Some illustrations of possibilistic correlation1
Robert Fullér
IAMSR, Åbo Akademi University,
Joukahaisenkatu 3-5 A, FIN-20520 Turku
e-mail: rfuller@abo.fi
József Mezei
Turku Centre for Computer Science,
Joukahaisenkatu 3-5 B, FIN-20520 Turku
e-mail: jmezei@abo.fi
Péter Várlaki
Budapest University of Technology and Economics,
Bertalan L. u. 2, H-1111 Budapest, Hungary
and
Széchenyi István University,
Egyetem tér 1, H-9026 Győr, Hungary
e-mail: varlaki@kme.bme.hu
Abstract: In this paper we will show some examples for possibilistic correlation.
In particular, we will show (i) how the possibilistic correlation coefficient of two
linear marginal possibility distributions changes from zero to -1/2, and from -1/2
to -3/5 by taking out bigger and bigger parts from the level sets of a their joint
possibility distribution; (ii) how to compute the autocorrelation coefficient of fuzzy
time series with linear fuzzy data.
1
Introduction
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.
1
Appeared in: Robert Fullér, József Mezei and Péter Várlaki, Proceedings of the Tenth International Symposium of Hungarian Researchers on Computational Intelligence and Informatic (CINTI
2009), [ISBN 978-963-7154-96-6], November 12-14, 2009, Budapest, Hungary, pp. 647-658.
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. Let A ∈ F be fuzzy number with a γlevel 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].
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.
Namely, we 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.
These weights (or importances) can be given by weighting functions. A function
f : [0, 1] → R is said to be a weighting function if f is non-negative,
monotone
R1
increasing and satisfies the following normalization condition 0 f (γ)dγ = 1.
Different weighting functions can give different (case-dependent) importances to
level-sets of possibility distributions.
In 2009 we introduced a new definition of possibilistic correlation coefficient (see
[2]) that improves the earlier definition introduced by Carlsson, Fullér and Majlender in 2005 (see [1]).
Definition ([2]). The f -weighted possibilistic correlation coefficient of A, B ∈ F
(with respect to their joint distribution C) is defined by
Z 1
ρf (A, B) =
ρ(Xγ , Yγ )f (γ)dγ
(1)
0
where
cov(Xγ , Yγ )
p
ρ(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.
In other words, the f -weighted possibilistic correlation coefficient is nothing else,
but the f -weighted average of the probabilistic correlation coefficients ρ(Xγ , Yγ )
for all γ ∈ [0, 1].
Consider the case, when A(x) = B(x) = (1 − x) · χ[0,1] (x), for x ∈ R, that
is [A]γ = [B]γ = [0, 1 − γ], for γ ∈ [0, 1]. Suppose that their joint possibility
distribution is given by F (x, y) = (1 − x − y) · χT (x, y), where
T = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 .
Figure 1: Illustration of joint possibility distribution F .
Then we have [F ]γ = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 − γ .
This situation is depicted on Fig. 1, where we have shifted the fuzzy sets to get a
better view of the situation. In this case the f -weighted possibilistic correlation of
A and B is computed as (see [2] for details),
Z 1
1
1
− f (γ)dγ = − .
ρf (A, B) =
2
2
0
Consider now the case when A(x) = B(x) = x · χ[0,1] (x) for x ∈ R, that is
[A]γ = [B]γ = [γ, 1], for γ ∈ [0, 1]. Suppose that their joint possibility distribution
is given by
W (x, y) = max{x + y − 1, 0}.
Then we get
Z
ρf (A, B) = −
0
1
1
1
f (γ)dγ = − .
2
2
We note here that W is nothing else but the Lukasiewitz t-norm, or in the statistical
literature, W is generally referred to as the lower Fréchet-Hoeffding bound for
copulas.
2
A transition from zero to -1/2
Suppose that a family of joint possibility distribution of A and B (where A(x) =
B(x) = (1 − x) · χ[0,1] (x), for x ∈ R) is defined by
n−1
y+x≤1
n
n
−
1
Cn (x, y) =
x+y ≤1
1 − n−1

n x − y, if 0 ≤ y ≤ 1, y ≤ x,

n

0,
otherwise



 1−x−
n−1
n y,
if 0 ≤ x ≤ 1, x ≤ y,
In the following, for simplicity, we well write C instead of Cn . A γ-level set of C
is computed by
[
n
n−1
(1 − γ), 0 ≤ y ≤ 1 − γ −
x
[C]γ = (x, y) ∈ R2 | 0 ≤ x ≤
2n − 1
n
n
n−1
2
(x, y) ∈ R |
(1 − γ) ≤ x ≤ 1 − γ, 0 ≤
y ≤1−γ−x .
2n − 1
n
The density function of a uniform distribution on [C]γ can be written as


