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Fuzzy data treated as functional data: A one-way ANOVA test approach

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  • González-Rodríguez, Gil
  • Colubi, Ana
  • Gil, María Ángeles

Abstract

The use of the fuzzy scale of measurement to describe an important number of observations from real-life attributes or variables is first explored. In contrast to other well-known scales (like nominal or ordinal), a wide class of statistical measures and techniques can be properly applied to analyze fuzzy data. This fact is connected with the possibility of identifying the scale with a special subset of a functional Hilbert space. The identification can be used to develop methods for the statistical analysis of fuzzy data by considering techniques in functional data analysis and vice versa. In this respect, an approach to the FANOVA test is presented and analyzed, and it is later particularized to deal with fuzzy data. The proposed approaches are illustrated by means of a real-life case study.

Suggested Citation

  • González-Rodríguez, Gil & Colubi, Ana & Gil, María Ángeles, 2012. "Fuzzy data treated as functional data: A one-way ANOVA test approach," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 943-955.
    • Handle: RePEc:eee:csdana:v:56:y:2012:i:4:p:943-955
    DOI: 10.1016/j.csda.2010.06.013
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    References listed on IDEAS

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    5. Gil, Maria Angeles & Montenegro, Manuel & Gonzalez-Rodriguez, Gil & Colubi, Ana & Rosa Casals, Maria, 2006. "Bootstrap approach to the multi-sample test of means with imprecise data," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 148-162, November.
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    8. Manteiga, Wenceslao Gonzalez & Vieu, Philippe, 2007. "Statistics for Functional Data," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4788-4792, June.
    9. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2004. "An anova test for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 111-122, August.
    10. Colubi, Ana & Gonzalez-Rodriguez, Gil, 2007. "Triangular fuzzification of random variables and power of distribution tests: Empirical discussion," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4742-4750, May.
    11. Cao, Jiguo & Ramsay, James O., 2009. "Generalized profiling estimation for global and adaptive penalized spline smoothing," Computational Statistics & Data Analysis, Elsevier, vol. 53(7), pages 2550-2562, May.
    12. Gonzalez-Rodriguez, Gil & Colubi, Ana & Angeles Gil, Maria, 2006. "A fuzzy representation of random variables: An operational tool in exploratory analysis and hypothesis testing," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 163-176, November.
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    Cited by:

    1. Colubi, Ana & Ramos-Guajardo, Ana Belén, 2023. "Fuzzy sets and (fuzzy) random sets in Econometrics and Statistics," Econometrics and Statistics, Elsevier, vol. 26(C), pages 84-98.
    2. Ramos-Guajardo, Ana Belén & Lubiano, María Asunción, 2012. "K-sample tests for equality of variances of random fuzzy sets," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 956-966.
    3. Sinova, Beatriz & Van Aelst, Stefan, 2015. "On the consistency of a spatial-type interval-valued median for random intervals," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 130-136.
    4. Nagy, Stanislav, 2017. "Integrated depth for measurable functions and sets," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 165-170.
    5. Pierpaolo D’Urso & María Ángeles Gil, 2017. "Fuzzy data analysis and classification," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 11(4), pages 645-657, December.
    6. Gholamreza Hesamian, 2016. "One-way ANOVA based on interval information," International Journal of Systems Science, Taylor & Francis Journals, vol. 47(11), pages 2682-2690, August.
    7. Gasser, Patrick, 2020. "A review on energy security indices to compare country performances," Energy Policy, Elsevier, vol. 139(C).
    8. Ana Belén Ramos-Guajardo, 2022. "A hierarchical clustering method for random intervals based on a similarity measure," Computational Statistics, Springer, vol. 37(1), pages 229-261, March.
    9. A. Blanco-Fernández & A. Ramos-Guajardo & A. Colubi, 2013. "Fuzzy representations of real-valued random variables: applications to exploratory and inferential studies," METRON, Springer;Sapienza Università di Roma, vol. 71(3), pages 245-259, November.
    10. María Casals & Norberto Corral & María Gil & María López & María Lubiano & Manuel Montenegro & Gloria Naval & Antonia Salas, 2013. "Bertoluzza et al.’s metric as a basis for analyzing fuzzy data," METRON, Springer;Sapienza Università di Roma, vol. 71(3), pages 307-322, November.
    11. González-Rodríguez, Gil & Colubi, Ana, 2017. "On the consistency of bootstrap methods in separable Hilbert spaces," Econometrics and Statistics, Elsevier, vol. 1(C), pages 118-127.
    12. Lubiano, María Asunción & Montenegro, Manuel & Sinova, Beatriz & de la Rosa de Sáa, Sara & Gil, María Ángeles, 2016. "Hypothesis testing for means in connection with fuzzy rating scale-based data: algorithms and applications," European Journal of Operational Research, Elsevier, vol. 251(3), pages 918-929.
    13. Sara de la Rosa de Sáa & María Asunción Lubiano & Beatriz Sinova & Peter Filzmoser, 2017. "Robust scale estimators for fuzzy data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 11(4), pages 731-758, December.
    14. Coletti, Giulianella & Gervasi, Osvaldo & Tasso, Sergio & Vantaggi, Barbara, 2012. "Generalized Bayesian inference in a fuzzy context: From theory to a virtual reality application," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 967-980.
    15. Choirat, Christine & Seri, Raffaello, 2014. "Bootstrap confidence sets for the Aumann mean of a random closed set," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 803-817.
    16. Miriam Alonso de la Fuente & Pedro Terán, 2020. "Harmonizing two approaches to fuzzy random variables," Fuzzy Optimization and Decision Making, Springer, vol. 19(2), pages 177-189, June.

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