×

Arithmetic and distance-based approach to the statistical analysis of imprecisely valued data. (English) Zbl 1348.62098

Borgelt, Christian (ed.) et al., Towards advanced data analysis by combining soft computing and statistics. Berlin: Springer (ISBN 978-3-642-30277-0/hbk; 978-3-642-30278-7/ebook). Studies in Fuzziness and Soft Computing 285, 1-18 (2013).
Summary: Most of the research developed in the last years by the SMIRE Research Group concerns the statistical analysis of imprecisely (set- and fuzzy set)-valued experimental data. This analysis has been based on an approach considering the usual arithmetic for these data as well as suitable metrics between them. The research perfectly fits into the research directions of the COST Action IC0702, which has been particularly helpful for scientific activities, discussions and exchanges associated with group members. The main objective of this paper is to summarize some of the main recent advances of the SMIRE Research Group.
For the entire collection see [Zbl 1254.68014].

MSC:

62F86 Parametric inference and fuzziness
62J86 Fuzziness, and linear inference and regression
62-02 Research exposition (monographs, survey articles) pertaining to statistics

Software:

SAFD
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aumann, R., Integrals of set-valued functions, J. Math. Anal. Appl., 12, 1-12 (1965) · Zbl 0163.06301 · doi:10.1016/0022-247X(65)90049-1
[2] Bertoluzza, C.; Corral, N.; Salas, A., On a new class of distances between fuzzy numbers, Mathware Soft. Comp., 2, 71-84 (1995) · Zbl 0887.04003
[3] Borgelt, C.; González-Rodríguez, G.; Trutschnig, W.; Lubiano, M. A.; Gil, M. A.; Grzegorzewski, P.; Hryniewicz, O., Combining Soft Computing and Statistical Methods in Data Analysis (2010), Heidelberg: Springer, Heidelberg · Zbl 1201.62003
[4] Borgelt, C.; Gil, M. A.; Sousa, J. M.C.; Verleysen, M., Towards Advanced Data Analysis by Combining Soft Computing and Statistics (2012), Heidelberg: Springer, Heidelberg · Zbl 1254.68014
[5] Blanco-Fernández, A., Casals, M.R., Colubi, A., Corral, N., García-Bárzana, M., Gil, M.A., González-Rodríguez, G., López, M.T., Lubiano, M.A., Montenegro, M., Ramos-Guajardo, A.B., de la Rosa de Sáa, S., Sinova, B.: Random fuzzy sets: a mathematical tool to develop statistical fuzzy data analysis. Iran J. Fuzzy Syst. (in press, 2012) · Zbl 1331.62055
[6] Blanco-Fernández, A.; Colubi, A.; González-Rodríguez, G., Confidence sets in a linear regression model for interval data, J. Statist. Planning Infer., 142, 6, 1320-1329 (2012) · Zbl 1242.62072 · doi:10.1016/j.jspi.2011.09.017
[7] Blanco-Fernández, A., Colubi, A., González-Rodríguez, G.: Linear regression analysis for interval data based on set arithmetic: a review. In: [4], pp. 19-32 (2012) · Zbl 1348.62191
[8] Blanco-Fernández, A.; Corral, N.; González-Rodríguez, G., Estimation of a flexible simple linear model for interval data based on set arithmetic, Comp. Stat. Data Anal., 55, 2568-2578 (2011) · Zbl 1464.62030 · doi:10.1016/j.csda.2011.03.005
[9] Blanco-Fernández, A., Corral, N., González-Rodríguez, G., Palacio, A.: On some confidence regions to estimate a linear regression model for interval data. In: [3], pp. 33-40 (2010)
