2 Weighted Least Square Support Vector Machines
for Regression (WLSSVR)
If the training samples have different importance to construct the LSSVR model, a
weighting function can be added to the LSSVR model to describe this difference. The
mathematical formula of the weighted LSSVR, named as WLSSVR, is
min
w;b
J p ðw; eÞ ¼
1
2
ww
T
þ
1
2
c
X N
k¼1
W k e
2
k
s.t. y k ¼ w
T
uðx k Þ þ b þ e k
ð1Þ
where W k is the weight at x k , e k is the error variable and c is the penalty constant.
The WLSSVR model can be given by the following solution
0
1
T
v
1 v X þ W
À1
=c
b
a
¼
0
y
ð2Þ
where W ¼ diagðW 1 ; W 2 ; . . .; W N Þ, y ¼ ½y 1 ; . . .; y N , 1 v ¼ ½1; . . .; 1, a ¼ ½a 1 ; . . .; a N
and X kj ¼ uðx k Þ; uðx j Þ
¼ Kðx k ; x j Þ:
3 Proposal Method
3.1 Sequential Sampling Strategies
Since the concerned probability of failure only relies on the sign of the limit state
function, it is essential to sample the points with a high potential to cross the failure
surface to the training sample set. The most straightforward way is to sample the points
on the approximate failure surface to the training sample set, but it is unrealistic to
obtain all the points on the approximate failure surface. For this reason, figure out some
important points on the approximate failure surface and sample them to the training
sample set is a better choice. Since the gradient information can help identify regions
with a high degree of nonlinearity, sampling the point with greatest gradient to the
training sample set can improve the global approximation quality [31]. The greatest
gradient point is named as GGP for simplicity. In this section, GGP-based sequential
sampling strategies are proposed.
In order to identify the nonlinear region of the failure surface, the relative derivative
is used here. Without loss of generality, we take the last design variable x n as the
dependent variable, then the relative derivative can be calculated as
@x n
@x i
¼ À
@gðxÞ
@x i
@gðxÞ
@x n
ði ¼ 1; . . .; n À 1Þ
ð 3Þ
Weighted Least Square Support Vector Regression Method …
213
for Regression (WLSSVR)
If the training samples have different importance to construct the LSSVR model, a
weighting function can be added to the LSSVR model to describe this difference. The
mathematical formula of the weighted LSSVR, named as WLSSVR, is
min
w;b
J p ðw; eÞ ¼
1
2
ww
T
þ
1
2
c
X N
k¼1
W k e
2
k
s.t. y k ¼ w
T
uðx k Þ þ b þ e k
ð1Þ
where W k is the weight at x k , e k is the error variable and c is the penalty constant.
The WLSSVR model can be given by the following solution
0
1
T
v
1 v X þ W
À1
=c
b
a
¼
0
y
ð2Þ
where W ¼ diagðW 1 ; W 2 ; . . .; W N Þ, y ¼ ½y 1 ; . . .; y N , 1 v ¼ ½1; . . .; 1, a ¼ ½a 1 ; . . .; a N
and X kj ¼ uðx k Þ; uðx j Þ
¼ Kðx k ; x j Þ:
3 Proposal Method
3.1 Sequential Sampling Strategies
Since the concerned probability of failure only relies on the sign of the limit state
function, it is essential to sample the points with a high potential to cross the failure
surface to the training sample set. The most straightforward way is to sample the points
on the approximate failure surface to the training sample set, but it is unrealistic to
obtain all the points on the approximate failure surface. For this reason, figure out some
important points on the approximate failure surface and sample them to the training
sample set is a better choice. Since the gradient information can help identify regions
with a high degree of nonlinearity, sampling the point with greatest gradient to the
training sample set can improve the global approximation quality [31]. The greatest
gradient point is named as GGP for simplicity. In this section, GGP-based sequential
sampling strategies are proposed.
In order to identify the nonlinear region of the failure surface, the relative derivative
is used here. Without loss of generality, we take the last design variable x n as the
dependent variable, then the relative derivative can be calculated as
@x n
@x i
¼ À
@gðxÞ
@x i
@gðxÞ
@x n
ði ¼ 1; . . .; n À 1Þ
ð 3Þ
Weighted Least Square Support Vector Regression Method …
213