Information Technology Reference
In-Depth Information
n
:[
,
]
ₒ[
,
]
(
x
1
,...,
x
n
)
O
0
1
0
1
is an OWA, if
O
is obtained under the process,
i
=
1
n
•
Select weights
p
1
,...,
p
n
in
[
0
,
1
]
, such that
p
i
=
1.
x
1
,...,
x
n
)
such that
x
1
x
n
)
•
(
x
1
,...,
x
n
)
(
···
Permute the n-pla
, to the n-pla
i
=
1
n
x
i
•
O
(
x
1
,...,
x
n
)
=
p
i
·
.
For example, if
n
=
2,
O
(
x
1
,
x
2
)
=
p
1
·
min
(
x
1
,
x
2
)
+
p
2
·
max
(
x
1
,
x
2
),
with
p
1
+
p
2
=
1
.
If
n
=
4, and the weights are
(
0
.
2
,
0
.
4
,
0
.
3
,
0
.
1
)
,itis
O
(
0
.
2
,
0
.
5
,
0
.
7
,
0
.
3
)
=
O
(
0
.
2
,
0
.
3
,
0
.
5
,
0
.
7
)
=
0
.
2
×
0
.
2
+
0
.
4
×
0
.
3
+
0
.
3
×
0
.
5
+
0
.
1
×
0
.
7
=
0
.
74
.
2.3.3 More on Aggregations
Because they are associative, continuous t-norms and continuous t-conorms can be
extended to n-dimensional aggregation functions. For example, with
n
=
3,
T
(
x
1
,
x
2
,
x
3
)
=
T
(
x
1
,
T
(
x
2
,
x
3
))
=
T
(
T
(
x
1
,
x
2
),
x
3
))
= ···
S
(
x
1
,
x
2
,
x
3
)
=
S
(
x
1
,
S
(
x
2
,
x
3
))
=
S
(
S
(
x
1
,
x
2
),
x
3
))
= ···
Nevertheless, not all aggregation functions are associative. For example, if
M
is the arithmetic mean,
M
2
x
1
+
x
2
+
x
3
(
x
1
,
M
(
x
2
,
x
3
))
=
,but
M
(
M
(
x
1
,
x
2
),
x
3
))
=
4
x
1
+
x
2
+
2
x
3
. Concerning means, the only associative are min, and max.
4
In
general,
Aggregation
Functions
are
not
commutative.
For
example,
a
2-dimensional quasi-linear mean
f
−
1
M
(
x
1
,
x
2
)
=
(
p
1
f
(
x
1
)
+
p
2
f
(
x
2
)),
p
1
+
p
2
=
1
,
1
is commutative if and only if
p
1
2
. Arithmetic and geometric means are
commutative, but weighted means in general are not.
If
T
is a continuous t-norm, and
S
a continuous t-conorm, the function
=
p
2
=
A
(
x
1
,
x
2
)
=
p
1
T
(
x
1
,
x
2
)
+
p
2
S
(
x
1
,
x
2
),
p
1
+
p
2
=
1
is an aggregation function that, since
T
min
max
S
, in general is not a
mean. The only exception is with
T
=
min, and
S
=
max, as it was said before. For
example,
Search WWH ::
Custom Search