Abstract
D-MORPH regression is a procedure for the treatment of a model prescribed as a linear superposition of basis functions with less observation data than the number of expansion parameters. In this case, there is an infinite number of solutions exactly fitting the data. D-MORPH regression provides a practical systematic means to search over the solutions seeking one with desired ancillary properties while preserving fitting accuracy. This paper extends D-MORPH regression to consider the common case where there is more observation data than unknown parameters. This situation is treated by utilizing a proper subset of the normal equation of least-squares regression to judiciously reduce the number of linear algebraic equations to be less than the number of unknown parameters, thereby permitting application of D-MORPH regression. As a result, no restrictions are placed on model complexity, and the model with the best prediction accuracy can be automatically and efficiently identified. Ignition data for a H 2/air combustion model as well as laboratory data for quantum-control-mechanism identification are used to illustrate the method.
Similar content being viewed by others
References
Bishop C.M.: Pattern Recognition and Machine Learning. Springer, New York, NY (2007)
Hastie T., Tibshirani R., Friedman J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York, NY (2001)
Tikhonov A.N.: Dokl. Akad. Nauk. SSSR 39, 195–198 (1943)
A.N. Tikhonov, Soviet Math. Dokl. 4, 1035–1038 (1963). English translation of Dokl. Akad. Nauk. SSSR 151, 501–504 (1963)
A.N. Tikhonov, V.A. Arsenin, Solution of Ill-Posed Problems. (Winston & Sons, Washington, 1977). ISBN 0-470-99124-0
Hansen P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia, PA (1998)
Hoerl A.E.: Chem. Eng. Prog. 58, 54–59 (1962)
Hoerl A.E., Kennard R.: Technometrics 12, 55–67 (1970)
Wahba G.: Spline models for observational data. SIAM, Philadelphia, PA (1990)
Wahba G.: Ann. Stat. 13, 1378–1402 (1985)
Wahba G., Wang Y.D., Gu C., Klein R., Klein B.: Ann. Stat. 23, 1865–1895 (1995)
Categories: linear algebra—estimation theory views, http://en.wikipedia.org/wiki/Tikhonov_regularization
Li G., Rabitz H.: J. Math. Chem. 48(4), 1010–1035 (2010)
N. Danielson, V. Beltrani, J. Dominy, H. Rabitz, Manuscript in preparation
Rothman A., Ho T.-S., Rabitz H.: Phys. Rev. A 72, 023416 (2005)
Rothman A., Ho T.-S., Rabitz H.: J. Chem. Phys. 123, 134104 (2005)
Rothman A., Ho T.-S., Rabitz H.: Phys. Rev. A 73, 053401 (2006)
Rao C.R., Mitra S.K.: Generalized Inverse of Matrix and Its Applications. Willey, New York, NY (1971)
Bellman R.: Introduction to Matrix Analysis, pp. 118. McGraw-hill Book Co., New York, NY (1970)
Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical Recipes in FORTRAN—The Art of Science Computing, pp. 51. Cambridge university press, New York, NY (1992)
Li J., Zhao Z.W., Kazakov A., Dryer F.L.: Int. J. Chem. Kinet. 36, 566–575 (2004)
Matlab [7.0R14], MathWorks, Inc. (2004)
Mitra A., Rabitz H.: Phys. Rev. A 67(3), 33407 (2003)
Rey-de-Castro R., Rabitz H.: Phys. Rev. A 81, 063422 (2010)
R. Rey-de-Castro, Z. Leghtas, H. Rabitz, Phys. Rev. Lett. (submitted)
Brif C., Chakrabarti R., Rabitz H.: New J. Phys. 12, 075008 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, G., Rey-de-Castro, R. & Rabitz, H. D-MORPH regression for modeling with fewer unknown parameters than observation data. J Math Chem 50, 1747–1764 (2012). https://doi.org/10.1007/s10910-012-0004-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-012-0004-z