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Generalized Weighted Fuzzy Expected Values in Uncertainty
Environment
ANNA SIKHARULIDZE
GIA SIRBILADZE
I. Javakhishvili Tbilisi State University
I. Javakhishvili Tbilisi State University
Department of Exact and Natural Sciences Department of Exact and Natural Sciences
University St. 2, 0143 Tbilisi
University St. 2, 0143 Tbilisi
GEORGIA
GEORGIA
[email protected]
[email protected]
NATIA SIRBILADZE
I. Javakhishvili Tbilisi State University
Department of Exact and Natural Sciences
University St. 2, 0143 Tbilisi
GEORGIA
[email protected]
Abstract: Two new versions of the most typical value (M T V ) [1, 2, 3, 5, 6, 8, 9] of the population (generalized
weighted averages) are introduced. The first version, W F EVg , is a generalization of the weighted fuzzy expected
value (W F EV ) [4]–[6] for any fuzzy measure g [7] on a finite set and it coincides with the W F EV when a
sampling probability distribution is used. The construction process is based on the Friedman-Schneider-Kandel
(FSK) [4]–[6] principle and the probability representation of a fuzzy measure [6]–[10]. The second, GW F EVg is
the generalization of W F EVg for any fuzzy measure. Furthermore, the generalized weighted fuzzy expected value
is expressed in terms of two monotone expectation (M E) [10] values. The convergence of iteration processes is
provided by an appropriate choice of a “weight” function. Example of the use of the new weighted averages is
discussed. In many cases these averages give better estimations than classical estimators of central tendencies such
as mean, median or the fuzzy “classical” estimators F EV and M E [4, 10].
Key–Words: Fuzzy measure, fuzzy statistics, fuzzy expected value, most typical value, weighted fuzzy expected
value
1
Introduction
representative value among the values of a compatibility function of a fuzzy set [2].
Different authors believe the M T V to be different fuzzy values. In our opinion, there cannot be any
preferable M T V in fuzzy statistics since all of them
are expert estimators and give better or worse results
depending on a specific problem. Our objective here is
to carry out analysis in order to establish which fuzzy
expected value gives a better representation and when.
In this paper we formulate the new definitions
of weighted fuzzy expected value with respect to
any fuzzy measure W F EVg` and the generalized
weighted fuzzy expected Value GW F EVg` and investigate their properties. An example illustrating the
application of generalized fuzzy expected values is
also discussed.
There exist two sources of fuzzy uncertainty: the first
one is when uncertainty is caused by some notion
and the second one when data are inexact, nonclear,
overlapped or when some individual (expert) is involved in representation or reception of data and in
most cases subjectively estimates or, what is worse,
appoints them. In that case data themselves contain
fuzziness. They are of dual probabilistic-possibilistic
nature. In such a situation only classical estimators of
central tendencies cannot be believed to be good estimators. They must be replaced by the corresponding
fuzzy statistical characteristics such as F EV , M E,
W F EV , F EI ([2]–[6], [8]–[10], etc.), by which the
objective-subjective (expert) data are to be evaluated.
Fuzzy statistics originates in the 80s when A.
Kandel [8] called Sugeno’s integral [7] fuzzy statistics
and when the notion of the most typical value (M T V )
of the population corresponding to the fuzzy notion
was introduced. The M T V became the most typical
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2
3
Weighted Fuzzy Expected Value
(W F EV )
M. Friedman, M. Schneider and A. Kandel proposed
a new scheme for calculating the M T V [4], which is
based on a two-factor principle. Consider, for example, the following two population groups:
Group #
i
j
χ
χi
χj
In this section we present a new version of W F EV .
the
variational
sampling
Suppose
(x1 , x2 , . . . , xk )
e
is given, χi = χAe(xi )
(n1 , n2 , . . . , nk )
are the compatibility values of some fuzzy set
e ⊂ {x1 , . . . , xn }. Let w be a non-negative monoA
tonically decreasing function defined over the interval
[0,1] and l > 1 be a real number.
Eq. (1) can be rewritten as
n
ni
nj
where χk is a compatibility level and nk is a frequency
of population element.
