INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS …naun.org/main/NAUN/ijmmas/16-235.pdfR. Full´er is with John von Neumann Faculty of Informatics, Obuda´ University, Becsi´ ´ut 96/B,

INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES 1

On Weighted Possibilistic Informational Coefficientof Correlation

Robert Fuller, Istvan A. Harmati, Peter Varlaki, Imre Rudas

Abstract—In their previous works Fuller et al. introducedthe notions of weighted possibilistic correlation coefficient andcorrelation ratio as measures of dependence between possibilitydistributions (fuzzy numbers). In this paper we introduce a newmeasure of strength of dependence between marginal possibilitydistributions, which is based on the informational coefficient ofcorrelation. We will show some examples that demonstrate somegood properties of the proposed measure.

Index Terms—Fuzzy number, Possibility distribution, Measureof dependence, Mutual information, Correlation.

I. INTRODUCTION

UNCERTAIN informations can be diveded into two maincategories: incomplete and imprecise information. Proba-

bility distributions can be interpreted as carriers of incompleteinformation [1], and possibility distributions can be interpretedas carriers of imprecise information. Measuring dependencebetween uncertain variables plays a fundamental role in bothcategories. In probability theory there are well-known mea-sures of dependence between random variables, such as corre-lation coefficient, correlation ratio, mean square contingency,and so on. In possibility theory there are several treatmentsfor characterizing dependence between fuzzy numbers, see forexample [2] and [3].

In this approach we use simple probability distributions tobuild up measures of dependence between possibility distribu-tions. Namely, we equip each level set of a possibility distri-bution with a uniform probability distribution, then determinea probabilistic measure of dependence, and then define mea-sures on possibility distributions by integrating these weightedprobabilistic notions over the set of all membership grades [4],[5]. These weights (or importances) can be given by weightingfunctions.

Definition 1. A function g : [0, 1] → R is said to be aweighting function if g is non-negative, monotone increasingand satisfies the ∫ 1

0

g(γ) dγ = 1

normalization condition

R. Fuller is with John von Neumann Faculty of Informatics, ObudaUniversity, Becsi ut 96/B, Budapest 1034, Hungary

I. A. Harmati is with the Department of Mathematics and ComputationalScience, Szechenyi Istvan University, Egyetem ter 1, Gyor 9026, Hungary,email: [email protected] (corresponding author)

P. Varlaki is with Szechenyi Istvan University, Egyetem ter 1, Gyor 9026,Hungary

I. Rudas is with John von Neumann Faculty of Informatics, ObudaUniversity, Becsi ut 96/B, Budapest 1034, Hungary

In other words a possibilistic measure of dependence isthe g-weighted average of the probabilistic measure of depe-nence. Different weighting functions can give different (case-dependent) importances to level-sets of possibility istributions.We should note here that the choice of uniform probabilitydistribution on the level sets of possibility distributions isnot without reason. We suppose that each point of a givenlevel set is equally possible and then we apply Laplace’sprinciple of Insufficient Reason: if elementary events areequally possible, they should be equally probable (for moredetails and generalization of principle of Insufficient Reasonsee [6], page 59). The uniform distribution is not the only way,for example we can deal with probabilty distributions whosedensity function is similar (has the same shape) to the jointpossibility distribution, see [7].

Definition 2. A fuzzy number A is a fuzzy set R with a normal,fuzzy convex and continuous membership function of boundedsupport.

Fuzzy numbers can be considered as possibility distribu-tions. A fuzzy set C in R2 is said to be a joint possibilitydistribution of fuzzy numbers A,B, if it satisfies the relation-ships

max{x | C(x, y)} = B(y),

andmax{y | C(x, y)} = A(x),

for all x, y ∈ R. Furthermore, A and B are called the marginalpossibility distributions of C. Marginal possibility distributionsare always uniquely defined by their joint possibility distribu-tion by the principle of falling shadows. A γ-level set (orγ-cut) of a fuzzy number A is a non-fuzzy set denoted by[A]γ and defined by

[A]γ = {t ∈ X | A(t) ≥ γ},

if γ > 0 and cl(suppA) if γ = 0, where cl(suppA) denotesthe closure of the support of A.