γ
 R 1
 2n − 1 , if (x, y) ∈ [C]γ
, if (x, y) ∈ [C]
f (x, y) =
=
n(1 − γ)2
γ dxdy
 [C]
 0
otherwise
0
otherwise
We can calculate the marginal density functions:

n
(2n − 1)(1 − γ − x)


,
if
(1 − γ) ≤ x ≤ 1 − γ

2

2n − 1
 (n − 1)(1 − γ)
(2n − 1)(1 − γ − n−1
n
f1 (x) =
n x)
, if 0 ≤ x ≤
(1 − γ)


2

n(1 − γ)
2n − 1


0
otherwise
and,

(2n − 1)(1 − γ − y)
n


,
if
(1 − γ) ≤ y ≤ 1 − γ

2

2n − 1
 (n − 1)(1 − γ)
(2n − 1)(1 − γ − n−1
n
f2 (y) =
n y)
, if 0 ≤ y ≤
(1 − γ)


2

n(1
−
γ)
2n
−1


0
otherwise
We can calculate the probabilistic expected values of the random variables Xγ and
Yγ , whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1] as,
Z n(1−γ)
2n−1
2n − 1
n−1
M (Xγ ) =
x(1 − γ −
x)dx
2
n(1 − γ) 0
n
Z 1−γ
2n − 1
(1 − γ)(4n − 1)
+
x(1 − γ − x)dx =
2
n(1−γ)
(n − 1)(1 − γ)
6(2n − 1)
2n−1
and, M (Yγ ) =
(1 − γ)(4n − 1)
.
6(2n − 1)
1−γ
, and for n → ∞ we
(We can easily see that for n = 1 we have M (Xγ ) =
2
1−γ
find M (Xγ ) →
.) We calculate the variations of Xγ and Yγ as,
3
M (Xγ2 )
Z n(1−γ)
2n−1
2n − 1
n−1
=
x)dx
x2 (1 − γ −
2
n(1 − γ) 0
n
Z 1−γ
2n − 1
+
x2 (1 − γ − x)dx
(n − 1)(1 − γ)2 n(1−γ)
2n−1
(1 − γ)2 ((2n − 1)3 + 8n3 − 6n2 + n)
=
.
12(2n − 1)3
(We can easily see that for n = 1 we get M (Xγ2 ) =
find M (Xγ2 ) →
(1 − γ)2
.) Furthermore,
6
(1 − γ)2
, and for n → ∞ we
3
(1 − γ)2 ((2n − 1)3 + 8n3 − 6n2 + n)
12(2n − 1)3
(1 − γ)2 (4n − 1)2
(1 − γ)2 (2(2n − 1)2 + n)
=
.
−
36(2n − 1)2
36(2n − 1)2
var(Xγ ) = M (Xγ2 ) − M (Xγ )2 =
And similarly we obtain
var(Yγ ) =
(1 − γ)2 (2(2n − 1)2 + n)
.
36(2n − 1)2
(We can easily see that for n = 1 we get var(Xγ ) =
(1 − γ)2
, and for n → ∞ we
12
find var(Xγ ) →
(1 − γ)2
.) And,
18
cov(Xγ , Yγ ) = M (Xγ Yγ ) − M (Xγ )M (Yγ )
=
(1 − γ)2 n(4n − 1) (1 − γ)2 (1 − n)(4n − 1)
−
.
12(2n − 1)2
36(2n − 1)2
(We can easily see that for n = 1 we have cov(Xγ , Yγ ) = 0, and for n → ∞ we
(1 − γ)2
find cov(Xγ , Yγ ) → −
.) We can calculate the probabilisctic correlation
36
of the random variables,
cov(Xγ , Yγ )
(1 − n)(4n − 1)
p
=
ρ(Xγ , Yγ ) = p
.
2(2n − 1)2 + n
var(Xγ ) var(Yγ )
(We can easily see that for n = 1 we have ρ(Xγ , Yγ ) = 0, and for n → ∞ we find
1
ρ(Xγ , Yγ ) → − .) And finally the f -weighted possibilistic correlation of A and
2
B is computed as,
Z 1
(1 − n)(4n − 1)
ρf (A, B) =
ρ(Xγ , Yγ )f (γ)dγ =
.
2(2n − 1)2 + n
0
1
We obtain, that ρf (A, B) = 0 for n = 1 and if n → ∞ then ρf (A, B) → − .
2
3
A transition from -1/2 to -3/5
Suppose that the joint possibility distribution of A and B (where A(x) = B(x) =
(1 − x) · χ[0,1] (x), for x ∈ R) is defined by
Cn (x, y) = (1 − x − y) · χTn (x, y),
where
[
1
Tn = (x, y) ∈ R | x ≥ 0, y ≥ 0, x + y ≤ 1,
x≥y
n−1
(x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1, (n − 1)x ≤ y .
2
In the following, for simplicity, we well write C instead of Cn . A γ-level set of C
is computed by
[
1
γ
2
[C] = (x, y) ∈ R | 0 ≤ x ≤ (1 − γ), (n − 1)x ≤ y ≤ 1 − γ − x
n
1
(x, y) ∈ R | 0 ≤ y ≤ (1 − γ), (n − 1)y ≤ x ≤ 1 − γ − y .
n
2
The density function of a uniform distribution on [C]γ can be written as