[10] Colubi, A.; Domínguez-Menchero, J. S.; López-Díaz, M.; Ralescu, D. A., On the formalization of fuzzy random variables, Inform. Sci., 133, 3-6 (2001) · Zbl 0988.28008 · doi:10.1016/S0020-0255(01)00073-1
[11] Colubi, A.; Domínguez-Menchero, J. S.; López-Díaz, M.; Ralescu, D. A., A D_E[0,1] representation of random upper semicontinuous functions, Proc. Am. Math. Soc., 130, 3237-3242 (2002) · Zbl 1005.28003 · doi:10.1090/S0002-9939-02-06429-8
[12] Colubi, A.; Fernández, C.; Gil, M. A., Simulation of random fuzzy variables: an empirical approach to statistical/probabilistic studies with fuzzy experimental data, IEEE Trans. Fuzzy Syst., 10, 384-390 (2002) · doi:10.1109/TFUZZ.2002.1006441
[13] Colubi, A.; González-Rodríguez, G.; Gil, M. A.; Trutschnig, W., Nonparametric criteria for supervised classification of fuzzy data, Int. J. Approx. Reas., 52, 1272-1282 (2011) · Zbl 1319.62125 · doi:10.1016/j.ijar.2011.05.007
[14] Colubi, A., González-Rodríguez, G., Trutschnig, W.: Discriminant analysis for fuzzy random variables based on nonparametric regression. In: Abst IFSA/EUSFLAT Conf. (2009) · Zbl 1170.68045
[15] Coppi, R.; D’Urso, P.; Giordani, P., A fuzzy clustering model for multivariate spatial time series, J. Classif., 27, 54-88 (2010) · Zbl 1337.62305 · doi:10.1007/s00357-010-9043-y
[16] Coppi, R.; D’Urso, P.; Giordani, P., Fuzzy and possibilistic clustering for fuzzy data, Comp. Stat. Data Anal., 56, 4, 915-927 (2012) · Zbl 1243.62089 · doi:10.1016/j.csda.2010.09.013
[17] Coppi, R.; D’Urso, P.; Giordani, P.; Santoro, A., Least squares estimation of a linear regression model with LR fuzzy response, Comp. Stat. Data Anal., 51, 267-286 (2006) · Zbl 1157.62460 · doi:10.1016/j.csda.2006.04.036
[18] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, Fuzzy Sets Syst., 100, 63-71 (1999) · doi:10.1016/S0165-0114(99)80007-4
[19] Dubois, D.; Prade, H., Operations on fuzzy numbers, Int. J. of Systems Science, 9, 6, 613-626 (1978) · Zbl 0383.94045 · doi:10.1080/00207727808941724
[20] Dubois, D.; Prade, H., Gradualness, uncertainty and bipolarity: making sense of fuzzy sets, Fuzzy Sets Syst., 192, 3-24 (2012) · Zbl 1238.03044 · doi:10.1016/j.fss.2010.11.007
[21] D’Urso, P.; De Giovanni, L., Midpoint radius self-organizing maps for interval-valued data with telecommunications application, Appl. Soft Comp., 11, 3877-3886 (2011) · doi:10.1016/j.asoc.2011.01.006
[22] 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) · doi:10.1016/j.chemolab.2003.11.005
[23] D’Urso, P.; Maharaj, E. A., Autocorrelation-based fuzzy clustering of time series, Fuzzy Sets Syst., 160, 3565-3589 (2009) · doi:10.1016/j.fss.2009.04.013
[24] D’Urso, P.; Maharaj, E. A., Wavelets-based clustering of multivariate time series, Fuzzy Sets and Systems, 193, 33-61 (2012) · Zbl 1237.62079 · doi:10.1016/j.fss.2011.10.002
[25] D’Urso, P.; Massari, R.; Santoro, A., A class of fuzzy clusterwise regression models, Inform. Sci., 180, 4737-4762 (2010) · Zbl 1204.62112 · doi:10.1016/j.ins.2010.08.018
[26] D’Urso, P.; Massari, R.; Santoro, A., Robust fuzzy regression analysis, Inform. Sci., 181, 4154-4174 (2011) · Zbl 1242.62073 · doi:10.1016/j.ins.2011.04.031
[27] Ferraro, M.B., Colubi, A., Giordani, P.: A linearity test for a simple regression model with LR fuzzy response. In: [3], pp. 251-258 (2010)