II. FORMER LEVEL-BASED MEASURES OF CORRELATION

A. Correlation Coefficient

In probability theory the correlation coefficient of randomvariables X and Y is defined by

ρ(X,Y ) =cov(X,Y )√

var(X)√

var(Y )=E(XY )− E(X) · E(Y )

D(X) ·D(Y ),

where E(X), E(Y ) and E(XY ) are expected value os X , Yand X ·Y respectively, and D(X), D(Y ) are the square rootsof the variances.

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Issue 4, Volume 6, 2012 592


In 2011 Fuller, Mezei and Varlaki introduced the principleof correlation (see [8]) that improves the earlier definitionintroduced by Carlsson, Fuller and Majlender in 2005 (see[4]). The main drawback of the earlier definition that it doesnot necessarily take its values from [−1; 1] if some level setsof the joint possibility distribution are not convex.

Definition 3. The g-weighted possibilistic correlation coeffi-cient of fuzzy numbers A and B (with respect to their jointdistribution C) is defined by

ρg(A,B) =

∫ 1

0

ρ(Xγ , Yγ)g(γ) dγ,

where

ρ(Xγ , Yγ) =cov(Xγ , Yγ)√

var(Xγ)√

var(Yγ),

and, where Xγ and Yγ are random variables whose jointdistribution is uniform on [C]γ for all γ ∈ [0, 1], andcov(Xγ , Yγ) denotes their probabilistic covariance.

If A and B are non-interactive fuzzy numbers then theirjoint possibility distribution is defined by C = A×B, so wethe membership function of the joint possibiliy distribution isdetermined from the membership functions of A and B by themin operator. A more general notion of independence of fuzzynumbers can be found in [9]. Since all [C]γ are rectangular andthe probability distribution on [C]γ is defined to be uniformwe get cov(Xγ , Yγ) = 0, for all γ ∈ [0, 1]. So the g-weightedpossibilistic covariance covg(A,B) = 0 and the g-weightedpossibilistic correlation coefficient ρg(A,B) = 0 for anyweighting function g. That is, non-interactivity entails zerocorrelation.

Zero correlation does not always implies non-interactivity.Let A,B be fuzzy numbers, let C be their joint possibility dis-tribution, and let γ ∈ [0, 1]. Suppose that [C]γ is symmetrical,i.e. there exists a ∈ R such that

C(x, y) = C(2a− x, y),

for all x, y ∈ [C]γ (the line defined by {(a, t)|t ∈ R} is theaxis of symmetry of [C]γ). In this case cov(Xγ , Yγ) = 0 andρg(A,B) = 0 for any weighting function g. (see [10]). Formore example on nonzero correlation see [11], [12] and [13].

B. Correlation RatioIn statistics, the correlation ratio is a measure of the re-

lationship between the statistical dispersion within individualcategories and the dispersion across the whole population orsample. The correlation ratio was originally introduced by KarlPearson [14] as part of analysis of variance and it was extendedto random variables by Andrei Nikolaevich Kolmogorov [15]as a square root of

η2(X|Y ) =D2[E(X|Y )]

D2(X),

where X and Y are random variables. If X and Y have a jointprobability density function, denoted by f(x, y), then we cancompute η2(X|Y ) using the following formulas

E(X|Y = y) =

∫ ∞−∞

xf(x|y) dx

andD2[E(X|Y )] = E(E(X|Y )− E(X))2,

and where,

f(x|y) =f(x, y)

f(y).

It measures a functional dependence between random variablesX and Y . It takes on values between 0 (no functionaldependence) and 1 (purely deterministic dependence).

In 2010 Fuller, Mezei and Varlaki introduced the defini-tion of possibilistic correlation ratio for marginal possibilitydistributions (see [16]).

