A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread

One of the most important aspects of the (statistical) analysis of imprecise data is the usage of a suitable distance on the family of all compact, convex fuzzy sets, which is not too hard to calculate and which reflects the intuitive meaning of fuzzy sets. On the basis of expressing the metric of Bertoluzza et al. [C. Bertoluzza, N. Corral, A. Salas, On a new class of distances between fuzzy numbers, Mathware Soft Comput. 2 (1995) 71-84] in terms of the mid points and spreads of the corresponding intervals we construct new families of metrics on the family of all d-dimensional compact convex sets as well as on the family of all d-dimensional compact convex fuzzy sets. It is shown that these metrics not only fulfill many good properties, but also that they are easy to calculate and easy to manage for statistical purposes, and therefore, useful from the practical point of view.

[1]  David García,et al.  Estimating the expected value of fuzzy random variables in the stratified random sampling from finite populations , 2001, Inf. Sci..

[2]  Edwin Hewitt,et al.  Real And Abstract Analysis , 1967 .

[3]  Bernd Möller,et al.  Safety assessment of structures in view of fuzzy randomness , 2003 .

[4]  Volker Krätschmer,et al.  Some complete metrics on spaces of fuzzy subsets , 2002, Fuzzy Sets Syst..

[5]  I. Bárány,et al.  convex sets , 2007 .

[6]  Lotfi A. Zadeh,et al.  Is there a need for fuzzy logic? , 2008, NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.

[7]  Ana Colubi,et al.  Statistical inference about the means of fuzzy random variables: Applications to the analysis of fuzzy- and real-valued data , 2009, Fuzzy Sets Syst..

[8]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[9]  Ana Colubi,et al.  Simulation of fuzzy random variables , 2009, Inf. Sci..

[10]  Ana Colubi,et al.  Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data , 2006, Fuzzy Sets Syst..

[11]  Carlo Bertoluzza,et al.  On a new class of distances between fuzzy numbers , 1995 .

[12]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[13]  P. Kloeden,et al.  Metric Spaces Of Fuzzy Sets Theory And Applications , 1975 .

[14]  M. Gil,et al.  Least squares fitting of an affine function and strength of association for interval-valued data , 2002 .

[15]  María Asunción Lubiano,et al.  Two-sample hypothesis tests of means of a fuzzy random variable , 2001, Inf. Sci..

[16]  Lotfi A. Zadeh,et al.  Toward a generalized theory of uncertainty (GTU)--an outline , 2005, Inf. Sci..

[17]  María Angeles Gil,et al.  Fuzzy random variables , 2001, Inf. Sci..

[18]  Volker Krätschmer,et al.  Limit theorems for fuzzy-random variables , 2002, Fuzzy Sets Syst..

[19]  Ana Colubi,et al.  On the formalization of fuzzy random variables , 2001, Inf. Sci..

[20]  Weiyin Fei,et al.  Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients , 2007, Inf. Sci..

[21]  Ana Colubi,et al.  Bootstrap approach to the multi-sample test of means with imprecise data , 2006, Comput. Stat. Data Anal..

[22]  Carlo Bertoluzza,et al.  A generalized real-valued measure of the inequality associated with a fuzzy random variable , 2001, Int. J. Approx. Reason..

[23]  Wolfgang Näther.  Random fuzzy variables of second order and applications to statistical inference , 2001, Inf. Sci..

[24]  Didier Dubois,et al.  Joint propagation of variability and imprecision in assessing the risk of groundwater contamination. , 2007, Journal of contaminant hydrology.

[25]  P. Kloeden,et al.  Metric spaces of fuzzy sets , 1990 .

[26]  R. A. Vitale.  Lp metrics for compact, convex sets , 1985 .