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Table 6.1(b). Training and forecasting performance of Takagi-Sugeno-type multi-input
multi-output neuro-fuzzy network with proposed Levenberg-Marquardt algorithm for
electrical load time series.
Sl. No.
Final SSE with pre-scaled data
(scale factor = 0.001)
Final SSE, MSE, and RMSE with
original (nonscaled) data
1.
SSE = 22.5777
SSE = 2.2578e+005
SSE1 = 2.6365
SSE2 = 6.7828
SSE3 = 13.1584
(with training data 1 to 1500)
MSE1 = 26.3650
MSE2= 67.8278
MSE3= 131.5837
RMSE1= 5.1347
RMSE2= 8.2358
RMSE3=11.471
(After training with Levenberg-
Marquardt algorithm)
2.
SSE = 42.3026
SSE1 = 5.0096
SSE2 = 11.7879
SSE3 = 25.5051
(with training and validation data
points 1 to 3489)
SSE = 4.2303e+005
(with training and validation data
points 1 to 3489)
Note that in the Table 6.1(a) and Table 6.1(b) SSE1, SSE2, and SSE3 indicate the
sum squared error values at the output nodes 1, 2 and 3 respectively of the Takagi-
Sugeno-type multi-input multi-output neuro-fuzzy network as formulated in
(Equation 6.11a), and SSE indicates the cumulative sum of the sum square error
values contributed by all three output nodes of the multi-input multi-output neuro-
fuzzy network as formulated in (Equation 6.11b).
6.7.2 Prediction of Chaotic Time Series
In the next application example the proposed neuro-fuzzy algorithm has been
tested for modelling and forecasting the Mackey-Glass chaotic time series,
generated by solving the Mackey-Glass time delay differential equation (6.44)
(M ATLAB , 1998).
(6.44)
dx dt
0.2
x t
G
1
10
t
G
0.1
x t
,
x
for
x
The equation describes the arterial CO 2 concentration in the case of normal and
abnormal respiration and belongs to a class of time-delayed differential equations
that are capable of generating chaotic behaviour. It is a well-known benchmark
problem in fuzzy logic and neural network research communities. Like in the
0
1
.
2
G
17
,
and
x
t
0
for
t
0
.
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