Geoscience Reference
In-Depth Information
Utilization of possibility theory allows also semantically describe fuzzy numbers
as possibility distributions [
16
]. This semantic than help us explaining what such
fuzzy numbers mean. The values with membership value 1 are believed to be
absolutely possible or unsurprising, thus they should cove the most likely result.
With decreasing degree of membership the possibility of obtaining given result
decreases and the surprise rises. When membership value reaches 0 then such
result is impossible (or almost impossible at some cases) and the surprise that such
result would present is maximal. Such semantics helps with practical explanation
what the results truly mean.
To asses position of fuzzy number
X to the fuzzy number Y four indices are
ne possibility and necessity that X is at least equal or
needed [
11
]. Two of them de
greater than Y:
Y
ð
½Y
;
1ÞÞ
¼ sup
x
min
ð
l
X
ð
x
Þ
;
sup
y
x
min
ð
l
Y
ð
y
ÞÞ
ð
2
Þ
X
N
X
ð
½Y
;
1ÞÞ
¼ inf
x
max
ð
1
l
X
ð
x
Þ
;
sup
y
x
ð
l
Y
ð
y
ÞÞ
ð
3
Þ
The other two determine if X is strictly greater than Y:
Y
ð
Y
;
1ÞÞ
¼ sup
x
min
ð
l
X
ð
x
Þ
;
inf
y
x
1
l
Y
ð
y
ÞÞ
ð
4
Þ
X
N
X
ð
Y
;
1ÞÞ
¼ inf
x
max
ð
1
l
X
ð
x
Þ
;
inf
y
x
1
l
Y
ð
y
ÞÞ
ð
5
Þ
Together these indices allow comparison of any two fuzzy numbers, based on
pairwise comparison any set of fuzzy numbers can be sorted.
For both set of indices there are four situations of the combinations of possibility
and necessity that can be outcome of the calculation. In this paragraph both rela-
tions
are referred as relation,
because the descriptions are valid for both pairs of indices. The
at
least equal or greater, and strictly greater
—
—
first situation is
¼
N
Y
¼
when
Q
X
Y
;
1
Y
;
1
0 which means that X is de
nitely does not
¼
Y. Then there is opposite situation
Q
X
Y
;
1
ful
l the given relation to
¼
1, in which X completely satisfy the relation. The other two rela-
tions contains some uncertainty, because they indicate certain results but they can
not provide them absolutely. The
Y
;
1
N
Y
first of those is situation when
Q
X
[
Y
;
1
0
¼
0. This means that there is possibility that X might satisfy the
relation, but it is not necessary. Obviously that means that the indicators are not
strong. The last possible combination of values
Y
;
1
and
N
Y
¼
is
Q
X
Y
;
1
1 and
[
Y
;
1
N
Y
0. In such case again it the relation is not satis
ed absolutely but
the indicators are much stronger than in previous case.
Search WWH ::
Custom Search