Latin hypercube techniques

There are several sampling techniques in Monte Carlo analyses, the most common being random, median Latin hypercube and random Latin hypercube. Latin hypercube techniques are usually preferred because they need fewer iterations and thus are more efficient. They are, however, inferior to random sampling if high percentiles of the output are of interest and if the exact shape of the output distribution is important (Cullen and Frey 1999). 

Uncertainty - analyzes the uncertainty of a system, sequence, or end state using either the Monte Carlo or Latin Hypercube simulation technique. 

The third step is to select the number of iterations or calculations of dose that are to be performed as a part of each simulation. For the analysis here, a total of 10,000 iterations based on the selection of input variables from each defined distribution were performed as part of each simulation. The large number of iterations performed, as well as the Latin hypercube sampling (non-random sampling) technique employed by the Crystal Ball simulation program, ensured that the input distributions were well characterized, that all portions of the distribution (such as the tails) were included in the analysis, and that the resulting exposure distributions were stable. 

The number of sampling iterations must be sufficient to give stable results for output distributions, especially for the tails. There are no simple rules, because the necessary number of runs depends on the number of variables entered as distributions, model complexity (mathematical structure), sampling technique (random or Latin hypercube), and the percentile of interest in the output distribution. There are formal methods to establish the number of iterations (Cullen and Frey 1999) however, the simulation iterations could simply be increased to a reasonable point of convergence. 

This section provides an overview of common methods for quantitative uncertainty analysis of inputs to models and the associated impact on model outputs. Furthermore, consideration is given to methods for analysis of both variability and uncertainty. In practice, commonly used methods for quantification of variability, uncertainty or both are typically based on numerical simulation methods, such as Monte Carlo simulation or Latin hypercube sampling. However, there are other techniques that can be applied to the analysis of uncertainty, some of which are non-probabilistic. Examples of these are interval analysis and fuzzy methods. The latter are briefly reviewed. Since probabilistic methods are commonly used in practice, these methods receive more detailed treatment here. The use of quantitative methods for variability and uncertainty is consistent with, or informed by, the key hallmarks of data... 

A Latin Hypercube sampling method was used in the Risk simulation to generate the input parameter values from the probability distribution functions. This method was chosen over the Monte Carlo technique, which samples randomly from the distribution function and causes clustering when low probability values are not sampled due to insufficient computational sampUng iterations. In contrast, the Latin Hypercube stratified sampling technique systematically samples all segments (stratifications) of the distribution just once, resulting in fewer computational iterations required to produce a representative probability curve. 

Among variance reduction techniques the importance sampling technique is most frequently applied in structural reliability analysis e.g. in [1,3,10-12] whereas other techniques e.g. stratified sampling [13], Latin hypercube sampling [14,15] and antithetic variates [11], also proved to be very powerful tools for structural reliability analyses. The concepts of these techniques will be discussed briefly below. 

For many types of problems the dominant variables are usually not known tn advance and, moreover, for time dependent problems dominant variables may even change with respect to time, i.e. a variable which is dominant in a particular period may not be dominant in other periods. Thus, it is not easy to indicate which of the variables are most Important. Latin hypercube sampling was developed [14] based on the stratified sampling technique... 

The Latin Hypercube Sampling (LHS) technique is employed to extract random values from each uniform distribution of leak rate range, and each sampling iteration forms a set of representative scenarios. For the target system, this iteration is repeated 100 times, and 100 sets of representative scenarios are randomly produced for estimating 100 different exceedance plots. 

This procedure is repeated for every component individually, but this represents just one possible system damage response. To reflect the full distribution of possible responses, a technique such as Monte Carlo simulation or Latin hypercube sampling should be used to repeat the damage assignment for all components and generate multiple system damage responses. The system performance and loss calculations should be calculated for each system damage response. 

The Latin hypercube sampling technique (LHS Helton and Davis 2003) can be employed in order to reduce the number of simulations, Ns, in addition to achieving an acceptable level of accuracy for the statistical characteristics of response. The LHS is a special type of MC simulation that uses the stratification of the theoretical CDFs of uncertain parameters. Stratification divides the CDF curve into Ns equal intervals on the probability scale (i.e., 0.0 to 1.0). A sample is then randomly drawn from each interval or stratification of the input CDFs based on the... 

Fig. 4 Approximation of the skewness by Latin hypercube sampling techniques...

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