Abstract
For the linear regression model, the data-fitting problem under the interval uncertainty of the data is studied. As an estimate of the linear function parameters, it is proposed to take their values that deliver the maximum for the so-called recognizing functional of the parameter set compatible with the data (the maximum compatibility method). The properties of the recognizing functional, its interpretation, and the properties of the estimates obtained using the maximum compatibility method are investigated. The relationships to other data analysis methods are discussed, and a practical electrochemistry problem is solved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Draper and H. Smith, Applied Regression Analysis (Wiley, New York, 1966), Vols. 1, 2.
Yu. V. Linnik, Method of Least Squares and Principles of the Theory of Observations (GIFML, Moscow, 1962; Univ. Liverpool, Liverpool, 1961).
L. V. Kantorovich, “On some new approaches to computational methods and observations processing,” Sib. Mat. Zh. 3, 701–709 (1962).
M. L. Lidov, “Minimax methods for estimation,” KIAM Preprint No. 71 (Keldysh Inst. Appl. Math., Moscow, 2010).
A. P. Voshchinin, A. F. Bochkov, and G. R. Sotirov, “Method of data analysis with interval non-statistical error,” Zavod. Lab. 56 (7), 76–81 (1990).
V. M. Kuntsevich, “Control and identification under uncertainty conditions: results and unsolved problems,” Radioelektron. Komp’yut. Sist., No. 5 (24), 34–46 (2007).
N. M. Oskorbin, A. V. Maksimov, and S. I. Zhilin, “Construction and analysis of empirical dependencies by the method of uncertaity center,” Izv. Altaisk. Univ., No. 1, 37–40 (1998).
S. I. Spivak, V. I. Timoshenko, and M. G. Slin’ko, “Using of Chebyshev alignment method in construction of kinetic model of complex chemical reaction,” Dokl. Akad. Nauk SSSR 192, 580–582 (1970).
S. I. Spivak, O. G. Kantor, D. S. Yunusova, S. I. Kuznetsov, and S. V. Kolesov, “Error estimates and measuring significance for linear models,” Inform. Primen. 9, 87–97 (2015).
V. A. Sukhanov, Study of Empirical Dependencies: Nonstatistical Approach (Altaisk. Univ., Barnaul, 2007) [in Russian].
S. I. Zhilin, “On fitting empirical data under interval error,” Reliable Comput. 11, 433–442 (2005).
B. T. Polyak and S. A. Nazin, “Parameter estimation in linear multidimensional systems with interval uncertaity,” Probl. Upravl. Inform., Nos. 1–2, 103–115 (2006).
A. L. Pomerantsev and O. Ye. Rodionova, “Construction of a multivariate calibration by the simple interval calculation method,” J. Anal. Chem. 61, 952–966 (2006).
S. I. Kumkov, “Working of experimental data of ionic condaction molten electrolyte by methods of interwal analysis,” Rasplavy, No. 3, 79–89 (2010).
A. A. Podruzhko and A. S. Podruzhko, Interval Representation of Polynomial Regressions (Editorial URSS, Moscow, 2003) [in Russian].
F. S. Schweppe, “Recursive state estimation: unknown but bounded errors and system inputs,” IEEE Trans. Autom. Control 13, 22–28 (1968).
P. L. Combettes, “Foundations of set-theoretic estimation,” Proc. IEEE 81, 182–208 (1993).
Bounding Approaches to System Identification, Ed. by M. Milanese, J. Norton, H. Piet-Lahanier, and E. Walter (Plenum, New York, 1996).
L. Jaulin, M. Kieffer, O. Didrit, and E. Walter, Applied Interval Analysis (Springer, London, 2001; Regular. Khaotich. Dinamika, Moscow, Izhevsk, 2007).
A. V. Kurzhanskii, “Identification problem–the theory of guaranteed estimations,” Avtom. Telemekh., No. 4, 3–26 (1991).
F. L. Chernous’ko, Estimation of Phase State of Dynamical Systems (Nauka, Moscow, 1988) [in Russian].
S. P. Shary, “Solvability of interval linear equations and data analysis under uncertainty,” Autom. Remote Control 73, 310–322 (2012).
V. Yu. Lemeshko and S. P. Postovalov, “On solving problems of statistical analysis of interval observations,” Vychisl. Tekhnol. 2 (1), 28–36 (1997).
A. I. Orlov, Applied Statistics (Ekzamen, Moscow, 2004) [in Russian].
Yu. I. Alimov and Yu. A. Kravtsov, “Is probability a 'normal' physical quantity?,” Sov. Phys. Usp. 35, 606–622 (1992).
