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Thereafter, the combined rule base and the rule grade table (RGT) are built, but the
conflicting rules with the maximum value of
D
(
Rule
) are selected, whereas the
conflicting rules with the lower value of
D
(
Rule
) are all rejected. It is to be noted
that besides the conflicting rules the redundant rules, that have both identical IF
parts as well as THEN parts, are also generated by this rule generation algorithm.
Since our final aim is to develop a fuzzy-logic-based predictor, or a fuzzy
model that is capable of forecasting the future values of a given time series, in the
following we will describe a fuzzy-rules generation algorithm based on a multi-
input single-output partitioning of the time series data.
Given a time series
X
= {
X
1
,
X
2
,
X
3
, ...,
X
q
}, at time points
t
= 1, 2, 3, . . . ,
q
; our
objective is to forecast the future values of this time series using a fuzzy-logic-
based predictor. For this forecasting problem, usually a set of known values of the
time series up to a point in time, say
t
, is used to predict the future value of the time
series at some point, say (
t
+
L
). The standard practice for this type of prediction is
to create a mapping from
D
sample data points, sampled every
d
units in time, to a
predicted future value of the time series at time point (
t
+
L
). Therefore, for each
t
,
the input data for the fuzzy logic predictor to be developed is a
D
-dimensional
vector of the form:
XI
(
t
)=[
X
{
t
-(
D
-1)
d
},
X
{
t
-(
D
-2)
d
}, …..,
X
{
t
}]
Following the conventional settings (for predicting the Mackey-Glass time series),
D
= 4 and
d
=
L
= 6 have been selected and, therefore, the input data of the fuzzy
predictor is a four-dimensional vector,
i.e.
XI
(
t
)=[
X
(
t
-18),
X
(
t
-12),
X
(
t
-6),
X
(
t
)].
The output data of the fuzzy predictor is a scalar and corresponds to the trajectory
prediction:
XO
(
t
)=[
X
(
t
+
L
)]
Therefore, for a four-input one-output fuzzy logic system the time series partition
can be obtained as:
(
X
11
,
X
12
,
X
13
,
X
14
,
Y
1
); ….; (
X
k
1
,
X
k
2
,
X
k
3
,
X
k
4
,
Y
k
);
etc.
which can be represented in
XIO
matrix (multi-input single output) form as
XXXXY
ª
º
11
12
13
14
1
«
»
XIO
«
# # # # #
(4.1)
»
«
»
X
XXXY
¬
¼
k
1
k
2
k
3
k
4
k
where,
X
k
1
,
X
k
2
,
X
k
3
,
X
k
4
are input values and
Y
k
as the corresponding output value
for
k
= 1, 2, 3, ...,
m
. Note that
XIO
stands for the time series data
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