Suppose ni > nj . We consider two postulates of
construction of M T V :
s = (χ1 w (|x1 − s|)
2. The effective location of the M T V with respect
to compatibility values: the distance between the
M T V and the compatibility value of the i-th
group |χi − M T V | participates in the definition
of the M T V with weight values proportional to
w (|χi − M T V |), where w is a strictly decreasing function.
2
k
 P
 x ∈A
l
(A) ≡  i
gsampling
k
ni
l
n l

A
,
 =
k
is called a fuzzy measure induced by a sampling distribution.
s = (χ1 w(|χ1 − s|)nl1 + χ2 w(|χ2 − s|)nl2
Then
+ · · · + χk w(|χk − s|)nlk )
× (w(|χ1 − s|)nl1 + w(|χ2 − s|)nl2
l
gsampling
({xi }) =
(1)
n l
i
k
, i = 1, 2, . . . , k.
(3)
It is obvious that during the weighting we conl
sider the values of the measure gsampling
in Eq. (2)
only on the sets of one element (fuzzy “weights” of
sets of one element).
Let X = {x1 , . . . , xk } be a finite set, X, 2X , g
be a fuzzy measure space, χAe be a compatibility
e
function of the
fuzzy subset A, χAe : X → [0; 1]
χi = χAe(xi ) ; let w be some “weight” function and
l > 1 be a real number.
By virtue of Eq. (2) and Definition 2 the following two new postulates of constructing the M T V with
respect to the fuzzy measure g on the set X can be formulated, which in what follows are referred to as the
Friedman-Schneider-Kandel (FSK) principles:
Definition 1 ([4]). The solution of Eq. (1) is called
the weighted fuzzy expected value (W F EV ) of order
l with the attached weight function w of compatibility
values (χ1 , . . . , χk ). M T V ≡ W F EV χAe, w .
The parameter l measures the dependence of
frequencies of population groups on the W F EV .
The rate at which the function w decreases defines
the“closeness” of the W F EV to higher compatibility values of χi . By virtue of the above-mentioned
two-factor principle, the mapping of weighting (1)
is invariant with respect to the M T V which is the
fixed point of the mapping. The examples of use
of W F EV and its composition with F EV are illustrated in [4], [6].
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n l
1
Definition 2. A fuzzy measure, which for any subset
A of the sampling X = {x1 , . . . , xk } is equal to the
l-th power of the frequency of A
a variational
sampling
Suppose
(x1 , x2 , . . . , xk )
is given, χi = χAe(xi )
e
(n1 , n2 , . . . , nk )
are the compatibility values of some fuzzy set
e ⊂ X = {x1 , x2 , . . . , xk }, w(x) is a non-negative
A
monotonically decreasing function defined over the
interval [0, 1] and l > 1 is a real number. Consider
the following equation with respect to s:
+ · · · + w(|χk −
n l
+χ2 w (|x2 − s|)
k
n l
k
+ · · · + χk w (|xk − s|)
)
k
n l
n l
2
1
+ w (|x2 − s|)
× (w (|x1 − s|)
k
k
n l
k
−1
+ · · · + w (|xk − s|)
) . (2)
k
1. Population effectiveness: the M T V must be
“less far” from χi than from χj since ni > nj .
s|)nlk )−1 .
Weighted Fuzzy Expected Value
with respect to a the Fuzzy Measure (W F EVgl )
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1. Fuzzy measure distribution effectiveness: the
M T V is “less far” from χi than from χj if g ({xi }) >
g ({xj }).
2. The effective location of the M T V with respect to compatibility values: a distance between the
M T V and the compatibility values χi of the element
xi ∈ X): |χi − M T V | participates in the definition of the M T V with weight values proportional
to w (|χi − M T V |), where w is a strictly decreasing
function.
Similarly to Eq. (2), let us consider the following
equation with respect to s:
k
P
s=
This is the probabilistic representation
of the W F EVgl by associated probabilities
(l)
(l)
(l)
Pτ1 , Pτ2 , . . . , Pτn of the fuzzy measure g.