(
n
 R 1
, if (x, y) ∈ [C]γ
, if (x, y) ∈ [C]γ
2
dxdy
(1
−
γ)
=
f (x, y) =
γ
 [C]
0
otherwise
0
otherwise
We can calculate the marginal density functions:

x
)
n(1 − γ − nx + n−1
1−γ



,
if 0 ≤ x ≤

2

(1 − γ)
n



nx
(1 − γ)
(n − 1)(1 − γ)

,
if
≤x≤
f1 (x) =
(1 − γ)2 (n − 1)
n
n


(n
−
1)(1
−
γ)
n(1
−
γ
−
x)


,
if
≤x≤1−γ



(1 − γ)2
n


0
otherwise
and,

y
)
n(1 − γ − ny + n−1
1−γ



,
if 0 ≤ y ≤

2

(1
−
γ)
n



ny
(1 − γ)
(n − 1)(1 − γ)

,
if
≤y≤
f2 (y) =
(1 − γ)2 (n − 1)
n
n


(n
−
1)(1
−
γ)
n(1
−
γ
−
y)


,
if
≤y ≤1−γ



(1 − γ)2
n


0
otherwise
We can calculate the probabilistic expected values of the random variables Xγ and
Yγ , whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1] as
n
M (Xγ ) =
(1 − γ)2
Z
n
+
(1 − γ)2
Z
=
1−γ
.
3
1−γ
n
x(1 − γ − nx +
0
(n−1)(1−γ)
n
1−γ
n
x
)dx
n−1
x2
n
dx +
n−1
(1 − γ)2
Z
1−γ)
(n−1)(1−γ)
n
x(1 − γ − x)dx
1−γ
. We calculate the variations of Xγ and Yγ as,
3
Z 1−γ
n
x
n
2
x2 (1 − γ − nx +
)dx
M (Xγ ) =
2
(1 − γ) 0
n−1
Z (n−1)(1−γ)
n
x3
n
+
dx
(1 − γ)2 1−γ
n−1
n
Z 1−γ)
n
+
x2 (1 − γ − x)dx
(1 − γ)2 (n−1)(1−γ)
That is, M (Yγ ) =
n
=
(1 −
γ)2 (3n2
− 3n + 2)
.
12n2
and,
(1 − γ)2 (3n2 − 3n + 2) (1 − γ)2
−
12n2
9
2
2
(1 − γ) (5n − 9n + 6)
=
.
36n2
And, similarly, we obtain
var(Xγ ) = M (Xγ2 ) − M (Xγ )2 =
var(Yγ ) =
(1 − γ)2 (5n2 − 9n + 6)
.
36n2
From
(1 − γ)2 (3n − 2) (1 − γ)2
−
,
12n2
9
we can calculate the probabilisctic correlation of the reandom variables:
cov(Xγ , Yγ ) = M (Xγ Yγ ) − M (Xγ )M (Yγ ) =
cov(Xγ , Yγ )
−3n2 + 7n − 6
p
ρ(Xγ , Yγ ) = p
=−
.
5n2 − 9n + 6
var(Xγ ) var(Yγ )
And finally the f -weighted possibilistic correlation of A and B:
Z 1
−3n2 + 7n − 6
ρf (A, B) =
ρ(Xγ , Yγ )f (γ)dγ = −
5n2 − 9n + 6
0
We obtain, that for n = 2
1
ρf (A, B) = − ,
2
and if n → ∞, then
3
ρf (A, B) → − .
5
We note that in this extremal case the joint possibility distribution is nothing else
but the marginal distributions themselves, that is, C∞ (x, y) = 0, for any interior
point (x, y) of the unit square.
4
A trapezoidal case
Consider the case, when




x,
if 0 ≤ x ≤ 1
1,
if 1 ≤ x ≤ 2
A(x) = B(x) =
3 − x, if 2 ≤ x ≤ 3



0,
otherwise
for x ∈ R, that is [A]γ = [B]γ = [γ, 3 − γ], for γ ∈ [0, 1]. Suppose that their joint