Definition 4. Let us denote A and B the marginal possibilitydistributions of a given joint possibility distribution C. Thenthe g-weighted possibilistic correlation ratio of marginalpossibility distribution A with respect to marginal possibilitydistribution B is defined by

η2g(A|B) =

∫ 1

0

η2(Xγ |Yγ)g(γ) dγ

where Xγ and Yγ are random variables whose joint distri-bution is uniform on [C]γ for all γ ∈ [0, 1], and η2(Xγ |Yγ)denotes their probabilistic correlation ratio.

III. RENYI’S POSTULATES FOR MEASURES OFDEPENDENCE

We use measures of dependence between random variablesto determine measures of dependence between possibilitydistributions, so it is natural to describe what conditions shouldsatisfy a good measure. In [17] A. Renyi gave seven postulateswhich should be fulfilled by a suitable measure of dependencebetween random variables X and Y (δ(X,Y )):A) δ(X,Y ) is defined for any pair of random variables X and

Y , neither of them being constant with probability 1.B) δ(X,Y ) = δ(Y,X).C) 0 ≤ δ(X,Y ) ≤ 1.D) δ(X,Y ) = 0 if and only if X and Y are independent.E) δ(X,Y ) = 1, if there is a strict dependence between X

and Y , i.e. either X = g(Y ) or Y = f(X), where g(x)and f(x) are Borel-measurable functions.

F) If the Borel-measurable functions f(x) and g(x) maps thereal axis in a one-to-one way ono itself, δ(f(X), g(Y )) =δ(X,Y ).

G) If the joint distribution of X and Y is normal, thenδ(X,Y ) = |ρ(X,Y )|, where ρ(X,Y ) is the correlationcoefficient of X and Y .

The correlation coefficient of the random variables X andY is defined only if D(X) and D(Y ) are finite and positve. Itmay be zero also if X and Y are not independent, moreover, itmay vanish inspite of functional dependence between X andY . For example if the distribution of X is symmetrical to zeroand Y = X2, then ρ(X,Y ) = 0. |ρ(X,Y )| = 1 is equal to 1if and only if there is a linear relationship between X and Y .The correlation coefficient satisfies postulates B and G, andits absolute value satisfies B, C and G.




TABLE IMEASURES OF DEPENDENCE AND FULFILLMENT OF THE POSTULATES

A B C D E F G

correlation coefficient X X

absolute value of corr. coeff. X X X

correlation ratio X X X

symmetrical correlation ratio X X X X

maximal correlation X X X X X X X

mutual information X X X X

informational coeff. of corr. X X X X X X X

The correlation ratio is defined provided that D(Y ) exist andis positive. It is not symmetric, but one can consider insteadof η(X|Y ) the quantity

max {η(X|Y ), η(Y |X)}

which is symmetric. The correlation ratio satisfies postulatesB, C, E and G, the symmatric correlation ratio satisfies B,C, D, E and G.

The maximal correlation introduced by H. Gebelein ([18])of random variables X and Y is defined by

S(X,Y ) = supf,g{ρ(f(X), g(Y ))}

where ρ(·, ·) is the correlation coefficient, and f(x) and g(x)run over all Borel-measurable functions such that f(X) andg(Y ) have finite and nonzero variance. It has all the propertiesA to G listed above, but unfortunately the maximal correlationis very difficult to determine and there does not always existsuch functions f0(x) and g0(x) that

S(X,Y ) = ρ(f0(X), g0(Y )) .

IV. INFORMATIONAL MEASURE OF CORRELATION

The well-known measures do not satisfy Renyi’s postulates.For a better candidate first we recall the definition of mutualinformation:

Definition 5. For any two continous random variables Xand Y (admitting a joint probability density), their mutualinformation is given by

I(X,Y ) =

∫ ∫f(x, y) ln

f(x, y)

f1(x) · f2(y)dxdy

where f(x, y) is the joint probability density function of X andY , and f1(x) and f2(y) are the marginal density functions ofX and Y , respectively.