Obviously, we can construct the iteration process
for Eq. (5) as we have done for Eq. (1) [6]:
k
P
sn =
w |(χi −
({xi })
4
Definition 3. The solution of Eq. (4) is called the
weighted fuzzy expected value of order l with the attached weight function w of the compatibility function
χ with respect to the fuzzy measure g.
It is denoted by W F EVgl χAe, w , (M T V =
W F EVgl ).
On the set {1, 2, . . . , k} there exist k! permutations.
Denote any permutation by σ =
(σ(1), σ(2), . . . , σ(k)) and the set of all possible permutations by Sk .
is called an associated probability distribution of the
fuzzy measure g l ;
(l)
(l)
(l)
{Pσ }σ∈Sk = {Pσ (xσ(1) ), . . . , Pσ (xσ(k) )}σ∈Sk is
called the class of associated probabilities of the fuzzy
measure g l .
It is known that ∀xi ⊂ X, ∃τi ∈ Sk permutation
such that [6]
(l)
g ` ({xi }) = Pτ(l)
(x
)
≡
P
x
.
i
τ
(1)
τ
i
i
i
Now Eq. (4) takes the form
s=
(l)
i=1
k
P
xτi (1)
.
(l)
w |(χi − s)| Pτi
i=1
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xτi (1)
,
w |(χi −
(l)
sn−1 )| Pτi
xτi (1)
Generalized Weighted Fuzzy Expected Value (GW F EV )
W EF Vg` (Eq. (5)) is constructed according to the
Friedman-Schneider-Kandel (FSK) principle when on
the finite set we consider not the probabilistic, but the
fuzzy measure g ` . The first postulate of FSK principle concerns effectiveness of distribution of the fuzzy
measure g in the “weighting” procedure. In (4) this is
represented by “fuzzy weights” g ` ({xi }) of singleton
sets {xi }, i = 1, 2, . . . , k. The second postulate states
that the M T V has a location such that it is close to the
values of “high” compatibility. This is represented in
the “weighting process” by weights w(|χi − M T V |).
Both postulates are combined into a normalized sum
χi w(|χi − s|)g ` ({xi }) presented in Eq. (4). This
means that M T V ≡ W F EVg` (the solution of (5)
is invariant with respect to the “weight” function constructed by the FSK principle which is certainly understandable and justified.
As the authors of the FSK principle [4] admit, the
two postulates of the “weighting” process cannot be
perfect. Some other postulates can also be added. We
think that in the case of the first postulate the “weighting” process has one drawback. The values g ` ({xi })
of the fuzzy measure g ` are considered only on singleton sets {xi }, i = 1, 2, . . . , k, while the values of
the same measure on the subsets of X with two or
more elements are not. The point is that the values
of compatibility χ1 , χ2 , . . . , χk are described by the
consonant body (Theorem of Decomposition), i.e., by
focal subsets of the form {xi ∈ X | χAe(xi ) ≥ α},
0 < α < 1. But if we consider the second postulate of the FSK principle, w(|χi − M T V |), i =
1, 2, . . . , n weights, then it becomes obvious that the
values of the fuzzy measure g ` on the cuts α-level
{xi ∈ X | χAe(xi ) ≥ α}, 0 ≤ α < 1 of participate
in the ”weighting” process. This strengthens the first
postulate. If we consider the condition of normalizing
Definition 4 ([10]). If σ ∈ Sk is a permutation, then
the following probability distribution
(l)
Pσ xσ(1) = g l xσ(1) ,
(l)
Pσ xσ(2) = g l xσ(1) , xσ(2) − g l xσ(1) ,
..................
(l)
Pσ xσ(i) = g l xσ(1) , . . . ,
xσ(i)
−g l xσ(1) , . . . , xσ(i−1) ,
..................
(l)
Pσ xσ(k) = 1 − g l xσ(1) , . . . , xσ(k−1) .
χi w |(χi − s)| Pτi
(4)
i=1
k
P
xτi (1)
where s0 = F EV χAe , F EV is Sugeno’s integral
of a χAe with respect to a fuzzy measure g on 2X .
