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Component Models for Fuzzy Data

Published online by Cambridge University Press:  01 January 2025

Renato Coppi ,
Paolo Giordani  and
Pierpaolo D’Urso
Renato Coppi
Affiliation:
Università degli Studi di Roma “La Sapienza”
Paolo Giordani*
Affiliation:
Università degli Studi di Roma “La Sapienza”
Pierpaolo D’Urso
Affiliation:
Università degli Studi del Molise
*
Requests for reprints should be sent to Paolo Giordani, Dipartimento di Statistica, Probabilità e Statistiche applicate, Università degli Studi di Roma ‘La Sapienza,’ P.le A. Moro, 5-00185 Roma, Italy. E-mail: paolo.giordani@uniroma1.it
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Abstract

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The fuzzy perspective in statistical analysis is first illustrated with reference to the “Informational Paradigm" allowing us to deal with different types of uncertainties related to the various informational ingredients (data, model, assumptions). The fuzzy empirical data are then introduced, referring to J LR fuzzy variables as observed on I observation units. Each observation is characterized by its center and its left and right spreads (LR1 fuzzy number) or by its left and right “centers" and its left and right spreads (LR2 fuzzy number). Two types of component models for LR1 and LR2 fuzzy data are proposed. The estimation of the parameters of these models is based on a Least Squares approach, exploiting an appropriately introduced distance measure for fuzzy data. A simulation study is carried out in order to assess the efficacy of the suggested models as compared with traditional Principal Component Analysis on the centers and with existing methods for fuzzy and interval valued data. An application to real fuzzy data is finally performed.

Keywords

informational paradigmfuzzy datadistance measure for fuzzy dataprincipal component analysisleast squares approach

Information

Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 4 , December 2006 , pp. 733 - 761
Copyright
Copyright © 2007 The Psychometric Society

Footnotes

We would like to express our gratitude to the Editor, the Associate Editor, and the Referees whose comments and suggestions improved significantly the quality of the paper.

