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We can perform two-way ANOVA to determine whether to reject the null hypothe-
ses with a predetermined significance level. See Nakama et al. [14, 15] for mathe-
matical details.
19.4
Extending Classical Statistical Procedures to Fuzzy Data
In this section, we outline a methodology for extending classical statistical proce-
dures to fuzzy data. Mathematical details are omitted.
We assume that for each
level
of each fuzzy set is a nonempty compact convex set. We define two basic arithmetic
operations on fuzzy sets: addition and scalar multiplication. There are several ways
to define them. Typically addition is defined as the Minkovski addition at each alpha
level, and scalar multiplication is also defined in a level-wise manner.
The Minkowski support function can be used to define a metric for fuzzy sets and
to transform them to Hilbert-space-valued functions. The transformation isometri-
cally embeds the class of fuzzy sets in a closed convex cone of a Hilbert space. As a
result, various convergence results for random fuzzy sets can be derived from essen-
tial theoretical results for Hilbert-space-valued random variables. For mathematical
details, see, for instance, Colubi [2].
α
greater than 0 and less than or equal to 1, the
α
19.5
Discussions
In many research areas, including medical research, we must often deal with ob-
servations that can be effectively represented by fuzzy sets. In addition to the
hypothesis-testing procedures that we have reviewed, many other types of statistical
analysis have also been developed for fuzzy data (for a review, see, for example,
Taheri [18]). These procedures are not only theoretically rigorous but also prac-
tical. For instance, Colubi and González-Rodríguez [4] and Colubi [2] conducted
statistical analyses of fuzzy data regarding forestry expert evaluations. Colubi et
al. [3] analyzed bank managers' investment aversion assessments represented by
fuzzy sets. Lubiano and Trutschnig [11] developed an R-package called SAFD,
with which one can easily perform one-way ANOVA for fuzzy data. We hope that
this paper will motivate the reader to use fuzzy measurements for statistical studies.
References
1. Chimka, J.R., Wolfe, H.: History of Ordinal Variables Before 1980. Scientific Research
and Essays 4, 853-860 (2009)
2. Colubi, A.: Statistical Inference about the Means of Fuzzy Random Variables: Appli-
cations to the Analysis of Fuzzy- and Real-valued Data. Fuzzy Sets and Systems 160,
344-356 (2009)
3. Colubi, A., Coppi, R., D'Urso, P., Gil, M.A.: Statisitcs with Fuzzy Random Variables.
Metron-International Journal of Statistics 65, 277-303 (2007)