.
s)| g l
i=1
k
P
i=1
χi w |(χi − s)| g l ({xi })
i=1
k
P
(l)
χi w |(χi − sn−1 )| Pτi
(5)
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of the “weighting” process, then the invariant form of
the “weighting” function is defined by the ratio of two
monotone expectations (see (8)). All this leads us to
the definition of a new weighted fuzzy average which
can be defined generally for any fuzzy measure space.
Suppose X is any universe, and (X, F, g) is any
fuzzy measure space, χAe is a measurable compatibile χe : X →
ity function F of the fuzzy subset A,
A
[0, 1]. The monotone expectation of χAe with respect
to the fuzzy measure g as Choquet integral is written
as [10]:
Z1
Eg (χAe) =
In practice, for a numerical solution of (8) the following simple iteration process is used:
s(N +1) = f (s(N ) ),
A
i.e. for Eq. (8) we have the iteration process
g {x ∈ X | χAe ≥ α} dα
s
0
g ` ({χβe(x)w(|χβe(x) − s|) ≥ α})dα
0
σ1
×
k
X
(i)
σ1
(i)
− s(N ) |)
)
(x
(N )
(N )
(i)
σ
(`)
σ1
1
w(|χβ (N )
i=1
σ2
·P
(i)
(`)
(N )
σ2
− s(N ) |)
−1
(xσ(N ) (i) )
, (10)
2
(N )
(N )
where the permutations σ1 = σ1 and σ2 = σ2
depend on the iteration step N and thus
A
A
R1
χβ (N ) w(|χβ (N )
·P
Definition 5. If w(·) is a positive “weight” function,
strictly decreasing, with values in the interval [0, 1]
and ` ≥ 1, β > 0 are real numbers, then the solution
of the equation (with respect to s)
s=
=
k
X
(6)
where Hα is a fuzzy subset of α level of χAe; g(Hα )
is said to be a measure function of χAe.
R1
(N +1)
i=1
g(Hα )dα,
A
X
Z
=
(9)
where s(0) is the fuzzy expected value
Z
β
(0)
s ≡ F EVg` (χ e) =6 χβe ◦ g ` ,
0
def
N = 0, 1, . . . ,
,
(7)
f (s(N ) ) ≡ Eg` χβew(|χβe − s(N ) |)
A
g ` ({w(|χβe(x) − s|) ≥ α})dα
A
Eg` w(|χβe − s(N ) |) . (11)
A
A
is called the generalized weighted fuzzy expected
value of power β of the compatibility function χAe with
weight w and with respect to the fuzzy measure g ` .
It can be easily shown that if the function w is chosen
effectively [6], the function f has compression in the
neighborhood of s(0) and the convergence of (11) is
guaranteed:
We denote this value by
lim s(N ) = GW F EVg`β .
GW F EVg`β ≡ GW F EVg` (χβe, w).
N →∞
A
In terms of the Monotone Expectation Eq. (7) can
be rewritten as
s = Ege χβew(|χβe − s|) Ege w(|χβe − s|) . (8)
A
A
Following [6], the function w(t) = e−λt is frequently
used in the role of w (where λ is some positive parameter).
A
Proposition 1 (without proof). If X
=
{x1 , x2 , . . . , xk } is a finite set and β > 1, ` > 1,
are real numbers, χi ≡ χAe(xi ) are values of the
e ⊂ X, then
compatibility function of a fuzzy subset A
there exist two probabilistic distributions P1 and P2
on X with respect to which Eq. (8) takes the form of
a ratio of mathematical expectations
EP1 χβew(|χβe − s|)
A
A
s=
.
EP2 w(|χβe − s|)
5
Let us consider an example of the use of the
GW F EVg` which is a solution of Eq. (8) (β = 1,
` = 2, w(|t|) = e−|t| , S0 = F EV ).
