Информация о публикации

Просмотр записей
Инд. авторы: Kreinovich V., Shary S.P.
Заглавие: Interval methods for data fitting under uncertainty: A probabilistic treatment
Библ. ссылка: Kreinovich V., Shary S.P. Interval methods for data fitting under uncertainty: A probabilistic treatment // Reliable Computing. - 2016. - Vol.23. - P.105-140. - ISSN 1385-3139. - EISSN 1573-1340.
Внешние системы: РИНЦ: 27143360; SCOPUS: 2-s2.0-84982703440;
Реферат: eng: How to estimate parameters from observations subject to errors and uncertainty? Very often, the measurement errors are random quantities that can be adequately described by the probability theory. When we know that the measurement errors are normally distributed with zero mean, then the (asymptotically optimal) Maximum Likelihood Method leads to the popular least squares estimates. In many situations, however, we do not know the shape of the error distribution; we only know that the measurement errors are located on a certain interval. Then the maximum entropy approach leads to a uniform distribution on this interval, and the Maximum Likelihood Method results in the so-called minimax estimates. We analyze specificity and drawbacks of the minimax estimation under essential interval uncertainty in data and discuss possible ways to solve the difficulties. Finally, we show that, for the linear functional dependency, the minimax estimates motivated by the Maximum Likelihood Method coincide with those produced by the Maximum Compatibility Method that originate from interval analysis. © 2016, Public Library of Science. All rights reserved.
Ключевые слова: Interval uncertainty; Maximum compatibility method; Maximum likelihood method; Regression; Data handling; Errors; Least squares approximations; Maximum entropy methods; Maximum likelihood; Measurement errors; Normal distribution; Probability; Random errors; Uncertainty analysis; Compatibility methods; Data fittings; Interval uncertainty; Maximum likelihood methods; Regression; Maximum likelihood estimation; Data fitting;
Издано: 2016
Физ. характеристика: с.105-140
Цитирование: 
1. W. CHENEY AND D. KINCAID, Numerical Mathematics and Computing. Third Edition. Brooks/Cole Publishing Company, Pacific Grove, California, 1994.
2. B. CHOKR AND V. KREINOVICH, How far are we from complete knowledge: complexity of knowledge acquisition in Dempster-Shafer approach. In:R. R. Yager, J. Kacprzyk, and M. Pedrizzi (eds.), Advances in the Dempster-Shafer Theory of Evidence. Wiley, New York, 1994, pp. 555-576.
3. E. Z. DEMIDENKO, Comment II to the Paper of A. P. Voshchinin, A. F. Bochkov, and G. R. Sotirov "A method of data analysis under interval nonstatistical error". Industrial Laboratory, vol. 56 (1990), No. 7, pp. 83-84. (in Russian) Accessible at http://www.nsc.ru/interval/Library/Thematic/DataProcs/VosBochSoti.pdf
4. I. M. GELFAND, S. V. FOMIN, Calculus of Variations. Mineola, New York, Dover Publications, 2000.
5. I. I. GORBAN, Phenomenon of statistical stability, Technical Physics, vol. 59 (2014), issue 3, pp 333-340. DOI: 10.1134/S1063784214030128
6. P. C. HANSEN, V. PEREYRA, AND G. SCHERER, Least Squares Data Fitting with Applications. The John Hopkins University Press, Baltimore, Maryland, 2013.
7. L. JAULIN, M. KIEFFER, O. DIDRIT, AND E. WALTER, Applied Interval Analysis. Springer, London, 2001.
8. E. T. JAYNES AND G. L. BRETTHORST, Probability Theory: the Logic of Science. Cambridge University Press, Cambridge, UK, 2003.
9. L. V. KANTOROVICH, On some new approaches to numerical methods and processing observation data. Siberian Math. Journal, vol. 3 (1962), No. 5, pp. 701-709. (in Russian) Electronic version is accessible at http://www.nsc.ru/interval/Introduction/Kantorovich62.pdf
10. S. I. KUMKOV, Processing experimental data of ion conductivity of melted electrolyte by interval analysis methods, Journal "Melts" (2010), No. 3, pp. 79-89. (in Russian)
11. E. L. LEHMANN AND G. CASELLA, Theory of Point Estimation. Springer, New York, 2003.
12. lintregr.m - a free Matlab code implementing the Maximum Compatibiity Method for interval data fitting. Available at http://www.nsc.ru/interval/Programing/MCodes/lintregr.m
13. M. MILANESE, J. NORTON, H. PIET-LAHANIER, AND E. WALTER, EDS., Bounding Approaches to System Identification. Plenum Press, New York, 1996.
14. R. B. MILLAR, Maximum Likelihood Estimation and Inference: With Examples in R, SAS and ADMB. Wiley, Chichester, UK, 2011.