V. N. Tutubalin, “Probability, computers, and the processing of experimental data,” Phys. Usp. 36, 628–641 (1993).
R. B. Kearfott, M. Nakao, A. Neumaier, B. Rump, S. P. Shary, and P. van Hentenryck, “Standardized notation in interval analysis,” Vychisl. Tekhnol. 15 (1), 7–13 (2010).
S. P. Shary, Finite-Dimensional Interval Analysis (Inst. Vychisl. Tekhnol. SO RAN, Novosibirsk, 2016) [in Russian]. http://www.nsc.ru/interval/Library/InteBooks.
M. Fiedler, J. Nedoma, J. Ramik, J. Rohn, and K. Zimmermann, Linear Optimization Problems with Inexact Data (Springer, New York, 2006; Regular. Khaotich. Dinamika, Moscow, Izhevsk, 2008).
E. Hansen and G. W. Walster, Global Optimization using Interval Analysis Methods: Revised and Expanded (CRC, Boca Raton, FL, 2003; Regular. Khaotich. Dinamika, Moscow, Izhevsk, 2012).
A. Neumaier, Interval Methods for Systems of Equations (Cambridge Univ. Press, Cambridge, 1990).
I. A. Sharaya, IntLinIncR3 Package for Visualization of Solutions Sets of Interval Linear Systems with Three Unknowns. http://www.nsc.ru/interval/Programing.
A. V. Lakeev and S. I. Noskov, “On solutions array of linear equation with intervally predetermined operators and right part,” Sib. Mat. Zh. 35, 1074–1084 (1994).
H. Beeck, “Uber die Struktur und Abschatzungen der Losungsmenge von linearen Gleichungssystemen mit Intervallkoeffizienten,” Computing 10, 231–244 (1972).
G. Alefeld and J. Herzberger, Introduction to Interval Computation (Elsevier, Amsterdam, 1987).
R. E. Moore, R. B. Kearfott, and M. J. Cloud, Introduction to Interval Analysis (SIAM, Philadelphia, 2009).
S. P. Sharyi and I. A. Sharaya, “Recognition of solvability of interval equations and its applications to data analysis,” Vychisl. Tekhnol. 18 (3), 80–109 (2013).
S. P. Shary and L. A. Sharaya, “On solvability recognition for interval linear systems of equations,” Optimiz. Lett. 10, 247–260 (2016).
E. Ya. Remez, Principles of Numerical Methods of Chebyshev Approximation (Nauk. Dumka, Kiev, 1969) [in Russian].
V. Kreinovich and S. P. Shary, “Interval methods for data fitting under uncertainty: a probabilistic treatment,” Reliable Comput. 23, 105–140 (2016).
A. S. Strekalovskii, Elements of Nonconvex Optimization (Nauka, Novosibirsk, 2003) [in Russian].
N. Z. Shor and N. G. Zhurbenko, “Minimization method using space stretching operation in the direction of two consistent gradients difference,” Kibernetika, No. 3, 51–59 (1971).
P. I. Stetsyuk, Ellipsoid Methods and r-Algorithms (Evrika, Kishinev, 2014) [in Russian].
E. A. Nurminski, “Separating plane algorithms for convex optimization,” Math. Programming. 76, 373–391 (1997).
E. Vorontsova, “Extended separating plane algorithm and NSO-solutions of PageRank problem,” in Proceedings of the International Conference on Discrete Optimization and Operations Research DOOR 2016, Vladivostok, Russia, September 19–23, 2016, Lect. Notes Comput. Sci. (Springer International, Cham, Switzerland, 2016), Vol. 9869, pp. 547–560.
T. E. Sutto, H. C. de Long, and P. S. Trulove, “Physical properties of substituted imidazolium based ionic liquids gel electrolytes,” Zeitschr. Naturforsch. A 57, 839–846 (2002).
F. P. Vasil’ev, Numerical Methods for Solving Extremal Problems (Fizmatlit, Moscow, 1988) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.P. Shary, 2017, published in Izvestiya Akademii Nauk, Teoriya i Sistemy Upravleniya, 2017, No. 6, pp. 3–19.
Rights and permissions
About this article
Cite this article
Shary, S.P. Maximum compatibility method for data fitting under interval uncertainty. J. Comput. Syst. Sci. Int. 56, 897–913 (2017). https://doi.org/10.1134/S1064230717050100
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064230717050100