Çelik, R., “A new test to detect monotonic and non-monotonic types of heteroscedasticity”, Journal of Applied Statistics, 44(2): 342-361, (2017).
14. Beal, S.L. and Sheiner, L.B., “Heteroscedastic nonlinear regression”, Technometrics, 30(3): 327-338, (1988).
15. Lin, J.G. and Wei, B.C., “Testing for heteroscedasticity in nonlinear regression models”, Communications in Statistics: Theory and Methods, 32(1): 171-192, (2003).
16. Cochrane, D. and Orcutt, G.H., “Application of least squares regression to relationships containing autocorrelated error terms”, Journal of the American Statistical Association, 44(245): 32-61, (1949).
17. Gallant, A.R. and Goebel, J.J., “Nonlinear regression with autocorrelated errors”, Journal of the American Statistical Association, 71(356): 961-967, (1976).
18. Glasbey, C.A., “Nonlinear regression with autoregressive time series errors”, Biometrics, 36(1): 135-139, (1980).
19. Aşıkgil, B. and Erar, A., “Modified two-stage least squares method”, XIIIth International Conference ASMDA, Vilnius, 124-128, (2009).
20. Aşıkgil, B. and Erar, A., “Polynomial tapered two-stage least squares method in nonlinear regression”, Applied Mathematics and Computation, 219: 9743-9754, (2013).
21. Rousseeuw, P.J. and Leroy, A.M., Robust Regression and Outlier Detection, John Wiley and Sons, New Jersey, (2003).
22. Sinha, S.K., Field, C.A. and Smith, B., “Robust estimation of nonlinear regression with autoregressive errors”, Statistics and Probability Letters, 63: 49-59, (2003).
23. Riazoshams, H., Midi, H.B. and Sharipov, O.S., “The performance of robust two-stage estimator in nonlinear regression with autocorrelated error”, Communications in Statistics: Simulation and Computation, 39(6): 1251-1268, (2010).
24. Jacquez, J.A., Compartmental Analysis in Biology and Medicine, Elsevier, New York, (1972).
25. Seber, G.A.F. and Wild, C.J., Nonlinear Regression, John Wiley and Sons, New York, (1989).
26. Cichocki, A., Zdunek, R., Phan, A.H. and Amari, S., Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation, John Wiley and Sons, Chichester, (2009).
27. Stein, P.E., Leslie, A.G., Finch, J.T. and Carrell, R.W., “Crystal structure of uncleaved ovalbumin at 1.95 A resolution”, Journal of Molecular Biology, 221(3): 941-959, (1991).