Abstract
A linear regression model for interval data based on the natural interval-arithmetic has recently been proposed. Interval data can be identified with 2-dimensional points in ℝ × ℝ + , since they can be parametrized by its mid-point and its semi-amplitude or spread, which is non-negative. The model accounts separately for the contribution of the mid-points and the spreads through a single equation. The least squares estimation becomes a quadratic optimization problem subject to linear constraints, which guarantee the existence of the residuals. Several estimators are discussed. Namely, a closed-form estimator, the restricted least-squares estimator, an empirical estimator and an estimator based on separate models for mids and spreads have been investigated. Real-life examples are considered. Simulations are performed in order to assess the consistency and the bias of the estimators. Results indicate that the numerical and the closed-form estimator are appropriate in most of cases, while the empirical estimator and the one based on separate models are not always suitable.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Billard, L., Diday, E.: From the Statistics of data to the Statistics of knowledge: Symbolic Data Analysis. J. Amer. Stat. Assoc. 98, 470–487 (2003)
Blanco-Fernández, A., Corral, N., González-Rodríguez, G.: Estimation of a flexible simple linear model for interval data based on set arithmetic. Comp. Stat. Data Anal. 55(9), 2568–2578 (2011)
Diamond, P.: Least squares fitting of compact set-valued data. J. Math. Anal. Appl. 147, 531–544 (1990)
Gil, M.A., Lubiano, A., Montenegro, M., López-García, M.T.: Least squares fitting of an affine function and strength of association for interval-valued data. Metrika 56, 97–111 (2002)
González-Rodríguez, G., Blanco, A., Corral, N., Colubi, A.: Least squares estimation of linear regression models for convex compact random sets. Adv. D. Anal. Class 1, 67–81 (2007)
Neto, E.d.A.L., de Carvalho, F.d.A.T., Freire, E.S.: Applying Constrained Linear Regression Models to Predict Interval-Valued Data. In: Furbach, U. (ed.) KI 2005. LNCS (LNAI), vol. 3698, pp. 92–106. Springer, Heidelberg (2005)
Lima Neto, E.A., de Carvalho, F.d.A.T.: Constrained linear regression models for symbolic interval-valued variables. Comp. Stat. Data Anal. 54, 333–347 (2010)
Trutschnig, W., González-Rodríguez, G., Colubi, A., Gil, M.A.: A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread. Inform. Sci. 179(23), 3964–3972 (2009)
Lawson, C.L., Hanson, R.J.: Solving least squares problems. Prentice-Hall, Englewood Cliffs (1974); Reprinted with a detailed “new developments” appendix in 1996 by SIAM Publications, Philadelphia
Gil, M.A., López, M.T., Lubiano, M.A., Montenegro, M.: Regression and correlation analyses of a linear relation between random intervals. Test 10(1), 183–201 (2001), doi:10.1007/BF02595831
Gil, M.A., González-Rodríguez, G., Colubi, A., Montenegro, M.: Testing linear independence in linear models with interval-valued data. Computational Statistics and Data Analysis 51(6), 3002–3015 (2007), doi:10.1016/j.csda.2006.01.015
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag GmbH Berlin Heidelberg
About this chapter
Cite this chapter
García-Bárzana, M., Colubi, A., Kontoghiorghes, E.J. (2013). On the Estimation of the Regression Model M for Interval Data. In: Borgelt, C., Gil, M., Sousa, J., Verleysen, M. (eds) Towards Advanced Data Analysis by Combining Soft Computing and Statistics. Studies in Fuzziness and Soft Computing, vol 285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30278-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-30278-7_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30277-0
Online ISBN: 978-3-642-30278-7
eBook Packages: EngineeringEngineering (R0)