Abstract
Many nonlinear models as e.g. models of chemical reactions are described by systems of differential equations which have no explicit solution. In such cases the statistical analysis is much more complicated than for nonlinear models with explicitly given response functions. Numerical approaches need to be applied in place of explicit solutions. This paper describes how the analysis can be done when the response function is only implicitly given by differential equations. It is shown how the unknown parameters can be estimated and how these estimations can be applied for model discrimination and for deriving optimal designs for future research. The methods are demonstrated with a chemical reaction catalyzed by the enzyme Benzaldehyde lyase.
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DEMIR, A. S.— EREN, E.— HOSRIK, B.— Şeşeoglu, Ö.— POHL, M.— JANZEN, E.— KOLTER, D.— FELDMANN, R.— DÜNKELMANN, P.— MÜLLER, M.: Enantioselective Synthesis of α-Hydroxy Ketones via Benzaldehyde Lyase-Catalyzed C-C Bond Formation Reaction, Advanced Synthesis & Catalysis 344(1), (2002), 96–103.
DORMAND, J. R.— PRINCE, P. J.: A family of embedded Runge-Kutta formulae, J. Comput. Math. 10, (1958), 517–534.
HURVICH, C. M.— TSAI, C. L.: Regression and time series model selection in small samples, Biometrika 76, (1989), 297–307.
STROMBERG, A. J.— RUPPERT, D.: Breakdown in nonlinear regression, J. Amer. Statist. Assoc. 87 (1992), 991–997.
ATKINSON, A. C.— DONEV, A. N.: Optimum Experimental Designs. Oxford Statistical Science Series, Oxford University Press, Oxford, 1992.
BUNKE, H.— BUNKE, O.: Nonlinear Regression, Functional Relations and Robust Methods, John Wiley & Sons, Inc., Berlin, 1989.
EFRON, B.— TIBSHIRANI, R. J.: An Introduction to the Bootstrap. Monogr. Statist. Appl. Probab. 57, Chapman & Hall/CRC, Boca Raton, 1998.
FAHRMEIR, L.— HAMERLE, A.: Multivariate statistische Verfahren, Walter deGruyter & Co., Berlin, 1995.
KÜHL, S.: Enzymkatalysierte C-C Knüpfung: Reaktionstechnische Untersuchungen zur Synthese pharmazeutischer Intermediate, Universitt Bonn, Bonn, 2007.
PÁZMAN, A.: Foundations of Optimum Experimental Design, Reidel, Dordrecht, 1986.
PÁZMAN, A.: Nonlinear Statistical Models, Kluwer, Dordrecht, 1993.
ROUSSEEUW, P. J.— LEROY, A. M.: Robust Regression and Outlier Detection, Wiley, New York, 1987.
SEBER, G. A. F.— WILD, C. J.: Nonlinear Regression, John Wiley & Sons, Inc., Hoboken, NJ, 2003.
STREHMEL, K.— WEINER, R.: Numerik gewhnlicher Differentialgleichungen, B. G. Teubner, Stuttgart, 1995.
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Communicated by Gejza Wimmer
Dedicated to Professor Andrej Pázman on the occasion of his 70th birthday
Research of third author supported by the SFB/TR TRR 30 Project D6.
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Siudak, A., Lieres, E.v. & Müller, C.H. Estimation, model discrimination, and experimental design for implicitly given nonlinear models of enzyme catalyzed chemical reactions. Math. Slovaca 59, 593–610 (2009). https://doi.org/10.2478/s12175-009-0150-3
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DOI: https://doi.org/10.2478/s12175-009-0150-3