[28] Ferraro, M. B.; Colubi, A.; González-Rodríguez, G.; Coppi, R., A determination coefficient for a linear regression model with imprecise response, Environmetrics, 22, 516-529 (2011) · doi:10.1002/env.1056
[29] Ferraro, M. B.; Coppi, R.; González-Rodríguez, G.; Colubi, A., A linear regression model for imprecise response, Int. J. Approx. Reas., 51, 759-770 (2010) · Zbl 1201.62086 · doi:10.1016/j.ijar.2010.04.003
[30] Ferraro, M.B., Giordani, P.: A multiple linear regression model for imprecise information. Metrika (in press, 2012), doi:10.1007/s00184-011-0367-3 · Zbl 1254.62079
[31] García-Bárzana, M., Colubi, A., Kontoghiorghes, E.: Least-squares estimation of a multiple regression model for interval data. In: Abst. 3rd Workshop ERCIM 2010 (2010) · Zbl 1337.62168
[32] García-Bárzana, M., Colubi, A., Kontoghiorghes, E.: A flexible multiple linear regression model for interval data. In: Abst. 4th Workshop ERCIM 2011 (2011) · Zbl 1348.62195
[33] Gil, M. A.; Lubiano, M. A.; Montenegro, M.; López-García, 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 · doi:10.1007/s001840100160
[34] Gil, M. A.; Montenegro, M.; González-Rodríguez, G.; Colubi, A.; Casals, M. R., Bootstrap approach to the multi-sample test of means with imprecise data, Comp. Stat. Data Anal., 51, 148-162 (2006) · Zbl 1157.62391 · doi:10.1016/j.csda.2006.04.018
[35] González-Rodríguez, G.; Blanco, A.; Colubi, A.; Lubiano, M. A., Estimation of a simple linear regression model for fuzzy random variables, Fuzzy Sets Syst., 160, 3, 357-370 (2009) · Zbl 1175.62073 · doi:10.1016/j.fss.2008.07.007
[36] González-Rodríguez, G.; Colubi, A.; D’Urso, P.; Montenegro, M., Multi-sample test-based clustering for fuzzy random variables, Int. J. Approx. Reas., 50, 721-731 (2009) · Zbl 1185.62114 · doi:10.1016/j.ijar.2009.01.003
[37] González-Rodríguez, G.; Colubi, A.; Gil, M. A., Fuzzy data treated as functional data. A one-way ANOVA test approach, Comp. Stat. Data Anal., 56, 4, 943-955 (2012) · Zbl 1243.62104 · doi:10.1016/j.csda.2010.06.013
[38] González-Rodríguez, G.; Colubi, A.; Trutschnig, W., Simulation of fuzzy random variables, Inform. Sci., 179, 642-653 (2009) · Zbl 1170.68045 · doi:10.1016/j.ins.2008.10.018
[39] González-Rodríguez, G., Trutschnig, W., Colubi, A.: Confidence regions for the mean of a fuzzy random variable. In: Abst. IFSA/EUSFLAT Conf. (2009) · Zbl 1170.68045
[40] Klement, E. P.; Puri, M. L.; Ralescu, D. A., Limit theorems for fuzzy random variables, Proc. R. Soc. Lond. A., 407, 171-182 (1986) · Zbl 0605.60038 · doi:10.1098/rspa.1986.0091
[41] Körner, R.; Näther, W.; Bertoluzza, C.; Gil, M. A.; Ralescu, D. A., On the variance of random fuzzy variables, Statistical Modeling, Analysis and Management of Fuzzy Data, 22-39 (2002), Heidelberg: Physica-Verlag, Heidelberg · Zbl 0980.00008
[42] Lubiano, M. A.; Gil, M. A.; López-Díaz, M.; López, M. T., The \(\overrightarrow\lambda \)-mean squared dispersion associated with a fuzzy random variable, Fuzzy Sets Syst., 111, 307-317 (2000) · Zbl 0973.60005 · doi:10.1016/S0165-0114(97)00389-8
[43] Lubiano, M.A., Trutschnig, W.: ANOVA for fuzzy random variables using the R-package SAFD. In: [3], pp. 449-456 (2010)
[44] Maharaj, E. A.; D’Urso, P., Fuzzy clustering of time series in the frequency domain, Inform. Sci., 181, 1187-1211 (2011) · Zbl 1215.62061 · doi:10.1016/j.ins.2010.11.031
