Let us note
.
The goal is to estimate the following probability:
LHS or Latin Hypercube Sampling is a sampling method enabling to
better cover the domain of variations of the input variables, thanks
to a stratified sampling strategy. This method is applicable in the
case of independent input variables. The sampling procedure is based
on dividing the range of each variable into several intervals of equal
probability. The sampling is undertaken as follows:
Step 1 The range of each input variable is stratified into
isoprobabilistic cells,
Step 2 A cell is uniformly chosen among all the available cells,
Step 3 The random number is obtained by inverting the Cumulative
Density Function locally in the chosen cell,
Step 4 All the cells having a common strate with the previous
cell are put apart from the list of available cells.
The estimator of the probability of failure with LHS is given by:
where the sample of is obtained as
described previously.
One can show that:
is the variance of the estimator of
the probability of exceeding a threshold computed by the LHS
technique,
is the variance of the estimator of
the probability of exceeding a threshold computed by a crude Monte
Carlo method.
With the notations
the asymptotic confidence interval of order associated
to the estimator is
where is the quantile from
the reduced standard gaussian law .
It gives an unbiased estimate for (reminding that all input
variables must be independent).
This method is derived from a more general method called ’Stratified
Sampling’.