A set arithmetic-based linear regression model for modelling interval-valued responses through real-valued variables. (English) Zbl 1321.62078

Summary: A new linear regression model for an interval-valued response and a real-valued explanatory variable is presented. The approach is based on the interval arithmetic. Comparisons with previous methods are discussed. The new linear model is theoretically analyzed and the regression parameters are estimated. Some properties of the regression estimators are investigated. Finally, the performance of the procedure is illustrated using both a real-life application and simulation studies.


62J05 Linear regression; mixed models
62F10 Point estimation
65C60 Computational problems in statistics (MSC2010)
65G30 Interval and finite arithmetic
Full Text: DOI


[1] Beresteanu, A.; Molinari, F., Asymptotic properties for a class of partially identified models, Econometrica, 76, 763-814, (2008) · Zbl 1274.62136
[2] Bertoluzza, C.; Corral, N.; Salas, A., On a new class of distances between fuzzy numbers, Mathware Soft Comput., 2, 71-84, (1995) · Zbl 0887.04003
[3] Billard, L.; Diday, E., Regression analysis for interval-valued data, (Kiers, H. A.L.; etal., Data Analysis, Classification, and Related Methods, Proc. 7th Conf. Int. Feder. Clas. Soc., (2000), Springer Berlin), 369-374 · Zbl 1026.62073
[4] Blanco-Fernández, A.; Corral, N.; González-Rodríguez, G., Estimation of a flexible simple linear model for interval data based on set arithmetic, Comput. Stat. Data Anal., 55, 9, 2568-2578, (2011) · Zbl 1464.62030
[5] Blanco-Fernández, A.; Colubi, A.; González-Rodríguez, G., Linear regression analysis for interval-valued data based on set arithmetic: a review, (Borgelt, C.; Gil, M. A.; Sousa, J.; Verleysen, M., Towards Advanced Data Analysis by Combining Soft Computing and Statistics, Studies in Fuzziness and Soft Computing, vol. 285, (2012), Springer), 19-81
[6] Blanco-Fernández, A.; Colubi, A.; González-Rodríguez, G., Confidence sets in a linear regression model for interval data, J. Stat. Plan Infer., 142, 6, 1320-1329, (2012) · Zbl 1242.62072
[7] C. Bontemps, T. Magnac, E. Maurin, Set Identified Linear Models, Working Paper, Toulouse School of Economics, 2007.
[8] Boukezzoula, R.; Galichet, S.; Bisserier, A., A revisited approach to linear fuzzy regression using trapezoidal fuzzy intervals, Inform. Sci., 180, 3653-3673, (2010) · Zbl 1205.68417
[9] Ĉerný, M.; Antoch, J.; Hladík, M., On the possibilistic approach to linear regression models involving uncertain, indeterminate or interval data, Inform. Sci., 244, 26-47, (2013) · Zbl 1357.62237
[10] M.G.C.A. Cimino, B. Lazzerini, F. Marcelloni, W. Pedrycz, Genetic interval neural networks for granular data regression, Inform. Sci. (2013). http://dx.doi.org/10.1016/j.ins.2012.12.049.
[11] Chuang, C. C., Extended support vector interval regression networks for interval input-output data, Inform. Sci., 178, 871-891, (2008) · Zbl 1126.68525
[12] Coppi, R.; D’Urso, P.; Giordani, P.; Santoro, A., Least squares estimation of a linear regression model with LR fuzzy response, Comput. Stat. Data Anal., 51, 267-286, (2006) · Zbl 1157.62460
[13] Coppi, R., Management of uncertainty in statistical reasoning: the case of regression analysis, Int. J. Approx. Reason., 47, 284-305, (2008) · Zbl 1184.62005
[14] Diamond, P., Fuzzy least squares, Inform. Sci., 46, 141-157, (1988) · Zbl 0663.65150
[15] Diamond, P., Least squares Fitting of compact set-valued data, J. Math. Anal. Appl., 147, 531-544, (1990)
[16] D’Urso, P., Linear regression analysis for fuzzy/crisp input and fuzzy/crisp output data, Comput. Stat. Data Anal., 42, 47-72, (2003) · Zbl 1429.62337
[17] D’Urso, P.; Giordani, P., A least squares approach to principal component analysis for interval valued data, Chem. Intel. Lab. Syst., 70, 179-192, (2004)
[18] D’Urso, P.; Massari, R.; Santoro, A., Robust fuzzy regression analysis, Inform. Sci., 181, 4154-4174, (2011) · Zbl 1242.62073
[19] Ferraro, M. B.; Coppi, R.; González-Rodríguez, G.; Colubi, A., A linear regression model for imprecise response, Int. J. Approx. Reason., 51, 7, 759-770, (2010) · Zbl 1201.62086
