Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 153

the OWA operator is defined by the expression:

n

OWA a 1 ,a 2 ,...,a n =

w i a j i . [8.72]

i =1

We can also consider fuzzy integrals to be included in this class of operators
[GRA 95], since Choquet and Sugeno integrals are idempotent, continuous, increas-
ing and included between the minimum and the maximum. It includes the specific
case of order statistics and therefore the minimum, the maximum and the median. The
Choquet integrals defined with respect to an additive measure μ are equivalent to a
weighted arithmetic mean, in which the weights w i assigned to the values x i are equal
to μ (

{ x i } ).

OWAs can also be interpreted as a particular class of Choquet integrals, where the
fuzzy measure is defined by:

i − 1

∀ A, | A | = i, μ ( A )= w n − j .

j =0

Conversely, any commutative Choquet integral is such that μ ( A ) only depends on

| A | and is equal to an OWA whose weights are given by:

n

w 1 =1 − w i ,

i =2

2 ,w i = μ A n − i +1 − μ A n − i , ∀ i ≥

where A i refers to any subset such that | A i | = i .

A more detailed study of the properties of these operators can be found in
[GRA 92, GRA 95].

8.5.4. Symmetric sums

Symmetric sums are defined by an auto-duality property, which corresponds to the
invariance of the result of the operation by inverting the scale of values to combine.
More specifically, a symmetric sum is a function σ :[0 , 1] × [0 , 1] → [0 , 1] such that:

- σ (0 , 0) = 0;
- σ is commutative;
- σ is increasing with respect to the two variables;
- σ is continuous;
- σ is self-dual:

∀ ∈

[0 , 1] 2 , σ ( x, y )=1

− − −

( x, y ) σ (1 x, 1 y ).