Analysis of the stability of nonlinear regression models to errors in measured data

Pattern Recognition and Image Analysis - Tập 26 - Trang 608-616 - 2016
G. I. Rudoy1
1Moscow Institute of Physics and Technology State University, Dolgoprudny, Russia

Tóm tắt

In order to reconstruct a nonlinear dependence of the refractive index of a medium on the wavelength, a set of inductively generated models for choosing the optimal one is considered. An algorithm for the inductive generation of admissible nonlinear models is applied. A criterion for determining the error in the coefficients of the generated models, which is referred to as stability, and a method for estimating the stability of the solution are proposed. The results of numerical simulation on the data obtained in an experiment on determining the composition of a mixture from its total dispersion are presented.

Tài liệu tham khảo

J. W. Davidson, D. A. Savic, and G. A. Walters, “Symbolic and numerical regression: experiments and applications,” in Developments in Soft Computing, Ed. by R. John and R. Birkenhead (De Montfort Univ., Leicester, 2001), pp. 175–182. C. Sammut and G. I. Webb, “Symbolic regression,” in Encyclopedia of Machine Learning, Ed. by C. Sammut, and G. I. Webb, (Springer, 2010), p. 954. doi 10.1007/978-0-387-30164-810.1007/978-0-387-30164-8 V. Strijov and G. W. Weber, “Nonlinear regression model generation using hyperparameter optimization,” Comput. Math. Appl. 60 (4), 981–988 (2010). doi 10.1016/j.camwa.2010.03.021 V. V. Strizhov, “Inductive producing methods for regressive models,” Preprint of Dorodnicyn Computing Centre of RAS (Moscow, 2008). G. I. Rudoi and V. V. Strizhov, “Algorithms for superposition inductive producing for approximating the measuring data,” Inf. Ee Primen. 7 (1), 44–53 (2013). D. W. Marquardt, “An algorithm for least-squares estimation of non-linear parameters,” J. Soc. Industr. Appl. Math. 11 (2), 431–441 (1963). J. J. More, “The Levenberg-Marquardt algorithm: implementation and theory,” in Lecture Notes in Mathematics, Ed. by G. A. Watson (Springer-Verlag, Berlin, 1978), pp. 105–116. ftp://math.liu.se in pub/references V. A. Vatutin, G. I. ivchenko, Yu. I. Medvedev, and V. P. Chistyakov, Probability Theory and Mathematical Statistics in Problems (Drofa, Moscow, 2005) [in Russian]. V. I. Malyshev, Introduction into Experimental Spectroscopy (Nauka, Moscow, 1979) [in Russian]. I. N. Zaidel, Spectroscopy: Technique and Practice (Nauka, Moscow, 1972) [in Russian]. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. (Springer-Verlag, 2009). C. McDiarmid, “On the method of bounded differences,” Surveys Combinat. 141 (1), 148–188 (1989). L. Devroye, “Exponential inequalities in nonparametric estimation,” in Nonparametric Functional Estimation and Related Topics (Springer, 1991), pp. 31–44. O. Bousquet and A. Elisseeff, “Stability and generalization,” J. Mach. Learning Res. 2, 499–526 (2002). C. Cortes, M. Mohri, D. Pechyony, and A. Rastogi, “Stability of transductive regression algorithms,” in Proc. 25th Int. ACM Conf. on Machine Learning (New York, 2008), pp. 176–183. C. Cortes, M. Mohri, and A. Talwalkar, “On the impact of kernel approximation on learning accuracy,” in Proc. Int. Conf. on Artificial Intelligence and Statistics (Sardinia, 2010), pp. 113–120. N. V. Serova, Polymeric Optical Materials (Nauchnye osnovy tekhnologii, St. Petersburg, 2001) [in Russian]. V. N. Vapnik, The Way to Restore Relationships According to Empirical Data (Nauka, Moscow, 1979) [in Russian].