For example 2, the initial number of training sample points is 70, and the termination condition is 0 ¼ 2 Â 10
À3 .
The absolute error rate of the tth iteration is defined as:
ðtÞ
a ¼
1
2N s
X N s
i¼1
signð~ g
ðtÞ
ðx i ÞÞ À signðgðx i ÞÞ
ð13Þ
Average absolute error rate is defined as the average of absolute error rate in five
consecutive iterations, similar to Eq. (10). The performance of these approximation
methods is evaluated by comparing the absolute error rate.
The results of sequential modeling are shown in Table 1, in which
ð0Þ
a is the initial
absolute error rate and
ðfinalÞ
a
is the average absolute error rate of the last iteration. In
addition, in the experiment, the approximate model was also constructed by directly
generating the same initial set of sample points as the number of sample points required
for sequential modeling, and the absolute error rate
direct
a
is obtained. The results show
that the proposed sequential modeling method reduces the absolute error rate of the
approximate model. For the nonlinear limit state equation, the GGP sequential modeling method is effective, such as Example 1 and 2. As the complexity of the model
increases, the number of iterations required for the convergence of the algorithm
increases.
The comparison between the WLSSVR approximate limit state equation and the
real limit state equation of Example 1 is shown in Fig. 1. The results show that the
approximate failure surface almost coincides with the real failure surface. The
sequential approximation modeling algorithm based on GGP only adds a few sample
points to achieve a good approximation effect. The GGP points are distributed on both
sides of the real limit state equation, and there are relatively more training samples at
the higher gradient.
Table 1. Sequential modeling results of WLSSVR based on GGP sequential sampling strategy
Example
Initial number of
training points
Final training
points number
ð0Þ
a
ðfinalÞ
a
direct
a
Example 1
40
57
0.0125
0.0017
0.0063
Example 2
70
282
0.2019
0.0344
0.0417
216
Y. Guo et al.
À3 .
The absolute error rate of the tth iteration is defined as:
ðtÞ
a ¼
1
2N s
X N s
i¼1
signð~ g
ðtÞ
ðx i ÞÞ À signðgðx i ÞÞ
ð13Þ
Average absolute error rate is defined as the average of absolute error rate in five
consecutive iterations, similar to Eq. (10). The performance of these approximation
methods is evaluated by comparing the absolute error rate.
The results of sequential modeling are shown in Table 1, in which
ð0Þ
a is the initial
absolute error rate and
ðfinalÞ
a
is the average absolute error rate of the last iteration. In
addition, in the experiment, the approximate model was also constructed by directly
generating the same initial set of sample points as the number of sample points required
for sequential modeling, and the absolute error rate
direct
a
is obtained. The results show
that the proposed sequential modeling method reduces the absolute error rate of the
approximate model. For the nonlinear limit state equation, the GGP sequential modeling method is effective, such as Example 1 and 2. As the complexity of the model
increases, the number of iterations required for the convergence of the algorithm
increases.
The comparison between the WLSSVR approximate limit state equation and the
real limit state equation of Example 1 is shown in Fig. 1. The results show that the
approximate failure surface almost coincides with the real failure surface. The
sequential approximation modeling algorithm based on GGP only adds a few sample
points to achieve a good approximation effect. The GGP points are distributed on both
sides of the real limit state equation, and there are relatively more training samples at
the higher gradient.
Table 1. Sequential modeling results of WLSSVR based on GGP sequential sampling strategy
Example
Initial number of
training points
Final training
points number
ð0Þ
a
ðfinalÞ
a
direct
a
Example 1
40
57
0.0125
0.0017
0.0063
Example 2
70
282
0.2019
0.0344
0.0417
216
Y. Guo et al.