Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 148

x Step 1: compute the degree of fulfilment of each rule by

l

ª º

x x ,1 dd l M ,

E P P

max

1
1

l

¬

G c G

¼

X

1

where is the min operator. For a crisp input x x

0 ,

which is equivalent

to a singleton fuzzy set, i.e.

P c x 1, for xx ;

1
1 G

0

and for all other points

xx z

0 ,

x 0.

P c

1
1 G

So the degree of fulfilment of the l th rule is reduced to

l

EP x .

l

G

0

1

x Step 2: compute the each rule consequent set as given by

l

l

E

l

c

F F

1

1

x Step 3: aggregate all consequent fuzzy sets as shown by

M

F c

l 1

* "*

2 M

* c c c c

FFF F

1 1 1 1

aggr

l 1

x Step 4: defuzzify the aggregated fuzzy set F c using the COG method.

aggr

The inferencing mechanism of the Mamdani type of fuzzy logic system can easily
be explained on an n -input single-output system described by M numbers of
Mamdani-type fuzzy rules

R 1 :

IF x 1 is

G and… and x n is

1 1

n

, THEN y is

1
1

G F

R 2 :

IF x 1 is

2
1

and… and x n is

2
n

, THEN y is

2
1

G G F

: : : : : :

R M : IF x 1 is

M

and… and x n is

M
n

, THEN y is

M

.

G G F

1 1

For a given set of rules and inputs x i , with " (also called the training

sample), the objective is to determine the crisp output value which is the
defuzzified value of the output fuzzy set. The inferencing of such a rule-based
fuzzy system proceeds as follows.

i 1, 2, , n ;