Some properties of the mutual information:• I(X,Y ) ≥ 0.• I(X,Y ) = 0 if and only if X and Y are independent.• I(X,Y ) =∞ if X and Y are conmtinous and there is a

functional relationship between X and Y .If the joint probability distribution is uniform on the γ-

levels, then the joint density function on a γ-level (f(x, y)) is

constant, so the formula above will be simpler. Let denote Tγthe area of the γ-level. Then

I(X,Y ) =

∫ ∫1

Tγln

1

Tγf1(x)f2(y)

dxdy

=

∫ ∫1

Tγln

1

Tγdxdy −

∫ ∫1

Tγln f1(x) dydx

−∫ ∫

1

Tγln f2(y) dxdy =

= ln1

Tγ−∫f1(x) ln f1(x) dx−

∫f2(y) ln f2(y) dy .

Moreover, if the joint probability distribution is uniform onthe γ-levels, and the marginal random variables X and Y hasthe same distribution, then

I(X,Y ) = ln1

Tγ− 2

∫f1(x) ln f1(x) dx .

Easy to check that the mutual information satisfies Renyi’spostulates except C, E and G. Based on the mutual informa-tion Linfoot ([19]) introduced a measure of dependence, whichsatisfies all of the postulates.

Definition 6. For two random variables X and Y , let denoteI(X,Y ) the mutual information between X and Y . Theirinformational coefficient of correlation is given by

L(X,Y ) =√

1− e−2I(X,Y ) .

Based on the definition above, we can define the following:

Definition 7. Let us denote A and B the marginal possibilitydistributions of a given joint possibility distribution C. Thenthe g-weighted possibilistic informational coefficient of corre-lation of marginal possibility distributions A and B is definedby

L(A,B) =

∫ 1

0

L(Xγ , Yγ)g(γ) dγ

where Xγ and Yγ are random variables whose joint distri-bution is uniform on [C]γ for all γ ∈ [0, 1], and L(Xγ , Yγ)denotes informational coefficient of correlation.

V. EXAMPLES

First we show that non-interactivity implies zero for infor-mational coefficient of correlation. In the second we show anexample for non-zero correlation. Then we give two exampleswhen the correlation coefficients and the correlation ratios arealso zero, but the marginal distributions are not independent.So in these cases the correlation coefficient and the correlationratio are not appropriate tools for measuring the dependence,but this problem not arises with the informational coefficientof correlation. Finally we show an example when the measuresof correlation depend on the γ-level sets.



0

0.5

1

0

0.5

10

0.5

1

Fig. 1. Joint possibility distribution defined by the min operator (Mamdanit-norm). This is the case of non-interactivity of A and B, all of the γ-levelsets are rectangulars.

A. Non-interactivity implies zero correlation

The joint possibility distribution is defined by the Mamdanit-norm ([20]), see Fig.1:

C(x, y) =

{min{x, y} if 0 ≤ x, y ≤ 1 ,

0 otherwise .

The marginal possibility distributions are

A(x) =

{x if 0 ≤ x ≤ 1 ,0 otherwise .

B(y) =

{y if 0 ≤ y ≤ 1 ,0 otherwise .

In this case the γ-level set is a square, with vertices (0, 0),(0, 1 − γ), (1 − γ, 0) and (1 − γ, 1 − γ). This is the case ofnon-interactivity of the marginal possibility distributions. Thearea of the γ-level set is

Tγ = (1− γ)2 .

The joint density function is:

f(x, y) =

1

Tγif γ ≤ x, y ≤ 1 ,

0 otherwise .

The marginal density function (we have the same expressionfor f2(y) with y) is:

f1(x) =

1

Tγ(1− γ) if γ ≤ x ≤ 1 ,

0 otherwise .

In this case Xγ and Yγ are independent (in probability sense),because:

f1(x) · f2(y) =1

Tγ(1− γ) · 1

Tγ(1− γ) = 1

(1− γ)2

=1

Tγ= f(x, y) .