possibility distribution is given by:

y, if 0 ≤ x ≤ 3, 0 ≤ y ≤ 1, x ≤ y, x ≤ 3 − y



1,
if 1 ≤ x ≤ 2, 1 ≤ y ≤ 2, x ≤ y
C(x, y) =
x, if 0 ≤ x ≤ 1, 0 ≤ y ≤ 3, y ≤ x, x ≤ 3 − y



0,
otherwise
Then [C]γ = (x, y) ∈ R2 | γ ≤ x ≤ 3 − γ, γ ≤ y ≤ 3 − x .
The density function of a uniform distribution on [F ]γ can be written as


2

 R 1
, if (x, y) ∈ [C]γ
, if (x, y) ∈ [C]γ
2
dxdy
=
f (x, y) =
(3
−
2γ)
γ

 [C]
0
otherwise
0
otherwise
The marginal functions are obtained as

 2(3 − γ − x)
, if γ ≤ x ≤ 3 − γ
f1 (x) =
(3 − 2γ)2

0
otherwise
and,

 2(3 − γ − y)
, if γ ≤ y ≤ 3 − γ
f2 (y) =
(3 − 2γ)2

0
otherwise
We can calculate the probabilistic expected values of the random variables Xγ and
Yγ , whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1] :
Z 3−γ
2
γ+3
x(3 − γ − x)dx =
M (Xγ ) =
2
(3 − 2γ) γ
3
and,
M (Yγ ) =
2
(3 − 2γ)2
Z
3−γ
y(3 − γ − y)dx =
γ
γ+3
3
We calculate the variations of Xγ and Yγ from the formula
var(X) = M (X 2 ) − M (X)2
as
M (Xγ2 ) =
2
(3 − 2γ)2
Z
3−γ
x2 (3 − γ − x)dx =
γ
2γ 2 + 9
6
and,
var(Xγ ) = M (Xγ2 ) − M (Xγ )2 =
2γ 2 + 9 (γ + 3)2
(3 − 2γ)2
−
=
.
6
9
18
And similarly we obtain
var(Yγ ) =
(3 − 2γ)2
.
18
Using the relationship,
cov(Xγ , Yγ ) = M (Xγ Yγ ) − M (Xγ )M (Yγ ) = −
(3 − 2γ)2
,
36
we can calculate the probabilisctic correlation of the random variables:
cov(Xγ , Yγ )
1
p
ρ(Xγ , Yγ ) = p
=− .
2
var(Xγ ) var(Yγ )
And finally the f -weighted possibilistic correlation of A and B is,
Z 1
1
1
ρf (A, B) = −
f (γ)dγ = − .
2
0 2
5
Time Series With Fuzzy Data
A time series with fuzzy data is referred to as fuzzy time series (see [3]). Consider
a fuzzy time series indexed by t ∈ (0, 1],
(
x
1 − , if 0 ≤ x ≤ t
1, if x = 0
At (x) =
and A0 (x) =
t
0
otherwise
0
otherwise
It is easy to see that in this case,
[At ]γ = [0, t(1 − γ)], γ ∈ [0, 1].
If we have t1 , t2 ∈ [0, 1], then the joint possibility distribution of the corresponding
fuzzy numbers is given by:
x
y
C(x, y) = 1 − −
· χT (x, y),
t1 t2
where
x
y
T = (x, y) ∈ R | x ≥ 0, y ≥ 0, +
≤1 .
t1 t2
y
x
γ
2
≤1−γ .
Then [C] = (x, y) ∈ R | x ≥ 0, y ≥ 0, +
t1 t2
The density function of a uniform distribution on [C]γ can be written as