[20] S. Ferson, V. Kreinovich, J. Hajagos, W. Oberkampf, L. Ginzburg, Experimental Uncertainty Estimation and Statistics for Data having Interval Uncertainty, Sandia Nat. Lab. Tech. Rep. SAND2007-0939, Setauket, New York, 2007.
[21] Gil, M. A.; Lubiano, A.; Montenegro, M.; López, M. T., Least squares Fitting of an affine function and strength of association for interval-valued data, Metrika, 56, 97-111, (2002) · Zbl 1433.60004
[22] Gil, M. A.; González-Rodríguez, G.; Colubi, A.; Montenegro, M., Testing linear independence in linear models with interval-valued data, Comput. Stat. Data Anal., 51, 3002-3015, (2007) · Zbl 1161.62358
[23] P. Giordani, Linear Regression Analysis for Interval-valued Data based on the Lasso Technique, Tech Report UniRoma. · Zbl 1414.62305
[24] González-Rodríguez, G.; Colubi, A.; Coppi, R.; Giordani, P., On the estimation of linear models with interval-valued data, (Proc. 17th IASC, (2006), Physica-Verlag Heidelberg)
[25] González-Rodríguez, G.; Blanco, A.; Corral, N.; Colubi, A., Least squares estimation of linear regression models for convex compact random sets, Adv. Data Anal. Classif., 1, 67-81, (2007) · Zbl 1131.62058
[26] Guo, P.; Tanaka, H., Dual models for possibilistic regression analysis, Comput. Stat. Data Anal., 51, 253-266, (2006) · Zbl 1157.62469
[27] Guo, J.; Li, W.; Gao, S., Standardization of interval symbolic data based on the empirical descriptive statistics, Comput. Stat. Data Anal., 56, 602-610, (2012) · Zbl 1239.62003
[28] Ham, J.; Hsiao, C., Two-stage estimation of structural labor supply parameters using interval data from the 1971 Canadian census, J. Econom., 24, 133-158, (1984) · Zbl 0525.62106
[29] Hong, D. H.; Song, J. K.; Young, H., Fuzzy least-squares linear regression analysis using shape preserving operations, Inform. Sci., 138, 185-193, (2001) · Zbl 1135.62356
[30] Huang, C-H., A reduced support vector machine approach for interval regression analysis, Inform. Sci., 217, 5664, (2012)
[31] Huber, C.; Solev, V.; Vonta, F., Interval censored and truncated data: rate of convergence of NPMLE of the density, J. Stat. Plan. Infer., 139, 1734-1749, (2009) · Zbl 1156.62353
[32] Hukuhara, M., Integration des applications measurables dont la valeur est un compact convexe, Funkcial. Ekvac., 10, 205-223, (1967) · Zbl 0161.24701
[33] Körner, R., On the variance of fuzzy random variables, Fuzzy Sets Syst., 92, 83-93, (1997) · Zbl 0936.60017
[34] Kreinovich, V.; Hung, T. N.; Berlin, W., On-line algorithms for computing mean and variance of interval data, and their use in intelligent systems, Inform. Sci., 177, 3228-3238, (2007) · Zbl 05174376
[35] Liew, C. K., Inequality constrained least-squares estimation, J. Am. Stat. Assoc., 71, 746-751, (1976) · Zbl 0342.62037
[36] Lima Neto, E. A.; de Carvalho, F. A.T., Constrained linear regression models for symbolic interval-valued variables, Comput. Stat. Data Anal., 54, 333-347, (2010) · Zbl 1464.62055
[37] Manski, C. F.; Tamer, E., Inference on regressions with interval data on a regressor or outcome, Econometrica, 70, 519-546, (2002) · Zbl 1121.62544
[38] Näther, W., Regression with fuzzy random data, Comput. Stat. Data Anal., 51, 235-252, (2006) · Zbl 1157.62463
[39] Rho, J-W.; Hwang, S-W.; Yi, B-K., Efficient bitmap-based indexing of time-based interval sequences, Inform. Sci., 194, 3856, (2012)
[40] Sinova, B.; Colubi, A.; Gil, M. A.; González-Rodríguez, G., Interval arithmetic-based simple linear regression between interval data: discussion and sensitivity analysis on the choice of the metric, Inform. Sci., 199, 109-124, (2012) · Zbl 06094584
[41] Tanaka, H.; Uejima, S.; Asai, K., Linear regression analysis with fuzzy model, IEEE Trans. SMC-2, 903-907, (1982) · Zbl 0501.90060
[42] Tanaka, H.; Hayashi, I.; Watada, J., Possibilistic linear regression analysis for fuzzy data, Eur. J. Oper. Res., 40, 389-396, (1989) · Zbl 0669.62054
[43] 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) · Zbl 1181.62016
[44] Vitale, R. A., L_{p} metrics for compact, convex sets, J. Approx. Theory, 45, 280-287, (1985) · Zbl 0595.52005
[45] Yamaguchi, D.; Li, G.-D.; Nagai, M., A grey-based rough approximation model for interval data processing, Inform. Sci., 177, 4727-4744, (2007) · Zbl 1126.68613
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