15. R. E. MOORE, Parameter sets for bounded-error data. Mathematics and Computers in Simulation, vol. 34 (1992), No. 2, pp. 113-119. DOI: 10.1016/0378-4754(92)90048-L
16. R. E. MOORE, R. B. KEARFOTT, M. J. CLOUD, Introduction to Interval Analysis. SIAM, Philadelphia, 2009.
17. H. T. NGUYEN, V. KREINOVICH, B. WU, AND G. XIANG, Computing Statistics under Interval and Fuzzy Uncertainty. Springer Verlag, Berlin, Heidelberg, 2012.
18. A. NEUMAIER, Interval Methods for Systems of Equations. Cambridge University Press, Cambridge, 1990.
19. P. V. NOVITSKII AND I. A. ZOGRAPH, Estimating the Measurement Errors. Energoatomizdat, Leningrad, 1991. (in Russian)
20. A. I. ORLOV, How often are the observations normal? Industrial Laboratory, vol. 57 (1991), No. 7, pp. 770-772. (in Russian)
21. N. M. OSKORBIN, A. V. MAKSIMOV, AND S. I. ZHILIN, Construction and analysis of empirical dependencies by uncertainty center method. The News of Altai State University, vol. 5 (1998), No. 1, pp. 37-40. (in Russian)
22. S. G. RABINOVICH, Measurement Errors and Uncertainty: Theory and Practice. Springer Verlag, Berlin, 2005.
23. J. ROHN, A Handbook of Results on Interval Linear Problems. Prague, Czech Academy of Sciences, 2005. An electronic book available at http://www.nsc.ru/interval/Library/Surveys/ILinProblems.pdf
24. F. C. SCHWEPPE, Recursive state estimation: unknown but bounded errors and system inputs. IEEE Trans. Autom. Control, AC-13 (1968), No. 1, pp. 22-28.
25. I. A. SHARAYA, IntLinInc2D - a software package for visualization of the solution sets to interval linear systems of relations with two unknowns. Version for Matlab. Available at http://www.nsc.ru/interval/sharaya/index.html#codes
26. S. P. SHARY, Interval analysis or Monte-Carlo methods? Computational Technologies, vol. 12 (2007), No. 1, pp. 103-115. (in Russian) Accessible at http://www.ict.nsc.ru/jct/getfile.php?id=946
27. S. P. SHARY, Solvability of interval linear equations and data analysis under uncertainty. Automation and Remote Control, vol. 73 (2012), No. 2, pp. 310-322. DOI: 10.1134/S0005117912020099
28. S. P. SHARY AND I. A. SHARAYA, Recognition of solvability of interval equations and its application to data analysis. Computational Technologies, vol. 18 (2013), No. 3, pp. 80-109. (in Russian) Accessible at http://www.ict.nsc.ru/jct/getfile.php?id=1551
29. S. P. SHARY AND I. A. SHARAYA, On solvability recognition for interval linear systems of equations. Optimization Letters, vol. 10 (2016), Issue 2, pp 247-260. DOI: 10.1007/s11590-015-0891-6
30. S. P. SHARY, New characterizations for the solution set to interval linear systems of equations. Applied Mathematics and Computation, vol. 265 (2015), pp. 570-573. DOI: 10.1016/j.amc.2015.05.029
31. S. P. SHARY, Finite-Dimensional Interval Analysis. Institute of Computational Technologies, Novosibirsk, 2015. (in Russian) Accessible at http://www.nsc.ru/interval/Library/InteBooks/SharyBook.pdf
32. S. P. SHARY, Maximum consistency method for data fitting under interval uncertainty. Journal of Global Optimization, published online July 18, 2015, pp. 1-16. DOI: 10.1007/s10898-015-0340-1
33. D. J. SHESKIN, Handbook of Parametric and Nonparametric Statistical Procedures. Chapman & Hall/CRC, Boca Raton, Florida, 2004.
34. T. E. SUTTO, H. C. DE LONG, AND P. C. TRULOVE, Physical properties of substituted imidazolium based ionic liquids gel electrolytes // Zeitschrift für Naturforschung A. 2002. Vol. 57, No. 11. P. 839-846.
35. A. B. TSYBAKOV, Introduction to Nonparametric Estimation. Springer, New York, 2009.
36. G. W. WALSTER AND V. KREINOVICH, For unknown-but-bounded errors, interval estimates are often better than averaging. ACM SIGNUM Newsletter, vol. 31 (1996), pp. 6-19.
37. S. I. ZHILIN, On fitting empirical data under interval error. Reliable Computing, vol. 11 (2005), pp. 433-442. DOI: 10.1007/s11155-005-0050-3 Accessaible at http://www.nsc.ru/interval/Library/Thematic/DataProcs/Zhilin.pdf
38. S. I. ZHILIN, Simple method for outlier detection in fitting experimental data under interval error, Chemometrics and Intelligent Laboratory Systems, vol. 88 (2007), pp. 60-68. DOI: 10.1016/j.chemolab.2006.10.004