Example 1. Suppose X = {x1 , x2 , x3 } and Fe is a
fuzzy subset
Fe = {0.4/x1 , 0.6/x2 , 0.8/x3 },
A
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Example
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Table 1: Associated probabilities of g `
(`)
σ ∈ S3
(1,2,3)
(1,3,2)
(2,1,3)
(2,3,1)
(3,1,2)
(3,2,1)
Pσ (x1 )
(1/3)`
(1/3)`
`
(2/3) − (1/3 − ε)` )
1 − (2/3 − ε)`
(2/3 − ε)` − (1/3)` )
1 − (2/3 − ε)`
(`)
Pσ (x2 )
(2/3)` − (1/3)` )
1 − (2/3 − ε)`
(1/3 − ε)`
(1/3 − ε)`
1 − (2/3 − ε)`
(2/3 − ε)` − (1/3)`
Calculations yield

1
,
if 0 < ε ≤ 15
 0.6
1
4
2
−
ε
if
<
ε
≤
F EV (χFe ) =
15
15 ,
 3
4
0.4
if 15
< ε ≤ 13 .
Table 2: Distributions of g on X
B⊂X
∅
{x1 }
{x2 }
{x3 }
{x1 , x2 }
{x1 , x3 }
{x2 , x3 }
{x1 , x2 , x3 }
g(B)
0
1/3
1/3 − ε
1/3
2/3
2/3 − ε
2/3 − ε
1
Note that when ε = 0, g is a probabilistic measure (a uniform distribution on X) and W F EVg =
W F EV . Because of small values of ε it makes sense
to calculate the mean:
1
X
− ε 0.6 .
mean =
g({xi })χFe (xi ) = 0.4 +
3
and also the fuzzy measure g is given on X, whose
associated probabilities are shown in the table below
(0 < ε < 1/3 is a parameter) (Table 1), if the distribution of g takes the form from Table 2.
Now according to (10) we obtain the
following iteration process for calculating
GW F EVg2 :
i
If we give a small step to the parameter ε
and gather calculations of mean, F EV , W F EVg2 ,
GW F EVg2 in one chart (see Chart 1), then it becomes obvious that for ε ≈ 0 all three M T V s
are close to 0.6 (except only GW F EVg2 ), but
χ−1 (M T V ) = x2 and thus x2 becomes the most
typical element of the set X (population). But here
a high level of the compatibility of x3 in Fe − 0.8 is
not taken into account. With the growth of ε, the values of mean and F EV decrease, which means that
M T V → 0.4 and χ−1
(M T V ) → x1 , or x1 beFe
comes a new typical element from the population X,
which does not consider levels of the compatibility of
x2 −0.6 and x3 −0.8 in Fe. We certainly cannot regard
this as a good representation. We have a different situation with the W F EVg2 because W F EVg2 → 0.56,
i.e., W F EVg2 retains closeness to 0.6 though it tends
to 0.5, i.e., it takes into account high levels of the compatibility of x2 and x3 . Obviously, in this case x1 ∪ x2
is the most typical element of the population X, which
is better seen in the W F EVg2 than in the F EV or the
mean. Note that here the fuzzy measure g corrects
the decision for which the fuzzy averaging has been
carried out.
For ε ≈ 0, x3 has the highest compatibility level
0.8, and the distribution of the fuzzy measure g gives
preference almost to none of the elements (Table 2).
This means that χ−1 (M T V ) must be shifted towards
x3 and the compatibility values of x1 and x2 must be
considered and therefore GW F EVg2 > W F EVg2 .
With the growth of ε the role of x3 decreases and the
sN +1 = (0.4e−|0.4−sN | Pσ(2)
(x1 )
1
+ 0.6e−|0.6−sN | Pσ(2)
(x2 ) + 0.8e−|0.8−sN | Pσ(2)
(x3 ))
1
1
× (e−|0.4−sN | Pσ(2)
(x1 ) + e−|0.6−sN | Pσ(2)
(x2 )
1
1
+ e−|0.8−sN | Pσ(2)
(x3 ))−1 ,
1
where s0 ≡ F EV (χFe ).
(2)
Following (8), the probabilities Pσj (xi ), j =
1, 2; i − 1, 2, 3, depend on SN and for any N we
(2)
choose different probabilities Pσj from the associated
probabilities of the fuzzy measure g 2 (Table 1).
