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6.7 Modelling and Identification of Nonlinear Dynamics
We would now like to illustrate the efficiency of the neuro-fuzzy approach
proposed in Section 6.4.1 on some forecasting examples.
6.7.1 Short-term Forecasting of Electrical Load
This application concerns the forecasting the electrical load demand, based on a
time series that predicts the values at time (
t + L
) using the available observation
data up to the time point
t
. For modelling purposes the time series data
X
= {
X
1
,
X
2
,
X
3
, …,
X
q
} have been rearranged in input-output form
XIO
. The neuro-fuzzy
predictor to be developed for time series modelling and forecasting is supposed to
operate with four inputs (
i.e
.
n =
4) and with three outputs (
i.e
.
m =
3). Taking both
the sampling interval and the lead time of forecast to be one time unit, then for
each
t
t
the input data have to be represented as a four-dimensional vector and
the output data as a three-dimensional vector
4
XI = [
X
(
t
-3),
X
(
t
-2),
X
(
t
-1),
X
(
t
)],
XO = [
X
(
t+
1),
X
(
t+
2),
X
(
t+
3)]
Furthermore, in order to have sequential output in each row, the values of
t
should
run as 4, 7, 10, 13, …, (
q
-3). The corresponding
XIO
matrix will then look like
(6.43), in which the first four columns represent the four inputs of the network and
the last three columns represent its output.
ª
,
,
X
,
X
,
,
X
º
XX
X
X
1
2
4
6
3
5
7
«
»
,
X
,
,
X
,
X
,
X
X
X
X
«
»
4
6
7
8
10
5
9
(6.43)
XIO
«
»
#
# # #
# # #
«
»
«
X
»
,
,
,
,
,
X
XX
XXX
q
3
¬
q
6
q
5
q
4
q
2
q
1
q
¼
In the selected forecasting example, 1163 input-output data were generated, from
which only the first 500 input-output data sets,
i.e
. the first 500 rows from the XIO
matrix, were used for the multi-input multi-output neuro-fuzzy network training.
The remaining 663 rows of the XIO matrix were used for verification of the
forecasting results. The training and forecasting performances achieved with the
neuro-fuzzy network are illustrated in Figures 6.7(a) - (d) and in Tables 6.1(a) and
6.1(b) respectively.
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