Recent Researches in Artificial Intelligence, Knowledge Engineering and Data Bases
On Possibilistic Correlation Coefficient and Ratio for Fuzzy
Numbers
PÉTER VÁRLAKI
JÓZSEF MEZEI
ISTVÁN Á. HARMATI
ROBERT FULLÉR
Bud. Univ. of Tech. Econ.
Åbo Akademi
Széchenyi István University
Åbo Akademi
Széchenyi István University
TUCS
Dep. of Math. Comp. Sci.
IAMSR
Egyetem tér 1, Győr
Joukahaisenkatu 3-5, Åbo
Egyetem tér 1, Győr
Joukahaisenkatu 3-5, Åbo
HUNGARY
FINLAND
HUNGARY
FINLAND
varlaki@kme.bme.hu
jmezei@abo.fi
harmati@sze.hu
rfuller@abo.fi
Abstract: In this paper we show some properties of possibilistic correlation coefficient and correlation ratio for
marginal possibility distributions. We will show some examples for their computation from some joint possibility
distributions.
Key–Words: Fuzzy number, Possibility distribution, Correlation coefficient, Correlation ratio
1
Introduction
Definition 1 A function g : [0, 1] → R is said to be
a weighting function if g is non-negative, monotone
increasingR and satisfies the following normalization
1
condition 0 g(γ)dγ = 1.
In probability theory the expected value of functions
of random variables plays a fundamental role in defining the basic characteristic measures of probability
distributions. For example, the variance, covariance
and correlation of random variables can be computed
as the expected value of their appropriately chosen
real-valued functions. In possibility theory we can
use the principle of expected value of functions on
fuzzy sets to define variance, covariance and correlation of possibility distributions. Marginal probability distributions are determined from the joint one by
the principle of ’falling integrals’ and marginal possibility distributions are determined from the joint possibility distribution by the principle of ’falling shadows’. Probability distributions can be interpreted as
carriers of incomplete information [12], and possibility distributions can be interpreted as carriers of imprecise information. In 1987 Dubois and Prade [3]
defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. They
also showed that this expectation remains additive in
the sense of addition of fuzzy numbers. In possibility
theory we can use the principle of average value of
appropriately chosen real-valued functions to define
mean value, variance, covariance and correlation of
possibility distributions. Namely, we can equip each
level set of a possibility distribution (represented by a
fuzzy number) with a uniform probability distribution,
then apply their standard probabilistic calculation, and
then define measures on possibility distributions by
integrating these weighted probabilistic notions over
the set of all membership grades [1, 4]. These weights
(or importances) can be given by weighting functions.
ISBN: 978-960-474-273-8
Different weighting functions can give different (casedependent) importances to level-sets of possibility
distributions. We should note here that the choice
of uniform probability distribution on the level sets
of possibility distributions is not without reason. We
suppose that each point of a given level set is equally
possible and then we apply Laplace’s principle of Insufficient Reason: if elementary events are equally
possible, they should be equally probable (for more
details and generalization of principle of Insufficient
Reason see [2], page 59). In this paper we will discuss
the concepts of possibilistic correlation coefficient and
ratio for marginal possibility distributions of joint possibility distributions.
Definition 2 A fuzzy number A is a fuzzy set R with
a normal, fuzzy convex and continuous membership
function of bounded support.
The family of fuzzy numbers is denoted by F.
Fuzzy numbers can be considered as possibility distributions. A fuzzy set C in R2 is said to be a joint
possibility distribution of fuzzy numbers A, B ∈ F,
if it satisfies the relationships
max{x | C(x, y)} = B(y),
and
max{y | C(x, y)} = A(x),
for all x, y ∈ R. Furthermore, A and B are called
the marginal possibility distributions of C. A γ-level
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Recent Researches in Artificial Intelligence, Knowledge Engineering and Data Bases
set (or γ-cut) of a fuzzy number A is a non-fuzzy set
denoted by [A]γ and defined by
In other words, the g-weighted possibilistic correlation coefficient is nothing else, but the g-weighted
average of the probabilistic correlation coefficients
ρ(Xγ , Yγ ) for all γ ∈ [0, 1].
In 2011 Harmati [10] proved that the correlation
coefficient depends only on the joint possibility distribution, but not directly on its marginal distributions.
In other words exact knowledge of the marginal possibility distributions does not give any restrictions for
the possibilistic correlation coefficient and also for the
g-weighted possibilistic correlation coefficient.
If A and B are non-interactive fuzzy numbers
then their joint possibility distribution is defined by
C = A × B. Since all [C]γ are rectangular and the
probability distribution on [C]γ is defined to be uniform we get cov(Xγ , Yγ ) = 0, for all γ ∈ [0, 1]. So
Covg (A, B) = 0 and ρg (A, B) = 0 for any weighting function g. That is, non-interactivity entails zero
correlation.
Fuzzy numbers A and B are said to be in perfect
correlation, if there exist q, r ∈ R, q 6= 0 such that
their joint possibility distribution is defined by [1]