So I(Xγ , Yγ) = 0, and then the informational coefficient ofcorrelation is

L(Xγ , Yγ) =√1− e−2I(Xγ ,Yγ) = 0 .

In this case the correlation measures are zeros for all γ, sothe g-weighted measures of correlation between the marginalpossibility distributions A and B for arbitrary weightingfunction g(γ):

Lg(A,B) =

∫ 1

0

L(Xγ , Yγ) · g(γ) dγ = 0

ρg(A,B) =

∫ 1

0

ρ(Xγ , Yγ) · g(γ) dγ = 0

ηg(A,B) =

∫ 1

0

η(Xγ , Yγ) · g(γ) dγ = 0

B. Łukasiewitz t-norm

The joint possibility distribution is defined by the well-known Łukasiewitz t-norm ([21]), see Fig.2:

C(x, y) =

max{x+ y − 1, 0} if 0 ≤ x, y ≤ 1and x+ y ≥ 1 ,

0 otherwise .

00.5

1

0

0.5

10

0.5

1

Fig. 2. Joint possibility distribution defined by the Łukasiewitz t-norm. Inthis case the γ-level sets are similar triangles.


A(x) =


B(y) =


The γ-level set:

[C]γ ={(x, y) ∈ R2| γ ≤ x, y ≤ 1, x+ y ≥ 1 + γ

}.

The joint probability distribution on [C]γ :

f(x, y) =

1

Tγif (x, y) ∈ [C]γ .

0 otherwise .




where Tγ is the area of the γ-level set:

Tγ =(1− γ)2

2.

The marginal density function (we have the same expressionfor f2(y) with y):

f1(x) =

1

Tγ(x− γ) if γ ≤ x ≤ 1 ,

0 otherwise .

For the mutual information I(Xγ , Yγ) we have to compute:

1∫γ

f1(x) ln f1(x) dx =

1∫γ

1

Tγ(x− γ) ln

1

Tγ(x− γ) dx

=[t =

x− γTγ

]=

1−γTγ∫0

t ln t · Tγ dt

= Tγ ·[t2

2ln t− t2

4

] 1−γTγ

0

= ln 2− ln(1− γ)− 1

2.

With this result the mutual information is

I(Xγ , Yγ) =

= ln1

Tγ−∫f1(x) ln f1(x) dx−

∫f2(y) ln f2(y) dy

= ln1

Tγ− 2 ·

(ln 2− ln(1− γ)− 1

2

)= ln

2

(1− γ)2− 2 ·

(ln 2− ln(1− γ)− 1

2

)=

= 1− ln 2 .

From this the informational coefficient of correlation:

L(Xγ , Yγ) =√1− e−2I(Xγ ,Yγ) =

√1− e−2(1−ln 2)

=√1− 4e−2 ≈ 0.6772 .

For this possibility distribution the correlation coefficient is(see [23])

ρ(Xγ , Yγ) = −1

2

and the correlation ratio is

η2(Xγ , Yγ) =1

4⇒ η(Xγ , Yγ) =

1

2

In this example η2 = ρ2, because of the linear relationshipbetween Xγ and Yγ . In this case the correlation measuresare not depend on the level γ, so the g-weighted measuresof correlation between the marginal possibility distributions Aand B for arbitrary weighting function g(γ):

Lg(A,B) =

∫ 1

0

L(Xγ , Yγ) · g(γ) dγ ≈ 0.6772

ρg(A,B) =

∫ 1

0

ρ(Xγ , Yγ) · g(γ) dγ = −0.5

ηg(A,B) =

∫ 1

0

η(Xγ , Yγ) · g(γ) dγ = 0.5

C. Pyramidal joint possibility distribution

Let the joint possibility distribution be a pyramid, whosevertices are (1, 0), (0, 1), (−1, 0) and (0,−1) on the xy-plane,see Fig.3. The marginal possibility distributions are

−1−0.5

00.5

1

−1−0.5

00.5

1

0

0.5

1

Fig. 3. Pyramid shaped joint possibility distribution. Because of symmetrythe correlation coefficient and the correlation ratio are both zeros, but A andB are not independent.

