Least squares fitting of an affine function and strength of association for interval-valued data

Alvarez, María Ángeles Gil; Lubiano, María Asunción; Montenegro, Manuel; López, María Teresa

doi:10.1007/s001840100160

Cited by 79 publications

(49 citation statements)

References 18 publications

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“…Many works follow the tradition of fuzzy least squares of Diamond [23,26], that relies on the use of a precise variance of interval or fuzzy set-valued data, and recommends minimizing a scalar mean squared error [41][42][43]. This is a direct adaptation of the classical regression methodology to set-valued data, that tries to fit a fuzzy set-valued linear function on the fuzzy data.…”

Section: Resultsmentioning

confidence: 99%

Random Fuzzy Sets as Ill-Perceived Random Variables

Couso

Dubois

Sánchez

2014

Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables

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Section: Resultsmentioning

confidence: 99%

Random Fuzzy Sets as Ill-Perceived Random Variables

Couso

Dubois

Sánchez

2014

Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables

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“…Bertoluzza et al's L 2 metric can be expressed in terms of the mid/spread representation for fuzzy numbers (see, for instance, Gil et al [50]). In the particular case of interval-valued data one can easily establish necessary and sufficient conditions for this representation to characterize a compact interval (given simply by the non-negativeness of the spread).…”

Section: Functions Satifying Coditions I) and Ii) Then There Exists mentioning

confidence: 99%

A parameterized metric between fuzzy numbers and its parameter interpretation

Sinova

Alvarez

López

et al. 2014

Fuzzy Sets and Systems

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When handling fuzzy number data, it is common practice to make use of a metric to quantify distances between fuzzy numbers. Several metrics have been suggested in the literature for this purpose. When statistically analyzing fuzzy number-valued data, L 2 metrics become especially useful. This paper introduces a new family of generalized L 2 metrics which take into account key features of the involved fuzzy numbers, namely, a measure of central location and two measures associated with the shape of the fuzzy numbers are used. A crucial property related to these three measures is that necessary and sufficient conditions can be established for them to characterize fuzzy numbers. Furthermore, the family of generalized L 2 metrics depends on one parameter. A discussion is provided regarding the interpretation of this parameter which can guide selection of its value in practice. * corresponding author: magil@uniovi.es, Fax: (+34) 985103354. Preprint submitted to Fuzzy Sets and SystemsJune 7, 2013 AbstractWhen handling fuzzy number data, it is common practice to make use of a metric to quantify distances between fuzzy numbers. Several metrics have been suggested in the literature for this purpose. When statistically analyzing fuzzy number-valued data, L 2 metrics become especially useful. This paper introduces a new family of generalized L 2 metrics which take into account key features of the involved fuzzy numbers, namely, a measure of central location and two measures associated with the shape of the fuzzy numbers are used. A crucial property related to these three measures is that necessary and sufficient conditions can be established for them to characterize fuzzy numbers. Furthermore, the family of generalized L 2 metrics depends on one parameter. A discussion is provided regarding the interpretation of this parameter which can guide selection of its value in practice.Keywords: fuzzy number, L 2 -metric, wabl/ldev/rdev representation of fuzzy numbers, L 2 wabl/ldev/rdev metric, weighting parameter

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“…In this paper, we formulate linear regression models for convex compact random sets in R p by following a set arithmetic approach (other approaches are discussed in Gil et al 2006). In fact, we present here a generalization of the models considered in Gil et al (2001Gil et al ( , 2002Gil et al ( , 2006 and González-Rodríguez et al (2006) for the particular case of interval data in R to the more general case of convex compact sets in R p .…”

Section: Introductionmentioning

confidence: 96%

“…This scenario has been generalized, from different points of view, to the case where the variables take intervals as their values (see, e.g., Diamond 1990;Gil et al 2001Gil et al , 2002Gil et al , 2006Billard and Diday 2003;Lima Neto et al 2004;De Carvalho et al 2004;González-Rodríguez et al 2006). In this paper, we formulate linear regression models for convex compact random sets in R p by following a set arithmetic approach (other approaches are discussed in Gil et al 2006).…”

Section: Introductionmentioning

confidence: 98%

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

González‐Rodríguez

Blanco

Corral

et al. 2007

ADAC

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Simple and multiple linear regression models are considered between variables whose "values" are convex compact random sets in 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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Least squares fitting of an affine function and strength of association for interval-valued data

Cited by 79 publications

References 18 publications

Random Fuzzy Sets as Ill-Perceived Random Variables

Random Fuzzy Sets as Ill-Perceived Random Variables

A parameterized metric between fuzzy numbers and its parameter interpretation

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

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