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Least squares estimation of linear regression models for convex compact random sets

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Abstract

Simple and multiple linear regression models are considered between variables whose “values” are convex compact random sets in \({\mathbb{R}^p}\) , (that is, hypercubes, spheres, and so on). We analyze such models within a set-arithmetic approach. Contrary to what happens for random variables, the least squares optimal solutions for the basic affine transformation model do not produce suitable estimates for the linear regression model. First, we derive least squares estimators for the simple linear regression model and examine them from a theoretical perspective. Moreover, the multiple linear regression model is dealt with and a stepwise algorithm is developed in order to find the estimates in this case. The particular problem of the linear regression with interval-valued data is also considered and illustrated by means of a real-life example.

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References

  • Aumann RJ (1965) Integrals of set-valued functions. J Math Anal Appl 12:1–12

    Article  MathSciNet  Google Scholar 

  • Bertoluzza C, Corral N, Salas A (1995) On a new class of distances between fuzzy numbers. Mathw Soft Comput 2:71–84

    MathSciNet  Google Scholar 

  • Billard L, Diday E (2003) From the statistics of data to the statistics of knowledge: symbolic data analysis. J Am Stat Assoc 98:470–487

    Article  MathSciNet  Google Scholar 

  • Bock HH, Diday E (2000) Analysis of symbolic data Exploratory methods for extracting statistical information from complex data Studies in classification, data analysis, and knowledge organization. Springer, Heidelberg

    Google Scholar 

  • Cressie NAC (1993) Statistics for spatial data. Wiley, New York

    Google Scholar 

  • De Carvalho FAT, Lima Neto EA, Tenorio CP (2004) A new method to fit a linear regression model for interval-valued data. In: Biundo S, Frühwirth T, Palm G (eds) Lectures Notes on artificial intelligence no. 3238. Proceedings of the 27th German conference on artificial intelligence (KI 2004, Ulm, Germany). Springer, Heidelberg, 295–306

  • Diamond P (1990) Least squares fitting of compact set-valued data. J Math Anal Appl 14:531–544

    MathSciNet  Google Scholar 

  • Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets. World Scientific, Singapore

    MATH  Google Scholar 

  • Gil MA, López-García MT, Lubiano MA, Montenegro M (2001) Regression and correlation analyses of a linear relation between random intervals. Test 10:183–201

    MathSciNet  Google Scholar 

  • Gil MA, Lubiano MA, Montenegro M, López-García MT (2002) Least squares fitting of an affine function and strength of association for interval-valued data. Metrika 56:97–111

    Article  MathSciNet  Google Scholar 

  • Gil MA, González-Rodrí guez G, Colubi A, Montenegro M (2006) Testing linear independence in linear models with interval-valued data. Comput Stat Data Anal (in press, doi:10.1016/j.csda.2006.01.015)

  • González-Rodríguez G, Colubi A, Coppi R, Giordani P (2006) On the estimation of linear models with interval-valued data. 17th conference of IASC-ERS (COMPSTAT’2006), Physica-Verlag, Heidelberg, pp 697–704

  • Körner R, Näther W (2002) On the variance of random fuzzy variables. In: Bertoluzza C, Gil MA, Ralescu DA (eds) Statistical modeling, analysis and management of fuzzy data. Physica-Verlag, Heidelberg, pp 22–39

  • Lima Neto EA, De Carvalho FAT, Tenorio CP (2004) Univariate and multivariate linear regression methods to predict interval-valued features. In: Webb GI, Xinghuo (eds) Lecture Notes on artificial intelligence no. 3339. Proceedings of 17th Australian joint conference on artificial intelligence (AI 2004, Cairns, Australia). Springer, Heidelberg, pp 526–537

  • Lubiano MA, Gil MA, López-Díaz M, López-García MT (2000) The \({\overrightarrow\lambda}\) -mean squared dispersion associated with a fuzzy random variable. Fuzzy Sets Syst 111:307–317

    Article  Google Scholar 

  • Matheron G (1975) Random sets and integral geometry. Wiley, New York

    MATH  Google Scholar 

  • Molchanov I (2005) Probability and its applications. Springer, London

    Google Scholar 

  • Stoyan D, Kendall WS, Mecke J (1987) Stochastic geometry and its applications. Wiley, Chichester

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dpto. de Estadística e I.O., Universidad de Oviedo, 33007, Oviedo, Spain

    Gil González-Rodríguez, Ángela Blanco, Norberto Corral & Ana Colubi

Authors
  1. Gil González-Rodríguez
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  2. Ángela Blanco
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  3. Norberto Corral
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  4. Ana Colubi
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Correspondence to Gil González-Rodríguez.

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González-Rodríguez, G., Blanco, Á., Corral, N. et al. Least squares estimation of linear regression models for convex compact random sets. ADAC 1, 67–81 (2007). https://doi.org/10.1007/s11634-006-0003-7

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  • DOI: https://doi.org/10.1007/s11634-006-0003-7

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