×

Fuzzy data treated as functional data: a one-way ANOVA test approach. (English) Zbl 1243.62104

Summary: The use of a fuzzy scale of measurements 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.

MSC:

62J10 Analysis of variance and covariance (ANOVA)
62G86 Nonparametric inference and fuzziness
62G10 Nonparametric hypothesis testing
62J86 Fuzziness, and linear inference and regression
46N30 Applications of functional analysis in probability theory and statistics
65C60 Computational problems in statistics (MSC2010)

Software:

fda (R)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Allen, I.E.; Seaman, C.A., Likert scales and data analyses, Qual. prog, 40, 64-65, (2007)
[2] Ammar, S.; Wright, R., Applying fuzzy-set theory to performance evaluation, Socio-econ. plan. sci., 34, 285-302, (2000)
[3] Cao, J.; Ramsay, J.O., Generalized profiling estimation for global and adaptive penalized spline smoothing, Comput. statist. data anal., 53, 2550-2562, (2009) · Zbl 1453.62056
[4] Castaing, C.; Valadier, M., ()
[5] Chimka, J.R.; Wolfe, H., History of ordinal variables before 1980, Scientific research and essays, 4, 853-860, (2009)
[6] Colubi, A., Statistical inference about the means of fuzzy random variables: applications to the analysis of fuzzy- and real-valued data, Fuzzy sets and systems, 160, 344-356, (2009) · Zbl 1175.62021
[7] 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
[8] Colubi, A.; Domínguez-Menchero, J.S.; López-Díaz, M.; Ralescu, D.A., A \(D_E [0, 1]\)-representation of random upper semicontinuous functions, Proc. amer. math. soc., 130, 3237-3242, (2002) · Zbl 1005.28003
[9] Colubi, A.; González-Rodríguez, G., Triangular fuzzification of random variables and power of distribution tests: empirical discussion, Comput. statist. data anal., 51, 4742-4750, (2007) · Zbl 1162.62342
[10] Cuevas, A.; Febrero, M.; Fraiman, R., An anova test for functional data, Comput. statist. data anal., 47, 111-122, (2004) · Zbl 1429.62726
[11] Cuevas, A.; Febrero, M.; Fraiman, R., On the use of the bootstrap for estimating functions with functional data, Comput. statist. data anal., 51, 1063-1074, (2006) · Zbl 1157.62390
[12] Fernández, E.; Fernández, M.; Anadón, S.; González-Rodríguez, G.; Colubi, A., Flood analysis: on the automation of the geomorphological-historical method, (), 239-246
[13] Ferraty, F.; Vieu, P., Nonparametric functional data analysis: theory and practice, (2006), Springer-Verlag New York · Zbl 1119.62046
[14] Ferraty, F.; Vieu, P., Additive prediction and boosting for functional data, Comput. statist. data anal., 53, 1400-1413, (2009) · Zbl 1452.62989
[15] 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, Comput. statist. data anal., 51, 148-162, (2006) · Zbl 1157.62391
[16] Giné, E.; Zinn, J., Bootstrapping general empirical measures, Ann. probab., 18, 851-869, (1990) · Zbl 0706.62017
[17] González-Manteiga, W.; Vieu, P., Guest editors of the special issue on statistics for functional data, Comput. statist. data anal., 51, 4788-5008, (2007)
[18] González-Rodríguez, G.; Colubi, A.; Gil, M.A., A fuzzy representation of random variables: an operational tool in exploratory analysis and hypothesis testing, Comput. statist. data anal., 51, 163-176, (2006) · Zbl 1157.62303
[19] González-Rodríguez, G.; Montenegro, M.; Colubi, A.; Gil, M.A., Bootstrap techniques and fuzzy random variables: synergy in hypothesis testing with fuzzy data, Fuzzy sets and systems, 157, 2608-2613, (2006) · Zbl 1119.62037
[20] Körner, R., An asymptotic \(\alpha\)-test for the expectation of random fuzzy variables, J. statist. plann. inference, 83, 331-346, (2000) · Zbl 0976.62013
[21] Körner, R.; Näther, W., On the variance of random fuzzy variables, (), 22-39
[22] Laha, R.G.; Rohatgi, V.K., Probability theory, (1979), Wiley New York · Zbl 0409.60001
[23] Likert, R., A technique for the measurement of attitudes, Arch. psychol., 140, 1-55, (1932)
[24] Lubiano, M.A.; Gil, M.A.; López-Díaz, M.; López-García, M.T., The \(\overset{\vec{}}{\lambda}\)-Mean squared dispersion associated with a fuzzy random variable, Fuzzy sets and systems, 111, 307-317, (2000)
[25] Montenegro, M.; Casals, M.R.; Lubiano, M.A.; Gil, M.A., Two-sample hypothesis tests of means of a fuzzy random variable, Inform. sci, 133, 89-100, (2001) · Zbl 1042.62012
[26] Montenegro, M.; Colubi, A.; Casals, M.R.; Gil, M.A., Asymptotic and bootstrap techniques for testing the expected value of a fuzzy random variable, Metrika, 59, 31-49, (2004) · Zbl 1052.62048
[27] Müller, H.-G., Functional modelling and classification of longitudinal data, Scandinavian J. statist., 32, 223-240, (2005) · Zbl 1089.62072
[28] Müller, H.-G.; Yang, W., Dynamic relations for sparsely sampled Gaussian processes (invited paper with discussions), Test, 19, 1-65, (2010)
[29] Nguyen, H.T., A note on the extension principle for fuzzy sets, J. math. anal. appl., 64, 369-380, (1978) · Zbl 0377.04004
[30] Puri, M.L.; Ralescu, D.A., Differentials of fuzzy functions, J. math. anal. appl., 91, 552-558, (1983) · Zbl 0528.54009
[31] Puri, M.L.; Ralescu, D.A., The concept of normality for fuzzy random variables, Ann. probab., 11, 1373-1379, (1985) · Zbl 0583.60011
[32] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
[33] Ramsay, J.O.; Silverman, B.W., Functional data analysis, (1997), Springer-Verlag New York · Zbl 0882.62002
[34] Ramsay, J.O.; Silverman, B.W., Applied functional data analysis, (2002), Springer-Verlag New York · Zbl 1011.62002
[35] Stevens, S.S., On the theory of scales of measurement, Science, 103, 2884, 677-680, (1946) · Zbl 1226.91050
[36] 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, 3964-3972, (2009) · Zbl 1181.62016
[37] Yao, F.; Müller, H.-G.; Wang, J.-L., Functional linear regression analysis for longitudinal data, Ann. statist., 33, 2873-2903, (2005) · Zbl 1084.62096
[38] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning, part 1, Inform. sci., 8, 199-249, (1975), Part 2. Inform. Sci. 8, 301-353; Part 3. Inform. Sci. 9, 43-80 · Zbl 0397.68071
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.