[45] Maharaj, E. A.; D’Urso, P.; Galagedera, D., Wavelets-based fuzzy clustering of time series, J. Classif., 27, 231-275 (2010) · Zbl 1337.62307 · doi:10.1007/s00357-010-9058-4
[46] Molchanov, I., Theory of Random Sets (2005), London: Springer, London · Zbl 1109.60001
[47] Montenegro, M., López, M.T., Lubiano, M.A., González-Rodríguez, G.: A dependent multi-sample test for fuzzy means. In: Abst. 2nd Workshop ERCIM 2009 (2009)
[48] Nakama, T., Colubi, A., Lubiano, M.A.: Two-way analysis of variance for interval-valued data. In: [3], pp. 475-482 (2010)
[49] Nakama, T., Colubi, A., Lubiano, M.A.: Factorial analysis of variance for fuzzy data. In: Abst. 3rd Workshop ERCIM 2010 (2010)
[50] Puri, M. L.; Ralescu, D. A., Strong Law of Large Numbers for Banach space valued random sets, Ann. Probab., 11, 222-224 (1983) · Zbl 0508.60021 · doi:10.1214/aop/1176993671
[51] Puri, M. L.; Ralescu, D. A., Fuzzy random variables, J. Math. Anal. Appl., 114, 409-422 (1986) · Zbl 0592.60004 · doi:10.1016/0022-247X(86)90093-4
[52] Ramos-Guajardo, A. B.; Colubi, A.; González-Rodríguez, G.; Gil, M. A., One sample tests for a generalized Fréchet variance of a fuzzy random variable, Metrika, 71, 185-202 (2010) · Zbl 1182.62103 · doi:10.1007/s00184-008-0225-0
[53] Ramos-Guajardo, A.B., González-Rodríguez, G., Montenegro, M., López, M.T.: Power analysis of the homoscedasticity test for random fuzzy sets. In: [3], pp. 537-544 (2010)
[54] Ramos-Guajardo, A. B.; Lubiano, M. A., K-sample tests for equality of variances of random fuzzy sets, Comp. Stat. Data Anal., 56, 4, 956-966 (2012) · Zbl 1243.62024 · doi:10.1016/j.csda.2010.11.025
[55] Sinova, B., Casals, M.R., Colubi, A., Gil, M.A.: The median of a random interval. In: [3], pp. 575-583 (2010)
[56] Sinova, B., de la Rosa de Sáa, S., Gil, M.A.: A generalized L^1-type metric between fuzzy numbers for an approach to central tendency of fuzzy data (submitted, 2012) · Zbl 1322.62186
[57] Sinova, B., Gil, M.A., Colubi, A., Van Aelst, S.: The median of a random fuzzy number. The 1-norm distance approach. Fuzzy Sets Syst. (in press, 2012), doi:10.1016/j.fss.2011.11.004 · Zbl 1260.60011
[58] Sinova, B., Van Aelst, S.: Comparing the medians of a random interval defined by means of two different L^1 metrics. In: [4] (to appear, 2012) · Zbl 1348.62164
[59] Trutschnig, W., Characterization of the sendograph-convergence of fuzzy sets by means of their L_p- and levelwise convergence, Fuzzy Sets Syst., 161, 1064-1077 (2010) · Zbl 1196.46059 · doi:10.1016/j.fss.2009.07.005
[60] 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 · doi:10.1016/j.ins.2009.06.023
[61] Trutschnig, W., Lubiano, M.A., Lastra, J.: SAFD — An R package for Statistical Analysis of Fuzzy Data. In: [4], pp. 107-118 (2012) · Zbl 1348.62007
[62] Vitale, R. A., Metrics for compact, convex sets, J. Approx. Theo., 45, 280-287 (1985) · Zbl 0595.52005 · doi:10.1016/0021-9045(85)90051-6
[63] Yang, M.-S.; Ko, C.-H., On a class of fuzzy c-numbers clustering procedures for fuzzy data, Fuzzy Sets Syst., 84, 49-60 (1996) · Zbl 0906.68136 · doi:10.1016/0165-0114(95)00308-8
[64] Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Part 1. Inform. Sci. 8, 199-249, Part 2. Inform. Sci. 8, 301-353, Part 3. Inform. Sci. 9, 43-80 (1975) · Zbl 0404.68075
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.