## Least squares estimation of linear regression models for convex compact random sets.(English)Zbl 1131.62058

Summary: 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.

### MSC:

 62J05 Linear regression; mixed models 60D05 Geometric probability and stochastic geometry 65C60 Computational problems in statistics (MSC2010) 65K05 Numerical mathematical programming methods
Full Text:

### References:

  Aumann RJ (1965) Integrals of set-valued functions. J Math Anal Appl 12:1–12 · Zbl 0163.06301  Bertoluzza C, Corral N, Salas A (1995) On a new class of distances between fuzzy numbers. Mathw Soft Comput 2:71–84 · Zbl 0887.04003  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  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 · Zbl 1039.62501  Cressie NAC (1993) Statistics for spatial data. Wiley, New York  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 · Zbl 0704.65006  Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets. World Scientific, Singapore · Zbl 0873.54019  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 · Zbl 0981.62062  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 · Zbl 1433.60004  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) · Zbl 1161.62358  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 · Zbl 0973.60005  Matheron G (1975) Random sets and integral geometry. Wiley, New York · Zbl 0321.60009  Molchanov I (2005) Probability and its applications. Springer, London · Zbl 1094.60500  Stoyan D, Kendall WS, Mecke J (1987) Stochastic geometry and its applications. Wiley, Chichester · Zbl 0622.60019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.