Neuro-fuzzy Approach - Computational Intelligence in Time Series Forecasting - page 258

Table 6.3(b). Comparison of training and evaluation performances of other fuzzy model and
Takagi-Sugeno-type multi-input single-output neuro-fuzzy networks trained with the
proposed Levenberg-Marquardt algorithm for Wang data (second-order nonlinear plant
data)

Method No. of rules No. of fuzzy

sets

Rules
conseq.

MSE
training

MSE
eval.

Wang and
Yen (1999)

40 (initial)
28 (optimized)

40 Gauss. (2D)
28 Gauss. (2D)

Singleton
Singleton

3.3e-4
3.3e-4

6.9e-4
6.0e-4

Yen and
Wang (1998)

36 (initial)
23 (optimized)
36 (initial)
24 (optimized)

12 B-splines
12 B-splines
12 B-splines
12 B-splines

Singleton
Singleton
Linear
Linear

2.8e-5
3.2e-5
1.9e-6
2.0e-6

5.1e-3
1.9e-3
2.9e-3
6.4e-3

Yen and
Wang (1999)

25 (initial)
20 (optimized)

25 Gauss. (2D)
20 Gauss. (2D)

Singleton
Singleton

2.3e-4
6.8e-4

4.1e-4
2.4e-4

Setnes and
Roubos
(2000)

7 (initial)
7 (optimized)
5 (initial)
5 (optimized)
4 (optimized)

14 Triangular
14 Triangular
10 Triangular
8 Triangular
4 Triangular

Singleton
Singleton
Linear
Linear
Linear

1.6e-2
3.0e-3
5.8e-3
7.5e-4
1.2e-3

1.2e-3
4.9e-4
2.5e-3
3.5e-4
4.7e-4

Roubos and
Setnes (2001)

5 (initial)
5 (optimized)
5 (optimized)

10 Triangular
10 Triangular
5 Triangular

Linear
Linear
Linear

4.9e-3
1.4e-3
8.3e-4

2.9e-3
5.9e-4
3.5e-4

Proposed
neuro-fuzzy
TS model

10 (initial,
non-optimized)
5 (initial,
non-optimized)

10 Gaussian
5 Gaussian

Linear
Linear

1.1866e-5
5.1866e-4

2.1268e-5
7.8003e-4

The plots of the finally tuned GMFs that made the fuzzy partitions of universes
of discourse of normalized input u ( k ) and input y ( k ) are shown in Figures 6.3(f)
and 6.3(g) respectively. The figures also show that there is further scope for
improving the accuracy, transparency and interpretability of neuro-fuzzy model
obtained through similarity measures and genetic-algorithm-based optimizations.
These issues, namely model transparency and interpretability, will be the main
subject of discussion in Chapter 7. The results obtained in this example also, in
general, summarize the excellent prediction performance of Takagi-Sugeno-type
multi-input single-output neuro-fuzzy networks when trained with the proposed
Levenberg-Marquardt Algorithm.

6.8 Other Engineering Application Examples

In the following, some engineering application examples are given in which the
systematic neuro-fuzzy modelling approach has been used to solve the problem of

x material property prediction

x pyrometer reading correction in temperature measurement of wafers, based

on prediction of wafer emissivity changes in a rapid thermal processing
system, such as chemical vapour deposition and rapid thermal oxidation