A(x) =

x+ 1 if −1 ≤ x ≤ 0 ,1− x if 0 < x ≤ 1 ,0 otherwise .

B(y) =

y + 1 if −1 ≤ y ≤ 0 ,1− y if 0 < y ≤ 1 ,0 otherwise .

Then the γ-level set is a square with vertices (1−γ, 0), (0, 1−γ), (−(1− γ), 0) and (0,−(1− γ)).

Because of symmetry the correlation coefficients and thecorrelation ratio of Xγ and Yγ are both zero, but Xγ and Yγare not independent:

f(x, y) 6= f1(x) · f2(y) .

The area of the γ-level set:

Tγ = 2(1− γ)2 .


f1(x) =

1

Tγ· 2(x+ 1− γ) if −1 + γ ≤ x ≤ 0 ,

1

Tγ· 2(1− γ − x) if 0 ≤ x ≤ 1− γ ,

0 otherwise .




We have to compute the following:1−γ∫

−(1−γ)

f1(x) ln f1(x) dx =

=

0∫−(1−γ)

1

Tγ· 2(x+ 1− γ) ln

1

Tγ· 2(x+ 1− γ) dx

+

1−γ∫0

1

Tγ· 2(1− γ − x) ln

1

Tγ· 2(1− γ − x) dx

= 2 ·(1

2ln

1

1− γ− 1

4

)= ln

1

1− γ− 1

2.

The mutual information:

I(Xγ , Yγ) = ln1

Tγ− 2 ·

(ln

1

1− γ− 1

2

)= ln

1

2(1− γ)2− 2 ln

1

1− γ+ 1 = 1− ln 2 .

From this we get the informational coefficient of correlationwhich is the same as in the previous example:

L(Xγ , Yγ) =√

1− e−2I(Xγ ,Yγ) ≈ 0.6772 .

In this case the correlation measures are not depend on thelevel γ, so the g-weighted measures of correlation betweenthe marginal possibility distributions A and B for arbitraryweighting function g(γ):

Lg(A,B) =

∫ 1

0

L(Xγ , Yγ) · g(γ) dγ ≈ 0.6772

ρg(A,B) =

∫ 1

0

ρ(Xγ , Yγ) · g(γ) dγ = 0

ηg(A,B) =

∫ 1

0

η(Xγ , Yγ) · g(γ) dγ = 0

D. Conical joint possibility distribution

Let the joint possibility distribution be a cone, whose axisis the z-axis, and the base is a circle with radius 1, see Fig.4.The marginal possibility distributions are:

A(x) =

x+ 1 if −1 ≤ x ≤ 0 ,1− x if 0 < x ≤ 1 ,0 otherwise .

B(y) =

y + 1 if −1 ≤ y ≤ 0 ,1− y if 0 < y ≤ 1 ,0 otherwise .

Then the γ-level set is a circle with centre (0, 0) and radius1 − γ. All of the γ-level sets are symmetrical to the x andthe y axis, so in this case the correlation coefficient and thecorrelation ratio are both zero, although Xγ and Yγ are notindependent, because

f(x, y) 6= f1(x) · f2(y) .

The area of the γ-level:

Tγ = π(1− γ)2 .

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

0

0.5

1

Fig. 4. Conical joint possibility distribution. Because of symmetry thecorrelation coefficient and the correlation ratio are both zeros, but A andB are not independent.


f1(x) =

1

Tγ· 2√(1− γ)2 − x2 if x ≤ |1− γ| ,

0 otherwise .

The integrals in this case are quite difficult, so we usednumerical methods. The mutual information is approximately:

I(Xγ , Yγ) = ln1

Tγ− 2

1−γ∫−(1−γ)

f1(x) ln f1(x) dx ≈ 0.1447 .

