Geoscience Reference
In-Depth Information
4 Mamdani
'
s Method
Let
'
s have a look at Mamdani
'
s method in detail [
7
].
be a knowledge base with k rules for n input variables
X
1
;
X
2
;
...
;
X
n
and one output variable Y. Each of the variables Xi
i
have the verbal
value A
i
;
j
Let B
¼
P
1
;
P
2
;
...
;
P
k
f
g
in j-th rule, variable Y has the verbal value B
j
, where
i
¼
1
2
;
...
;
n,
;
j
¼
1
2
;
...
;
k. For Mamdani
'
s regulator are de
ned:
;
Rules
P
1
:
if X
1
is A
11
and X
2
is A
21
and
and X
n
is A
n1
;
then Y is B
1
P
2
:
if X
1
is A
12
and X
2
is A
22
and
and X
n
is A
n2
;
then Y is B
2
P
k
:
if X
1
is A
1k
and X
2
is A
2k
and
...
and X
n
is A
nk
then Y is B
k
X
1
is A
0
1
and X
2
is A
0
2
and
and X
n
is A
0
n
Observing
...
Y is B
0
Conclusion
Because the effort with the whole of the relation is numerically arduous, it is pref-
erable to use the approach FITA (
(first inference then aggregation), which means rea-
final aggregate conclusion is B
0
¼
k
j
¼
1
B
j
.
soning of conclusion rule-by-rule, where the
,
k
j
¼
1
k
j
¼
1
min w
j
;
l
B
j
ðÞ
Therefore
l
B
0
ðÞ
can be presented as
l
B
0
ðÞ¼
max
l
B
j
ðÞ¼
max
is the total weight of
where
w
j
¼
min w
1j
;
w
2j
;
...
;
w
nj
j-th rule, numbers
w
1j
;
w
2j
;
...
;
w
nj
are particular degrees of ful
lment of the premises in j-th rule
X
1
is A
1j
;
X
2
is A
2j
;
...
;
X
n
is A
nj
.
We can generalize the properties to t-norm T.
Consider the generalisation of t-norm T for an intersection and t-norm T
for an
assignment of the relation (Fig.
1
). The membership function for degrees w
j
¼
Tw
1j
;
w
2j
;
...
;
w
nj
.
is de
k
j
¼
1
k
j
¼
1
T
ned as
l
B
0
ðÞ¼
max
l
B
j
ðÞ¼
max
w
j
;
l
B
j
ðÞ
is method is written T
¼
T
M
and T
¼
T
P
.
For Larsen
'
5 Defuzzi
cation
If we apply crisp inputs, the results of inference are fuzzy outputs. We often need to
cation. There are several methods
to defuzzify. We can distribute them to methods searching the most acceptable
solution and methods of the best compromise [
8
].
The methods of the most acceptable solution are presented by the methods of the
most important maximum with selection of the biggest value of the membership
functions placed leftmost, middlemost or rightmost
find the particular real value of output by defuzzi
—
Left of Maximum (LoM),
Mean of Maximum (MoM), Right of Maximum (RoM).
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