Now we obtain the following iteration process for
calculating the W F EVg2 :
sN +1 = (0.4e−|0.4−sN | (1/3)2
+ 0.6e−|0.6−sN | (1/3 − ε)2 + 0.8e−|0.8−sN | (1/3)2 )
× (e−|0.4−sN | (1/3)2 + e−|0.6−sN | (1/3 − ε)2
+ e−|0.8−sN | (1/3)2 )−1 ,
where s0 = F EV .
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(`)
Pσ (x3 )
1 − (2/3)`
(2/3 − ε)` − (1/3)`
1 − (2/3)`
(2/3 − ε)` − (1/3 − ε)` )
(1/3)`
(1/3)`
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average value becomes χ−1 (M T V ) = x2 . When
ε → 1/3, the role of x2 decreases, the role of x3 is
also slightly weakened, though its compatibility value
is the highest one and must be considered anyway. But
the role of x1 increases. This is best of all described
in GW F EVg2 .
(Grant #GNSF/ST08/1-361). Any idea in this publication is possessed by the author and may not represent
the opinion of the Georgian National Science Foundation itself.
References:
[1] D. Dubois, H. Prade, Théorie des Possibilités.
Applications á la représentation des connaissances en informatique, Paris, Milan, Barcelone,
Mexico, 1988.
[2] M. Friedman, M. Henne, A. Kandel, Most typical values for fuzzy sets. Fuzzy sets and Systems
87, 1997, pp. 27–37.
[3] M. Schneider, M. Friedman, A. Kandel, On
fuzzy reasoning in expert systems. Proc. 1987
International Symposium on Multiple-Valued
Logic, Boston, MA, 1987.
[4] M. Friedman, M. Schneider, A. Kandel, The use
of weighted fuzzy expected value (WFEV) in
fuzzy expert systems. Fuzzy Sets and Systems
31, 1989, pp. 37–45.
[5] M. Schneider, A. Kandel, Properties of the fuzzy
expected value and the fuzzy expected interval in
fuzzy environment. Fuzzy Sets and Systems 28,
1988, pp. 55–68.
[6] G. Sirbiladze, A. Sikharulidze, Weighted Fuzzy
Averages in Fuzzy Environment, Parts I, II,
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2(2), 2003,
pp. 139–157, 159–172.
[7] M. Sugeno, Theory of fuzzy integrals and its applications. Ph. D. Thesis, Tokyo Institute of Technology, 1974.
[8] A. Kandel, Fuzzy statistics and forecast evaluation. IEEE Trans. Systems Man Cybernet. SMC8, 1978, no. 5, 396–401.
[9] A. Kandel, On the control and evaluation of uncertain processes. IEEE Trans. Automat. Control
25, 1980, no. 6, 1182–1187.
[10] I. Campos, C. Bolanos, Representation of Fuzzy
Measures Through Probabilities. Fuzzy Sets and
Systems 31, 1989, pp. 23–36.
Chart 1.
6
Conclusion
The paper is mainly of descriptive character. The existing weighted fuzzy expected value is generalized
for any fuzzy measure (GW F EVg`β ). The obtained
new weighted expected values actually represent iteration processes whose convergence is guaranteed by
an appropriate choice of a weight function. Propositions on the correctness of generalization are formulated.
Generalized weighted average may be defined using Sugeno integral [7] as an aggregation instrument
with respect to FSK principle
s=
R
(s) {χβe(x)w(|χβe(x) − s|)} ◦ g ` (·)
A
X
A
(s) {w(|χβe(x) − s|)} ◦ g ` (·)
R
X
A
or using any other aggregation instrument. An interesting problem is to investigate all aggregated
GW EF V on examples and choose an optimal statistics.
Generalized weighted averages can be successfully used in expert systems (for example [3]), where
F EV and W F EV are used but cannot be estimated
because of data scarcity.
Acknowledgement
The designated project has been fulfilled by financial
support of the Georgian National Science Foundation
ISSN: 1790-5109
64
ISBN: 978-960-474-154-0