With this approximation the informational coefficient of cor-relation is:

L(Xγ , Yγ) =√

1− e−2I(Xγ ,Yγ) ≈ 0.5013 .

In this case the correlation measures are not depend on thelevel γ, so the g-weighted measures of correlation betweenthe marginal possibility distributions A and B for arbitraryweighting function g(γ):

Lg(A,B) =

∫ 1

0

L(Xγ , Yγ) · g(γ) dγ ≈ 0.5013

ρg(A,B) =

∫ 1

0

ρ(Xγ , Yγ) · g(γ) dγ = 0

ηg(A,B) =

∫ 1

0

η(Xγ , Yγ) · g(γ) dγ = 0

E. Larsen t-norm

Let the joint possibility distribution defined by the productt-norm (Larsen t-norm, [22]), see Fig.5 (for more general case,when the joint possibility distribution is defined by the productof triangular fuzzy numbers see [12]):

C(x, y) =

{xy if 0 ≤ x, y ≤ 1 ,0 otherwise .




00.5

1

0

0.5

10

0.5

1

Fig. 5. Joint possibility distribution defined by the product t-norm (Larsent-norm). In this case the correlation measures are depend on the γ-level sets.The possibilistic correlation can be determined by an appropriate weightingfunction.

A(x) =


B(y) =


Then a γ-level set of C is,

[C]γ ={(x, y) ∈ R2| 0 ≤ x, y ≤ 1, xy ≥ γ

}.

The area of the γ-level:

Tγ = 1− γ + γ ln γ .


f1(x) =

1

Tγ·x− γx

if γ ≤ x ≤ 1 ,

0 otherwise .

In this case the correlation coefficient, the correlation ratio,the mutual information and the informational coefficient ofcorrelation depend on the level γ. The integrals are quitedifficult (see [23]), so we used numerical integration methods.The results are shown on Fig.6. The limits in zero and in 1are the following (see [23], [24] )

limγ→0

ρ = 0 , limγ→1

ρ = −1/2 ,

limγ→0

η = 0 , limγ→1

η = 1/2 ,

limγ→0

L = 0 , limγ→1

L ≈ 0.6772 .

In this case the measures of correlation depend on γ, sothe weighted possibilistic correlation can be determined bychoosing a weighting function g(γ). If the weighting functionis g(γ) ≡ 1 then:

Lg(A,B) =

∫ 1

0

L(Xγ , Yγ) · 1 dγ ≈ 0.5786

ρg(A,B) =

∫ 1

0

ρ(Xγ , Yγ) · 1 dγ ≈ 0.3888

ηg(A,B) =

∫ 1

0

η(Xγ , Yγ) · 1 dγ ≈ 0.4010

If the weighting function is g(γ) = 2γ then:

Lg(A,B) =

∫ 1

0

L(Xγ , Yγ) · 2γ dγ ≈ 0.6282

ρg(A,B) =

∫ 1

0

ρ(Xγ , Yγ) · 2γ dγ ≈ 0.4441

ηg(A,B) =

∫ 1

0

η(Xγ , Yγ) · 2γ dγ ≈ 0.4490

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

correlation coefficientcorrelation ratioinformational coeff.

Fig. 6. The absolute value of the correlation coefficient, the correlation ratioand the informational coefficient of correlation in function of γ, when thejoint possibility distribution is defined by Larsen t-norm.

VI. CONCLUSION

We have introduced the notion of weighted possibilisticinformational coefficient of correlation between marginal dis-tributions of a joint possibility distribution. This coefficient isequal to zero if and only if the marginal possibility distribu-tions are independent (which does not hold for possibilisticcorrelation coefficient and ratio).

ACKNOWLEDGMENT

This work was supported in part by the project TAMOP421B at the Szechenyi Istvan University, Gyor.

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INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS …naun.org/main/NAUN/ijmmas/16-235.pdfR. Full´er is with John von Neumann Faculty of Informatics, Obuda´ University, Becsi´ ´ut 96/B,