r - The Hong Kong Polytechnic University

complex structure, the reliability analysis method based on SVM with uniform design method (UDM) was
proposed here. UDM is adopted to select training data and SVM is used as response surface. Structural
reliability index is calculated in combination with the traditional reliability analysis methods (such as, the
first-order reliability method (FORM), the second-order reliability method (SORM) or Monte Carlo simulation
method (MCSM)). It is shown through numerical examples that by introducing uniform design method, the
number of training data for establishing a SVM model is reduced and SVM can approximate complex response
surface well. The analysis results show that the method is feasible and has a good prospect, which provides a
new way for structural reliability analysis.
METHOD OF SOLUTION
The Uniform Design Method
UDM (Fang K. T. et al.1978) considers that within the scope of the study the experimental points are evenly
spread. Its mathematical theory is consistent distribution theory in number-theoretic method. UDM is ideal for
the situations of testing with multi-factor, multi-level and system models completely unknown.
This method develops some well-designed tables to carry out tests, taking into account the test points within test
scope are uniformly dispersed. By using this experimental design method, the number of trials can be reduced
s
and the efficiency can be improved. The tables are named uniform design tables which are expressed as U
n
( q ) .
Where, U means the uniform design. n is number of tests. q is number of layers. s is possible factors.
These methods, such as, good lattice point method, the Latin method, orthogonal expansion method and
stochastic optimization method, are used to construct uniform design. The steps of how to construct uniform
distribution design table by good lattice point method are as follows:
1) Given a positive integer n , to find a positive integer h ( h < n), of which the greatest common divisor of
n and h is 1. The positive integers meeting this condition make up of a vector H = ( h , h , L , h
1 2 m
) .
2) Generate the j column of the uniform design table, u = ih ( mod n
ij j
) . If ih exceeds n , ih subtract
j
j
an appropriate multiple of n , making the difference fall into [1, n ]. Thus, U = ( u ij ) is a matrix with size of
n× m.
D X is introduced to check the construction of uniform design tables.
3) The discrepancy ( )
D
X
N
S
X∈C
n
X (1)
( ) = max X
−V([0, ))
Where, V ([0, X))
is the volume of [0, X ) . N X
is the number of x 1
, x 2
, L , x points which belong to
m
[0, X ) .
S
C is the domain defined.
Uniform design tables constructed should be used with corresponding accessory table. The experimental points,
determined by the accessory table distribute uniformly in the test area, are the most representative test points.
Support Vector Machines
SVM is a machine learning method based on statistical learning theory. In accordance with limited samples of
information, SVM finds the best between the complexity and learning ability in the model in order to get the
best generalization. SVM algorithm will eventually transform into a quadratic optimization issue.
n
Given a set of data points{ x , y}
, i = 1,2, L , l , such that x ∈ R is an input and y ∈ R is a target output, the
i i
i
i
standard form of support vector regression can be expressed as
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