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Similarly, let the computed values of
E
and
E
for the second and the third
rules, using a similar procedure, be 0.6 and 0.7 respectively. Therefore, the
computed row vector will be
>
@
ª
123
,
º
E EEE
,
0.5, 0.6, 0.7
. Furthermore, for
¬
¼
this example the relational matrix
R
is of size
M k
u , where the number of rules
M
= 3 and the number of output fuzzy sets (
e.g.
HIGH, MEDIUM, and LOW)
k
=
3. The relational matrix is formulated using the degree of membership of each rule
output in the output fuzzy set. Therefore,
0.9
0.1
0.0
ª
º
«
»
R
«
0.1
0.8
0.0
,
»
«
»
0.0
0.7
0.2
¬
¼
P
y
and
l
1, 2,
"
,
M
;
j
1, 2,
"
,
k
.
R
l
lj
F
j
Now applying the max-min relational composition, the output fuzzy set can be
computed as follows:
T
ª
º
max min 0.5, 0.9 , min 0.6, 0.1 , min 0.7, 0.0
0.9
0.1
0.0
ª
º
«
»
«
»
«
>
@
Z
0.5
0.6
0.7
D
0.1
0.8
0.0
max min 0.5, 0.1 , min 0.6, 0.8 , min 0.7, 0.7
»
.
«
»
«
»
«
»
0.0
0.7
0.2
¬
¼
«
max min 0.5, 0.0 , min 0.6, 0.0 , min 0.7, 0.2
»
¬
¼
>
@
This finally results in
Z
0.5
0.7
0.2
.
Supposing now that the COGs of the output fuzzy sets are known,
i.e.
if the
COG
l
j
;
j
1, 2,
"
,
k
; and noting that
1
2
3
;
are given respectively as
F
FFF
j
j
j
1
2
3
then the crisp output from the inference of the
relational fuzzy-rule-based system will be
y
30,
y
20 and
y
10,
0.5
u u u
30
0.7
20
0.2
10
31
y
22.142.
0
0.5
0.7
0.2
1.4
The various fuzzy inferencing mechanisms described in the Sections 4.4.1 to 4.4.3
can similarly be applied to time series forecasting applications when the
corresponding fuzzy model (fuzzy rules) of a given time series is available.
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