Abstract
The question of whether two random variables describing two features under study are independent arises in many statistical studies and practical applications. Many parametric and nonparametric tests of bivariate independence can be found in the literature, like the chi-square test for contingency tables, tests based on Pearson’s, Kendall’s, Spearman’s correlation coefficients and so on. The problem of testing independence becomes much more difficult when the available data are imprecise, incomplete or vague. Although fuzzy modeling provide appropriate tools for dealing with uncertain data, some limitations of fuzzy random variables inhibit the straightforward generalization of the classical tests of independence into fuzzy framework. At the same time, this situation imposes the quest for new solutions that may be applied in fuzzy environment. In this contribution we propose a new permutation test of independence for fuzzy data based on the so-called distance covariance.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bakirov, N.K., Rizzo, M.L., Székely, G.J.: A multivariate nonparametric test of independence. J. Multivar. Anal. 97, 1742–1756 (2006)
Ban, A.I., Coroianu, L., Grzegorzewski, P.: Fuzzy Numbers: Approximations. Ranking and Applications. Polish Academy of Sciences, Warsaw (2015)
Blanco-Fernández, A., et al.: A distance-based statistic analysis of fuzzy number-valued data (with Rejoinder). Int. J. Approximate Reason. 55, 1487–1501, 1601–1605 (2014)
Chukhrova, N., Johannssen, A.: Fuzzy regression analysis: systematic review and bibliography. Appl. Soft Comput. 84, 105708 (2019)
Gibbons, J.D., Chakraborti, S.: Nonparametric Statistical Inference. Marcel Dekker, Inc. (2003)
Gil, M.A., Lubiano, M.A., Montenegro, M., López, M.T.: Least squares fitting of an affine function and strength of association for interval-valued data. Metrika 56, 97–111 (2002)
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 Syst. 157, 2608–2613 (2006)
Grzegorzewski, P.: Two-sample dispersion problem for fuzzy data. In: Lesot, M.-J., et al. (eds.) IPMU 2020. CCIS, vol. 1239, pp. 82–96. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-50153-2_7
Grzegorzewski, P.: Permutation k-sample goodness-of-fit test for fuzzy data, In: Proceedings of the 2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1–8 (2020)
Grzegorzewski, P., Gadomska, O.: Nearest neighbor tests for fuzzy data, In: Proceedings of the 2021 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1–6 (2021)
Hryniewicz, O.: Selection of variables for systems analysis, application of a fuzzy statistical test for independence. In: Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2014), Proceedings, Part III, pp. 2197–22043. Springer (2004). https://doi.org/10.1007/978-3-319-08855-6
Hryniewicz, O.: Goodman-Kruskal \(\gamma \) measure of dependence for fuzzy ordered categorical data. Comput. Stat. Data Anal. 51, 323–334 (2006)
Hukuhara, M.: Integration des applications measurables dont la valeur est un compact convexe. Funkcialaj Ekvacioj 10, 205–223 (1967)
Lubiano, M.A., Gil, M.A., López-Díaz, M., López-García, M.T.: The \(\lambda \)-mean squared dispersion associated with a fuzzy random variable. Fuzzy Sets Syst. 111, 307–317 (2000)
Lubiano, M.A., Montenegro, M., Sinova, B., de la Rosa de Sáa, S., Gil, M.A.: Hypothesis testing for means in connection with fuzzy rating scale-based data: algorithms and applications. Eur. J. Oper. Res. 251 918–929 (2016)
Lyons, R.: Distance covariance in metric spaces. Ann. Probab. 41, 3284–3305 (2013)
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)
Puri, M.L., Ralescu, D.A.: Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986)
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)
Székely, G.J., Rizzo, M.L., Bakirov, N.K.: Measuring and testing dependence by correlation of distances. Ann. Stat. 35, 2769–2794 (2007)
Székely, G.J., Rizzo, M.L., Bakirov, N.K.: The distance correlation t-test of independence in high dimension. J. Multivar. Anal. 117, 193–213 (2013)
Taheri, S.M., Hesamian, G.: Goodman-Kruskal measure of association for fuzzy-categorized variables. Kybernetika 47, 110–122 (2011)
Taheri, S.M., Hesamian, G., Viertl, R.: Contingency tables with fuzzy information. Commun. Stat. Theory Meth. 45, 5906–5917 (2016)
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. Inf. Sci. 179, 3964–3972 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Grzegorzewski, P. (2022). Testing Independence with Fuzzy Data. In: Ciucci, D., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2022. Communications in Computer and Information Science, vol 1602. Springer, Cham. https://doi.org/10.1007/978-3-031-08974-9_42
Download citation
DOI: https://doi.org/10.1007/978-3-031-08974-9_42
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-08973-2
Online ISBN: 978-3-031-08974-9
eBook Packages: Computer ScienceComputer Science (R0)