[A]γ = {t ∈ X | A(t) ≥ γ},
if γ > 0 and cl(suppA) if γ = 0, where cl(suppA)
denotes the closure of the support of A. Let A ∈ F
be fuzzy number with a γ-level set denoted by [A]γ =
[a1 (γ), a2 (γ)], γ ∈ [0, 1] and let Uγ denote a uniform
probability distribution on [A]γ , γ ∈ [0, 1].
2
A Correlation Coefficient for
Marginal Possibility Distributions
In 2004 Fullér and Majlender [4] introduced the notion of covariance between marginal distributions of a
joint possibility distribution C as the expected value
of their interactivity function on C. That is, the gweighted measure of interactivity between A ∈ F and
B ∈ F (with respect to their joint distribution C) is
defined by their measure of possibilistic covariance
[4], as
C(x1 , x2 ) = A(x1 ) · χ{qx1 +r=x2 } (x1 , x2 ),
1
Z
Covg (A, B) =
cov(Xγ , Yγ )g(γ)dγ,
0
where χ{qx1 +r=x2 } , stands for the characteristic function of the line
where Xγ and Yγ are random variables whose joint
distribution is uniform on [C]γ for all γ ∈ [0, 1], and
cov(Xγ , Yγ ) denotes their probabilistic covariance.
It is easy to see that the possibilistic covariance is
an absolute measure in the sense that it can take any
value from the real line. To have a relative measure
of interactivity between marginal distributions Fullér,
Mezei and Várlaki introduced the principle of correlation (normalized covariance) in 2011 (see [9]) that improves the earlier definition introduced by Carlsson,
Fullér and Majlender in 2005 (see [1]).
{(x1 , x2 ) ∈ R2 |qx1 + r = x2 }.
If A and B have a perfect positive (negative) correlation then from ρ(Xγ , Yγ ) = 1 (ρ(Xγ , Yγ ) = −1) (see
[1] for details), for all γ ∈ [0, 1], we get ρg (A, B) = 1
(ρg (A, B) = −1) for any weighting function g.
We emphasize here that zero correlation does not
always imply non-interactivity. Let A, B ∈ F be
fuzzy numbers, let C be their joint possibility distribution, and let γ ∈ [0, 1]. Suppose that [C]γ is symmetrical, i.e. there exists a ∈ R such that
Definition 3 The g-weighted possibilistic correlation
coefficient of A, B ∈ F (with respect to their joint
distribution C) is defined by
Z
C(x, y) = C(2a − x, y),
1
ρg (A, B) =
for all x, y ∈ [C]γ (the line defined by {(a, t)|t ∈
R} is the axis of symmetry of [C]γ ). In this case
cov(Xγ , Yγ ) = 0 and ρg (A, B) = 0 for any weighting function g. (see [5]).
We should note here that there exist several other
ways to define correlation coefficient for fuzzy numbers, e.g. Liu and Kao [15] used fuzzy measures to
define a fuzzy correlation coefficient of fuzzy numbers and they formulated a pair of nonlinear programs
to find the α-cut of this fuzzy correlation coefficient,
then, in a special case, Hong [11] showed an exact calculation formula for this fuzzy correlation coefficient.
ρ(Xγ , Yγ )g(γ)dγ,
0
where
ρ(Xγ , Yγ ) = p
cov(Xγ , Yγ )
p
,
var(Xγ ) var(Yγ )
and, where Xγ and Yγ are random variables whose
joint distribution is uniform on [C]γ for all γ ∈ [0, 1],
and cov(Xγ , Yγ ) denotes their probabilistic covariance.
ISBN: 978-960-474-273-8
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Recent Researches in Artificial Intelligence, Knowledge Engineering and Data Bases
3
A Correlation Ratio for Marginal
Possibility Distributions
4
In this Section we will compute the correlation coefficient and the correlation ratio for A and B when
their joint possibility distribution, C, is defined by
C(x, y) = T (A(x), B(y)), where T is a t-norm
[20]. We will consider the Mamdani t-norm [17],
Łukasiewicz t-norm [16] and Larsen t-norm [14]. For
simplicity we will consider fuzzy numbers of same
shape,
x if 0 ≤ x ≤ 1
A(x) =
,
0 otherwise
y if 0 ≤ y ≤ 1
B(y) =
.
0 otherwise
In statistics, the correlation ratio is a measure of the
relationship between the statistical dispersion within
individual categories and the dispersion across the
whole population or sample. The correlation ratio was
originally introduced by Karl Pearson [18] as part of
analysis of variance and it was extended to random
variables by Andrei Nikolaevich Kolmogorov [13] as,
η 2 (X|Y ) =
Examples
D2 [E(X|Y )]
,
D2 (X)
where X and Y are random variables. If X and Y
have a joint probability density function, denoted by
f (x, y), then we can compute η 2 (X|Y ) using the following formulas
Z ∞
E(X|Y = y) =
xf (x|y)dx
Recall that X and Y are random variables then their
correlation coefficient is computed from,
ρ(X, Y ) = p
−∞
and
E(XY ) − E(X)E(Y )
p
− E 2 (X) E(Y 2 ) − E 2 (Y )
E(X 2 )
and their correlation ratio is defined by,
D2 [E(X|Y )] = E(E(X|y) − E(X))2 ,
and where,
η 2 (X, Y ) =
f (x, y)
.
f (y)
It measures a functional dependence between random
variables X and Y . It takes on values between 0
(no functional dependence) and 1 (purely deterministic dependence). In probability theory the joint distribution function is always connected with its marginals
by a copula (Sklar theorem [19]), but it is not always
true for joint possibility distributions [6]. In 2010
Fullér, Mezei and Várlaki introduced the definition of
possibilistic correlation coefficient for marginal possibility distributions (see [6]).
E(E 2 (X|Y )) − E 2 (X)
.
E(X 2 ) − E 2 (X)
f (x|y) =
4.1
In this case the joint possibility distribution C(x, y) is
defined by the min operator,
C(x, y) = min{A(x), B(y)}.
Then a γ-level set of C is,
[C]γ = (x, y) ∈ R2 | γ ≤ x, y ≤ 1 .
The joint density function of a uniform distribution on
[C]γ is

