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Monte Carlo Analysis
Latin Hypercube Sampling
The quasi-Monte Carlo efficiency relies on the empirical convergence rate of commonly used
sequences. This convergence rate is governed by the discrepancy that is asymptotically given
by:
Asymptotically this is smaller than the standard error of a pseudorandom estimate, the error
converges in O(n −1 ) instead of O(n −1/2 ) for traditional Monte Carlo. We then have a method
that can potentially compete with MC to run the simulations. Unfortunately the convergence
rate depends on the dimension d. For example, with d = 10, we have:
for n ≤ 10 34
We conclude that the current implementation of the QMC techniques should not be used as an
option to accelerate the Monte Carlo analysis when only a moderate number of simulations is
allowed and when the number of parameters is greater than 5.
The estimators of the mean, yield, variance, and so on, are the same as in the Monte Carlo
method. However, a disadvantage of this approach is that the use of a deterministic source
means that the unbiased and the usual error estimation methods are lost. Therefore it is difficult
to assess the final value of the integral (the mean value).
For a detailed overview of QMC methods, refer to: P. L’Ecuyer and C. Lemieux, Recent
Advances in Randomized Quasi-Monte Carlo Methods in Modeling Uncertainty: An
Examination of Stochastic Theory, Methods and Applications, Kluwer, 2001.
Related Topics
Sampling Plan Methods
Latin Hypercube Sampling
Latin hypercube sampling (LHS) may be considered as a particular case of stratified sampling.
The purpose of stratified sampling is to achieve a better coverage of the sample space of the
input factors. In the latin hypercube, sampling is undertaken as follows:
• The range of each input factor x j is divided into n intervals of equal marginal probability
1/n.
• One observation of each input factor is made in each interval using random sampling
within that interval.
Eldo® User's Manual, 15.3 457