Quality & Quantity (2007) 41:333–340 © Springer 2006
Multiple and Partial Correlation Coefﬁcients
of Fuzzy Sets
and JONG-WUU WU
Department of Mathematics Education, National Hsinchu Teachers College, Hsin-Chu,
Department of Applied Mathematics, National Chiayi University, Chiayi
City 60004, Taiwan, R.O.C.
Abstract. In many applications, multiple correlation and partial correlation for three or
more fuzzy sets are very important, but Chiang and Lin (1999, Fuzzy Sets and Systems 102:
221–226) do not solve this problem. Here, we propose a method to calculate the multiple
correlation and partial correlation for fuzzy data, by adopting the concepts from the mul-
tivariate correlation model. In order to ﬁt into normal framework, we use empirical logit
transform (see, Agresti, [1990, Categorical Data Analysis. New York: Wiley]; Johnson and
Wichern, [1992, Applied Multivariate Statistical Analysis 3rd edn. Engelwood Cliffs; Prentice-
Hall.]) for membership function grades to achieve this.
Key words: correlation model, empirical logit transform, fuzzy sets, multiple correlation,
In the study of multivariate correlation models one is naturally very inter-
ested in relationships among the variables. One set of measures useful to this
end consists of the coefﬁcients of multiple correlation and the coefﬁcients of
partial correlation. All partial correlation coefﬁcients measure the correlation
between two variables. What we are interested in, ﬁnding the coefﬁcients of
multiple correlation and the coefﬁcients of partial correlation between fuzzy
sets (FSs), which can tell us the relationships among the FSs.
Chiang and Lin (1999) discussed the correlation coefﬁcient between two
FSs, by adopting the concepts from the conventional statistics. Similar
works have been done by Bustince and Burillo (1995), Gerstenkorn and
Manko (1991), Hong and Hwang (1995) and Yu (1993). In many appli-
cations, partial correlation and multiple correlation for three or more FSs
Author for correspondence: Jong-Wuu-wu, Department of Applied Mathematics,
National Chiayi University, Chiayi City 60004, Taiwan, R.O.C., E-mail: email@example.com.