Abstract
The work is devoted to application of global optimization in data fitting problem under interval uncertainty. Parameters of the linear function that best fits intervally defined data are taken as the maximum point for a special (“recognizing”) functional which is shown to characterize consistency between the data and parameters. The new data fitting technique is therefore called “maximum consistency method”. We investigate properties of the recognizing functional and present interpretation of the parameter estimates produced by the maximum consistency method.
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References
Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)
Beeck, H.: Über die Struktur und Abschätzungen der Lösungsmenge von linearen Gleichungssystemen mit Intervallkoeffizienten. Computing 10, 231–244 (1972)
Draper, N.R., Smith, H.: Applied Regression Analysis, 3rd edn. Wiley, New York (1998)
Fiedler, M., Nedoma, J., Ramik, J., Rohn, J., Zimmerman, M.: Linear Optimization Problems with Inexact Data. Springer, Berlin (2006)
Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis, with Examples in Parameter and State Estimation. Robust Control and Robotics. Springer, London (2001)
Kantorovich, L. V.: On some new approaches to numerical methods and processing observation data. Sib. Math. J. 3(5), 701–709 (1962) (in Russian). Electronic version is accessible at http://www.nsc.ru/interval/Introduction/Kantorovich62.pdf
Kearfott, R.B., Nakao, M., Neumaier, A., Rump, S., Shary, S.P., van Hentenryck, P.: Standardized notation in interval analysis. Comput. Technol. 15(1), 7–13 (2010)
Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1996)
Lakeyev, A.V., Noskov, S.I.: A description of the set of solutions of a linear equation with intervally defined operator and right-hand side. Russian Acad. Sci. Dokl. Math. 47(3), 518–523 (1993)
Milanese, M., Norton, J., Piet-Lahanier, H., Walter, E. (eds.): Bounding Approaches to System Identification. Plenum Press, New York (1996)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)
Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)
Nurminski, E.A.: Separating plane algorithms for convex optimization. Math. Progr. 76(3), 373–391 (1997)
Rao, C.R., Toutenburg, H., Shalabh, Heumann, C.: Linear Models and Generalizations. Least Squares and Alternatives. Springer, New York (2008)
Remez, E.Ya.: General computational methods of Chebyshev approximation: The problems with linear real parameters. Oak Ridge, U.S. Atomic Energy Commission. Translation 4491 (1962)
Schweppe, F.C.: Recursive state estimation: unknown but bounded errors and system inputs. IEEE Trans. Autom. Control 13(1), 22–28 (1968)
Shary, S.P.: Solvability of interval linear equations and data analysis under uncertainty. Autom. Remote Control 73(2), 310–322 (2012). doi:10.1134/S0005117912020099
Shary, S.P.: Finite-dimensional interval analysis. Institute of Computational Technologies SB RAS, Novosibirsk (2013). Electronic book accessible at http://www.nsc.ru/interval/Library/InteBooks
Shary, S.P., Sharaya, I.A.: Recognition of solvability of interval equations and its application to data analysis. Comput. Technol. 18(3), 80–109 (2013) (in Russian)
Shary, S.P., Sharaya, I.A.: On solvability recognition for interval linear systems of equations. Optim. Lett. (2015). Prepublished on May 6, 2015. doi: 10.1007/s11590-015-0891-6
Shor, N.Z.: Nondifferentiable Optimization and Polynomial Problems. Kluwer, Boston (1998)
Shor, N.Z., Stetsyuk, P.I.: Modified \(r\)-algorithm to find the global minimum of polynomial functions. Cybern. Syst. Anal. 33(4), 482–497 (1997)
Strekalovsky, A.S.: On the minimization of the difference of convex functions on a feasible set. Comput. Math. Math. Physics 43(3), 380–390 (2003)
Vorontsova, E.A.: A projective separating plane method with additional clipping for non-smooth optimization. WSEAS Trans. Math. 13, 115–121 (2014)
Zhilin, S.I.: On fitting empirical data under interval error. Reliab. Comput. 11(5), 433–442 (2005)
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Shary, S.P. Maximum consistency method for data fitting under interval uncertainty. J Glob Optim 66, 111–126 (2016). https://doi.org/10.1007/s10898-015-0340-1
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DOI: https://doi.org/10.1007/s10898-015-0340-1