Latin Hypercube sampling method

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. 

Coefficient p (9) was evaluated using the Latin Hypercube Sampling method (McKey et al. 1979) for 100 simulation runs. The dependence of p on t is shown in Figure 6. For clarity the Gauss probability density function and scale Pp which is evaluated from p, is shown on the right side of the figure. 

Isukapalli et al. (2000) coupled the Stochastic Response Surface Method (SRSM) with ADIFOR. The ADIFOR method (see Sect. 5.2.5) is used to transform the model code into one that calculates the derivatives of the model outputs with respect to inputs or transformed inputs. The calculated model outputs and the derivatives at a set of sample points are used to approximate the unknown coefficients in the series expansions of outputs. The coupling of the SRSM and ADIFOR methods was applied for an atmospheric photochemical model. The results obtained agree closely with those of the traditional Monte Carlo and Latin hypercube sampling methods whilst reducing the required number of model simulations by about two orders of magnitude. 

Novak D, Teply B, Kersner Z (1998) The role of Latin Hypercube Sampling method in reliability engineering. In Shiraishi N, Shinozuka M, Wen YK (eds) The 7th international conference on structural safety and reliability, ICOSSAR 97, vol 2. Balkema, Rotterdam (Kyoto, Japan)... 

Local sensitivity analysis is of limited value when the chemical system is non-linear. In this case global methods, which vary the parameters over the range of their possible values, are preferable. Two global uncertainty methods have been used in this work, a screening method, the so-called Morris One-At-A-Time (MOAT) analysis and a Monte Carlo analysis with Latin Hypercube Sampling (Saltelli et al., 2000 Zador et al., submitted, 20041). The analyses were performed by varying rate parameters, branching ratios and constrained concentrations within their uncertainty interval,... 

Finally, a Monte Carlo method coupled with the Latin Hypercube Sampling (LHS) was used to assess the overall model uncertainty. The 2a standard deviation of the model was estimated to be 30-40% for OH and 25-30% for HO2, which is comparable to the instrumental uncertainty. 

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... 

Model variance was propagated using the factorial, DPD, Monte Carlo and Latin hypercube sampling (LHS) methods. Table A2.6 provides a summary comparison of the outputs—the arithmetic mean, arithmetic standard deviation, coefficient of variation (CV), geometric mean (GM), geometric standard deviation (GSD), 5th percentile and 95th percentile outcomes— from each method. 

Figure A2.4 compares the CDFs for intake obtained from factorial design and DPD methods with the exact analytical solution for the CDF of intake. The 27 data points from the DPD and factorial methods were used to plot the empirical CDF shown in Figure A2.4. Figure A2.5 compares the CDF for intake obtained from 2000 Monte Carlo simulations with the exact analytical solution for the CDF of intake. Figure A2.6 compares the CDF obtained from 200 Latin hypercube sampling Monte Carlo simulations with the exact analytical solution for the CDF of intake. The Monte Carlo and Latin hypercube sampling empirical CDFs were plotted using all simulation outcomes.

This relatively simple model illustrates the viability of the straightforward analytical analysis. Most models, unfortunately, involve many more input variables and proportionally more complex formulae to propagate variance. Fortunately, the Latin hypercube sampling and Monte Carlo methods simplify complex model variance analysis. 

We tested the EASI implementation against a RED implementation, as both methods allow the specification of a sample size. We performed 150 runs per sample size (100, 300, 1000, 3000, 10000, 30000). The results are reported in Figures 3 and 4. EASI was used with input data from a Latin hypercube sampling algorithm while RED used random permutations of equidistant samples from the zig-zag function. We conclude that both methods perform equally well. Moreover, they exhibit the same flaws For small sample sizes the true value is over-estimated, while for large sample sizes an imder-estimation occurs. 

To analyze the aggregation effect of the different monitoring systems and their significance in the pasteurization stage, a base case (all controls inactive, i.e. no control) has been simulated using Monte Carlo method with Latin Hypercube Sampling see case 1 in table 5. Thus, the simulation procedure has been built as a spreadsheet model in Microsoft Excel with adds on Risk 5.0 (Palisade Newfield) and integrating the predictive and probabihstic models and input data described above. 

The Latin Hypercube Sampling (LHS) method was used for the evaluation of sensitivity indices (McKey et al. 1979, Iman et al. 1980). The model output Y is the load carrying capacity evaluated in each simulation run of the LHS method. The conditional random arithmetical mean was evaluated for 330 simulation runs. The variance was also calculated for 330 simulation runs. All input random imperfections were considered as random quantities for the evaluation of the variance V(T) of the load carrying capacity. 9000 simulation runs were applied. The second-order sensitivity indices S.J were calculated analogously. The influence of imperfections on the load carrying capacity changes with increasing slenderness of the beam. Non-dimensional slenderness can be calculated acc. to EUROCODE 3 as ... 

Fig. 5.10 A comparison of samples produced by different sampling methods for a 2-parameter model (a) 1,024 random sampling points, (b) 1,024 Latin hypercube sampling points, (c) 1,024 points of the Halton sequence and (d) 1,024 points of the Sobol sequence. All sampling procedures are based on a uniform distribution in the domain [0, 1] x [0, 1]. Adapted from (Ziehn 2008)...

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. 

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