 1
if (x, y) ∈ [C]γ ,
f (x, y) =
Tγ

0 otherwise ,
Definition 4 Let us denote A and B the marginal
possibility distributions of a given joint possibility distribution C. Then the g-weighted possibilistic correlation ratio of marginal possibility distribution A with
respect to marginal possibility distribution B is defined by
Z 1
2
ηg (A|B) =
η 2 (Xγ |Yγ )g(γ)dγ
where
Tγ = (1 − γ)2
denotes the area of the γ-level set. The marginal density functions are obtained from f (x, y) by
Z
f1 (x) = f (x, y) dy
0
where Xγ and Yγ are random variables whose joint
distribution is uniform on [C]γ for all γ ∈ [0, 1], and
η 2 (Xγ |Yγ ) denotes their probabilistic correlation ratio.
that is,
So the g-weighted possibilistic correlation ratio
of the fuzzy number A on B is nothing else, but the
g-weighted average of the probabilistic correlation ratios η 2 (Xγ |Yγ ) for all γ ∈ [0, 1].
ISBN: 978-960-474-273-8
Mamdani t-norm

 1
(1 − γ) if γ ≤ x ≤ 1 ,
f1 (x) =
T
 γ
0
otherwise .
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Recent Researches in Artificial Intelligence, Knowledge Engineering and Data Bases
The expected values are,
1
Z
E(Xγ ) = E(Yγ ) =
The joint density function of a uniform distribution on
[C]γ is

 1
if (x, y) ∈ [C]γ ,
f (x, y) =
Tγ

0 otherwise ,
1+γ
.
2
xf1 (x) dx =
γ
and,
where
E(Xγ2 ) = E(Yγ2 ) =
1
Z
x2 f1 (x) dx =
γ
1+γ+
3
γ2
(1 − γ)2
Tγ =
2
denotes the area of the γ-level set. The marginal density function