References

Bezdek, J.C. (1993). Fuzzy models—What are they, and why?. IEEE Transactions on Fuzzy Systems, 1, 1–6.CrossRefGoogle Scholar
Bock, H.H., Diday, E. (2000). Analysis of symbolic data: Exploratory methods for extracting statistical information from complex data, Heidelberg: Springer Verlag.CrossRefGoogle Scholar
Boulard, H., Kamp, Y. (1988). Auto-association by multilayer perceptrons and singular value decomposition. Biological Cybernetics, 59, 291–294.CrossRefGoogle Scholar
Cazes, P. (2002). Analyse factorielle d’un tableau de lois de probabilité. Revue de Statistique Appliqué, 50, 5–24.Google Scholar
Cazes, P., Chouakria, A., Diday, E., Schektman, Y. (1997). Extension de l’analyse en composantes principales à des donnés de type intervalle. Revue de Statistique Appliqué, 45, 5–24.Google Scholar
Chouakria, A. (1998). Extension des méthodes d’analyse factorielle à des donnés de type intervalle. Unpublished doctoral dissertation. University of Paris 9 Dauphine.Google Scholar
Chouakria, A., Diday, E., Cazes, P. (1998). Vertices principal component analysis with an improved factorial representation. In Rizzi, A., Vichi, M., Bock, H.H. (Eds.), Advances in data science and classification (pp. 397–402). Barlin: Springer-Verlag.CrossRefGoogle Scholar
Colubi, A., Fernández-García, C., Gil, M.A. (2002). Simulation of random fuzzy variables: An empirical approach to statistical/probabilistic studies with fuzzy experimental data. IEEE Transactions on Fuzzy Systems, 10, 384–390.CrossRefGoogle Scholar
Coppi, R. (2002). A theoretical framework for data mining: The “Informational Paradigm. Computational Statistics and Data Analysis, 38, 501–515.CrossRefGoogle Scholar
Denœux, T., Masson, M. (2004). Principal component analysis of fuzzy data using autoassociative neural networks. IEEE Transactions on Fuzzy Systems, 12, 336–349.CrossRefGoogle Scholar
Dubois, D., Prade, H. (1988). Possibility theory, New-York: Plenum Press.Google Scholar
D’Urso, P., Giordani, P. (2005). A possibilistic approach to latent component analysis for symmetric fuzzy data. Fuzzy Sets and Systems, 150, 285–305.CrossRefGoogle Scholar
Giordani, P., Kiers, H.A.L. (2004). Principal component analysis of symmetric fuzzy data. Computational Statistics and Data Analysis, 45, 519–548.CrossRefGoogle Scholar
Gower, J.C. (1975). Generalized Procrustes analysis. Psychometrika, 40, 33–51.CrossRefGoogle Scholar
Helton, J.C., Johnson, J.D., Oberkampf, W.L. (2004). An exploration of alternative approaches to the representation of uncertainty in model predictions. Reliability Engineering and System Safety, 85, 39–71.CrossRefGoogle Scholar
Herrera, F., Herrera-Viedma, E., Verdegay, J.L. (1997). Linguistic measures based on fuzzy coincidence for researching consensus in group decision making. International Journal of Approximate Reasoning, 16, 309–334.CrossRefGoogle Scholar
Huskisson, E.C. (1983). Visual analogue scales. In Melzack, R. (Eds.), Pain measurement and assessment (pp. 33–37). New York: Raven.Google Scholar
Jaulin, L., Kieffer, M., Didrit, O., Walter, E. (2001). Applied interval analysis, New York: Springer-Verlag.CrossRefGoogle Scholar
Kiers, H.A.L. (2004). Bootstrap confidence intervals for three-way methods. Journal of Chemometrics, 18, 22–36.CrossRefGoogle Scholar
Kiers, H.A.L., Van Mechelen, I. (2001). Three-way component analysis: Principles and illustrative application. Psychological Methods, 6, 84–110.CrossRefGoogle ScholarPubMed
Kroonenberg, P.M. (1983). Three-mode principal component analysis: Theory and applications, Leiden: DSWO Press.Google Scholar
Kroonenberg, P.M. (1994). The TUCKALS line—A suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.CrossRefGoogle Scholar
Lauro, C., Palumbo, F. (2000). Principal component analysis of interval data: A symbolic data analysis approach. Computational Statistics, 15, 73–87.CrossRefGoogle Scholar
Little, R.J.A., Rubin, D.B. (2002). Statistical analysis with missing data, New York: Wiley.CrossRefGoogle Scholar
Montenegro, M., Colubi, A., Casals, M.R., Gil, M.A. (2004). Asymptotic and bootstrap techniques for testing the expected value of a fuzzy random variable. Metrika, 59, 31–49.CrossRefGoogle Scholar
Näther, W., Körner, R. (2002). Linear regression with random fuzzy variables. In Bertoluzza, C., Gil, M.A., Ralescu, D.A. (Eds.), Statistical modeling, analysis and management of fuzzy data (pp. 282–305). Heidelberg: Physica-Verlag.CrossRefGoogle Scholar
Palumbo, F., Lauro, C. (2003). A PCA for interval-valued data based on midpoints and radii. In Yanai, H., Okada, A., Shigemasu, K., Kano, Y., Meulman, J. (Eds.), New developments in psychometrics (pp. 641–648). Tokyo: Springer Verlag.CrossRefGoogle Scholar
Puri, M.L., Ralescu, D.A. (1986). Fuzzy random variables. Journal of Mathematical Analysis and Application, 114, 409–422.CrossRefGoogle Scholar
Rodriguez Rojas, O. (2000). Classification et modèles linéaires de l’analyse des donnés symboliques. Unpublished doctoral dissertation. University of Paris 9 Dauphine.Google Scholar
Ruspini, E.H., Bonissone, P., Pedrycz, W. (1998). Handbook of fuzzy computation, Bristol: Institute of Physics.CrossRefGoogle Scholar
Sii, H.S., Ruxton, T., Wang, J. (2001). A fuzzy-logic-based approach to qualitative safety modelling for marine systems. Reliability Engineering and System Safety, 73, 19–34.CrossRefGoogle Scholar
Tanaka, H., Guo, P. (1999). Possibilistic data analysis for operations research, Heidelberg: Physica-Verlag.Google Scholar
Tang Ahanda, B. (1998). Extension de méthodes d’analyse factorielle sur des donnés symboliques. Unpublished doctoral dissertation. University of Paris 9 Dauphine.Google Scholar
Timmerman, M.E., Kiers, H.A.L. (2002). Three-way component analysis with smoothness constraints. Computational Statistics and Data Analysis, 40, 447–470.CrossRefGoogle Scholar
Timmerman, M.E., Kiers, H.A.L., & Smilde, A.K. (in press). Estimating confidence intervals in principal component analysis: A comparison between the bootstrap and asymptotic results. British Journal of Mathematical and Statistical Psychology.Google Scholar
Yang, M.S., Ko, C.H. (1996). On a class of c-numbers clustering procedures for fuzzy data. Fuzzy Sets and Systems, 84, 49–60.CrossRefGoogle Scholar
Zadeh, L. A. (1965). Fuzzy sets. Information Control, 8, 338–353.CrossRefGoogle Scholar
Zadeh, L.A. (1973). Outline of a new approach to the analysis of complex system and decision processes. IEEE Transactions on Systems, Man and Cybernetics, 3, 28–44.CrossRefGoogle Scholar
Zadeh, L.A. (2002). Toward a perception-based theory of probabilistic reasoning with imprecise probabilities. Journal of Statistical Planning and Inference, 105, 233–264.CrossRefGoogle Scholar
Zadeh, L.A. (2005). Toward a generalized theory of uncertainty (GTU)—An outline. Information Sciences, 172, 1–40.CrossRefGoogle Scholar
Zimmermann, H.J. (2001). Fuzzy set theory and its applications, Kluwer Academic: Dordrecht.CrossRefGoogle Scholar