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dividing every variable forming the multivariate distribution into k equiprobable
intervals, and sampling once from each interval of the variable. Then these samples
are paired randomly to form k random vectors from the multivariate distribution.
Figure 6 depicts the strati
ed sampling in both forms, traditional and LHS, where
the difference is in the pairing process. In the traditional strati
ed sampling, samples
from every interval of variable i is paired with every other samples from all
intervals of variable j; whereas in the LHS, one sample from an interval of variable
i is paired only once with any one of the sample from an interval of variable j. The
pairing in LHS can also be done in such a way as to account for the mutual
correlation of the variables by preserving their rank correlation (Wyss and
Jorgensen 1998 ), and hence capturing the inter-dependence structure of the mul-
tivariate distribution.
Similarly, LHS of homothetic stress directions is performed from historical data
by dividing the load stress factor variables into k equidistant intervals (i.e., equal
width; a modi
cation to traditional LHS that partitions into equiprobable intervals),
sampling once from each interval of the variable, and pairing them preserving their
rank correlation, to form k homothetic stress directions. Figure 7 shows (a) traditional
strati
ed sampling and (b) LHS of homothetic stress directions in 3-dimensional state
space. In the case of LHS, for k intervals per dimension, irrespective of state space
size the uniform strati
cation of stress direction is achieved with k samples; com-
pared to the stratified sampling that produces k n 1 samples for k intervals per
dimension, in a state space of dimension n. The ideal number of k is found in an
incremental fashion until there is no further improvement in the boundary limits.
Hence computation to
find the boundary region can be decreased drastically by using
the proposed method based on LHS of stress directions and linear sensitivities.
Fig. 6 Stratified sampling
a traditional, b LHS
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