 1
(x − γ) if γ ≤ x ≤ 1 ,
f1 (x) =
Tγ

0
otherwise .
.
The expected value of the product is
E(Xγ Yγ ) =
Z
γ
1Z
γ
1
(1 + γ)2
xyf (x, y) dy dx =
.
4
The expected values are:
Z 1
1
2+γ
E(Xγ ) = E(Yγ ) =
x (x − γ) dx =
.
Tγ
3
γ
The conditional expected value is
1
Z
E(X|Y ) =
xf (x|y) dx
γ
1
Z
=
γ
1
Z
=
γ
E(Xγ2 )
f (x, y)
dx
x
f2 (y)
1
x
dx
1−γ
=
E (X|Y )f2 (y) dy
Z
1
=
γ
1+γ
2
!2
1
(1 − γ) dy
Tγ
(1 + γ)2
.
4
(γ + 5)(γ + 1)
.
12
The conditional expected value:
Z 1
2+γ−y
1
E(X|Y ) =
x
dx =
.
2
1+γ−y y − γ
Łukasiewicz t-norm
In this example we define the joint possibility distribution by the Łukasiewitz t-norm, that is,
C(x, y) = max{A(x) + B(y) − 1, 0}.
Z
2
E(E (X|Y )) =
γ
Then a γ-level set of C is
[C]γ = (x, y) ∈ R2 | γ ≤ x, y ≤ 1, x + y ≥ 1 + γ .
ISBN: 978-960-474-273-8
1
(x − γ) dx
Tγ
From these we can compute the correlation coefficient:
1
ρ(Xγ , Yγ ) = −
2
The conditional density function is:

f (x, y)  1
if γ ≤ y ≤ 1 ,
f (x|y) =
=
y
−
γ

f2 (y)
0
otherwise .
From these results it follows the correlation ratio is
also equal to zero. It is not surprising, since in this
case A and B are non-interactive fuzzy numbers.
4.2
x2
+ 2γ + 3
.
6
=
2
γ
=
γ2
1
E(E (X|Y )) =
1
=
The expected value of the product is,
Z 1Z 1
1
E(Xγ Yγ ) =
xy dy dx
T
γ
γ
1+γ−x
and,
Z
Z
γ
1+γ
.
=
2
2
=
E(Yγ2 )
266
=
1
2+γ−y
2
!2
3γ 2 + 10γ + 11
.
24
1
(y − γ) dy
Tγ
Recent Researches in Artificial Intelligence, Knowledge Engineering and Data Bases
With this result we can compute the correlation ratio
and we get
η 2 (Xγ , Yγ ) =
1
4
⇒
η(Xγ , Yγ ) =
0.5
0.4
1
2
0.3
In this example η 2 = ρ2 , because of the linear relationship between A and B.
0.2
4.3
Larsen t-norm
In this case we define the joint possibility distribution
C by the product t-norm, C(x, y) = A(x)B(y). Then
a γ-level set of C is,
[C]γ = (x, y) ∈ R2 | 0 ≤ x, y ≤ 1, xy ≥ γ .
0
0
E(E 2 (X|Y )) =
=
1 (1 −
Tγ
2
x
γ
2
γ)
γ
0.6
0.8
1
11 3
(γ + 2γ 2 − 5γ + 2 − 2γ ln γ) .
8 Tγ
From these expression we could determine the correlation coefficient and the correlation ratio for any γ.
The final formulas are very long and difficult, so they
are omitted here.
The expected values are,
1
0.4
and, furthermore,
where Tγ = 1 − γ + γ ln γ denotes the area of the
γ-level set. The marginal density function

 1 x−γ
if γ ≤ x ≤ 1 ,
f1 (x) =
Tγ x

0
otherwise .
Z
0.2
Figure 1: The absolute value of the correlation coefficient (ρ) and the correlation ratio (η) as the function
of γ.
The joint density function of a uniform distribution on
[C]γ is

 1
if (x, y) ∈ [C]γ ,
f (x, y) =
Tγ

0 otherwise ,
E(Xγ ) = E(Yγ ) =
|ρ|
η
0.1
Acknowledgements: This work was supported in
part by the project TAMOP 421B at the Széchenyi
István University, Győr.
1 x−γ
dx
Tγ x
.
References:
E(Xγ2 ) = E(Yγ2 ) =
Z
1
x2
γ
=
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1
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Tγ
γ
γ/x
=
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(1 − γ 2 + 2γ 2 ln γ) .
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E(X|Y ) =
ISBN: 978-960-474-273-8
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