r - The Hong Kong Polytechnic University

Sigmoid function
Another kernel, that might seem appealing, is the hyperbolic tangent kernel
K xc , = tanh v( xc T
) + c
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( )
i
(
i
)
Supposing y = 1+ exp( − 2 ( v( xc ) ))
+ c
i
i
z
m
⎛ 2
( − α + α )
i i
1
i=
1
i
⎞
, then
= ∑ ⎜ − ⎟+
b
y
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⎝
∂y
i
∂x
j
⎠
( ( ))
( j ) T
2 exp 2 ( ) 2 ( j
vc v c vc )
( y 1)
i i i i
=− − xc + =− −
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m m m
∂z
∂z
∂y
1 1 1
= = − − − −
∂x ∂y
∂x y
y y
i
*
( )( ( j ) *
) ( j
∑ 2∑ − α + α 2 vc ( y 1) = 4 vc
)
2
( α α )( )
i i i i ∑ − +
i
i i
2 (15)
j i= 1 i j i= 1 i i=
1
i i
2
m
m
∂ z ∂ ⎛
( j) *
1 1 ⎞
( j)
*
∂ 1 1
= ⎜4 ∑vc
( − α + α )( − ) 4
2
( ) ( )
i
i i ⎟ = ∑vc
− α + α −
i
i i
2
∂x ∂x ∂x j k k ⎝ i= 1 y y
i i ⎠ i=
1
∂x y y
k i i
2 1 ∂y
2 1
= − = − − + − −
SVM-based reliability analysis method with UDM
m
( j) 2 ( j) ( k) *
4 ∑vc ( ) 8 v c c ( )( y 1)
i 3 2 i i i i 3 2 i
i= 1 y y ∂x i i k i=
1
y y
i i
m
*
i
( − α + α
i i
) ∑ ( α α )
Once the first-order and second-order derivatives of the structural response have been calculated, FORM or
SORM can be used to get the failure probability.
The calculation steps of SVM-based reliability method can be described as follows:
1) Select the random variables, specify their probabilistic characters, and define the limit state function g ( x ) = 0;
2) Generate the samples from the distribution functions of the basic variables with UDM and compute the
corresponding value of the performance function g(x) and divide the samples into two groups, namely the
training set and the validation set;
3) Establish the SVM model of the performance function;
4) Calculate the support vectors according to the training set and test SVM according to the validation set. If the
tolerance defined before is met, the next step will be executed. Otherwise, repeat steps 3 through 4 until the
error bellows the tolerance;
5) Choose the reliability method, such as FORM, SORM or MCSM and compute failure probability according
to the functional approximation by SVM.
6) Calculate the reliability index β and failure probability P
f
.
APPLICATIONS
Example 1
The limit state function is defined as in [17]
g ( x ) = exp(0.2× x + 1.4) −x
1 2
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where x and x are assumed to be independent and have a standard normal distribution with zero mean and
1 2
unit standard deviation.
2
Select the uniform design table U
25 ( 25 ). Take 20 samples which are uniformly distributed within[ − 3, 3]
and compose the experimental designs with 2 factors and 20 levels. The results are shown in table 1. It can be
seen that the results obtained by UDM-SVM RSM are in good agreement with reference(Kim et al. 1997; Zheng
et al. 2000; Elhewy et al. 2006). It suggests that even in the case of small sample sizes, SVM is also more
accurate fitting of the limit state surface. Moreover, for the same number of training samples taken (eg 25),
UDM selected samples obtained better results than the orthogonal design.
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