 R 1
, if (x, y) ∈ [C]γ
dxdy
f (x, y) =
γ
 [C]
0
otherwise
That is,
2


2
, if (x, y) ∈ [C]γ
f (x, y) =
t1 t2 (1 − γ)2
 0
otherwise
The marginal functions are obtained as

x
 2(1 − γ − t1 )
, if 0 ≤ x ≤ t1 (1 − γ)
f1 (x) =
t1 (1 − γ)2

0
otherwise
and,

y
 2(1 − γ − t2 )
, if 0 ≤ y ≤ t2 (1 − γ)
f2 (y) =
t2 (1 − γ)2

0
otherwise
We can calculate the probabilistic expected values of the random variables Xγ and
Yγ , whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1] :
2
t1 (1 − γ)2
Z
2
M (Yγ ) =
t2 (1 − γ)2
Z
M (Xγ ) =
t1 (1−γ)
x(1 − γ −
x
t1 (1 − γ)
)dx =
t1
3
y(1 − γ −
y
t2 (1 − γ)
)dx =
.
t2
3
0
and
0
t2 (1−γ)
We calculate now the variations of Xγ and Yγ as,
M (Xγ2 )
2
=
t1 (1 − γ)2
t1 (1−γ)
Z
x2 (1 − γ −
0
x
t2 (1 − γ)2
)dx = 1
t1
6
and,
var(Xγ ) = M (Xγ2 ) − M (Xγ )2 =
t2 (1 − γ)2
t21 (1 − γ)2 t21 (1 − γ)2
−
= 1
.
6
9
18
And, in a similar way, we obtain,
var(Yγ ) =
t22 (1 − γ)2
.
18
From,
t1 t2 (1 − γ)2
,
36
we can calculate the probabilisctic correlation of the reandom variables,
cov(Xγ , Yγ ) = −
cov(Xγ , Yγ )
1
p
=− .
ρ(Xγ , Yγ ) = p
2
var(Xγ ) var(Yγ )
The f -weighted possibilistic correlation of At1 and At2 ,
Z
ρf (At1 , At2 ) =
0
1
1
1
− f (γ)dγ = − .
2
2
So, the autocorrelation function of this fuzzy time series is constant. Namely,
1
R(t1 , t2 ) = − ,
2
for all t1 , t2 ∈ [0, 1].
References
[1] C. Carlsson, R. Fullér and P. Majlender, On possibilistic correlation, Fuzzy
Sets and Systems, 155(2005) 425-445.
[2] R. Fullér, J. Mezei and P. Várlaki, An improved index of interactivity for fuzzy numbers, Fuzzy Sets and Systems (to appear). doi:
10.1016/j.fss.2010.06.001
[3] B. Möller, M. Beer and U. Reuter, Theoretical Basics of Fuzzy Randomness
- Application to Time Series with Fuzzy Data, In: Safety and Reliability of
Engineering Systems and Structures - Proceedings of the 9th Int. Conference
on Structural Safety and Reliability, edited by G. Augusti and G.I. Schueller
and M. Ciampoli. Millpress, Rotterdam, pages 1701-1707.