Latin hypercube sampling

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. 

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. 

Latin hypercube Hammersley sequence sampling (LHSS), 26 1013-1015 Latin hypercube sampling (LHS), 26 1005, 1007-1011, 1012, 1013, 1014, 1015 future trends in, 26 1047 in process synthesis and design,... 

Permonosulfuric acid (PMS), 26 392 Permselective diaphragms, 9 656-657 Permutations, in Latin hypercube sampling, 26 1009-1010 Pernicious anemia, vitamin B12 and, 25 804 Perovskite carbides, 4 692 Perovskite ferrites, 22 55, 56t, 57 Perovskite material, mercury-base superconducting, 23 801 Perovskites, 5 590-591 22 94-96, 97 ... 

Stein M. 1987. Large sample properties of simulations using Latin hypercube sampling. Technometrics 29 143-151. 

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.

Figure A2.6 Comparison of the CDF for intake obtained from 200 Latin hypercube sampling Monte Carlo simulations with the CDF from the exact analytical solution for intake...

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. 

In a companion paper (Kleijnen et al., 2003), we changed the metamodel in (1) after the screening phase, as follows. For those controllable factors found to be important by sequential bifurcation, we augmented (1) with quadratic effects to form a predictive model for optimization. For those environmental or noise factors identified by sequential bifurcation as important, we created environmental scenarios through Latin hypercube sampling for robustness analysis. 

Helton J. C. and Davis F. J. (2002) Latin Hypercube Sampling and the Propagation of Uncertainty in Analyses Complex Systems. Sandia National Laboratiories. 

Luxmoore, R.J., G.V. Wilson, P.M. Jardine, and R.H, Gardner. 1990. Use of percolation theory and latin hypercube sampling in field-scale solute transport investigations, p. 437-439. In D.-Q. Lin (ed.) Proc. 1st Int. Symp. For. Soils, Harbin, People s Republic of China. 22-27 July. Northeast Forestry Univ., Harbin, People s Republic of China. 

For example, Wu (2000) computed the AUC, AUMC, and MRT for a 1-compartment model and then showed what impact changing the volume of distribution and elimination rate constant by plus or minus their respective standard errors had on these derived parameters. The difference between the actual model outputs and results from the analysis can then be compared directly or expressed as a relative difference. Alternatively, instead of varying the parameters by some fixed percent, a Monte Carlo approach can be used where the model parameters are randomly sampled from some distribution. Obviously this approach is more complex. A more comprehensive approach is to explore the impact of changes in model parameters simultaneously, a much more computationally intensive problem, possibly using Latin hypercube sampling (Iman, Helton, and Campbell, 1981), although this approach is not often seen in pharmacokinetics. 

PRISM (Gardner et al., 1983) as outer shell performs uncertainty analysis, using the Latin Hypercube Sampling for input parameters, allowing arbitrary parameter distributions and correlations ... 

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. 

Hypercube Sampling LHS, Latin Hypercube Sampling using conditional Median values LHS-M). A third group investigates the influence of different subsample strategies A scheme chosen to resemble the rule-of-thumb I = as close as possible from... 

Model 1 was used to compare results obtained using two different sampling schemes Simple Monte Carlo and Latin Hypercube sampling (LHS). No significant difference in the sensitivity indices was found. The effect of sample size was studied using as sample sizes 1000, 5000 and 25000. Completely irrelevant input parameters got closer and closer (with less dispersion) to null sensitivity indices estimates, while input parameters with some real impact on the output stayed far from zero for all sample sizes, with stable values. 

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 full Exposure Assessment model combines the food pathway characteristics, the predictive microbiological models for growth and reduction of Listeria and the probabilistic models in each step of food chain studied, which has been built as a spreadsheet model in Microsoft Excel with add on Risk 5.0 (Palisade Newfield). For one iteration of the Monte Carlo model, 350 batch of milk are simulated. Per simulation 10000 iterations are run using Latin Hypercube sampling, representing 10000 independent industrial batches and 105.000.000 packages of one litre of milk per year. 

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. 

The Latin Hypercube sampling aims at spreading the design points evenly on the basis of various geometric criteria. The reader may find it beneficial to learn about these designs using the function Ihsdesign of the Matlab Statistics Toolbox [40]. 

Florian, A. 1992. An efficient sampling scheme Updated Latin Hypercube Sampling. Proha-bilistic Engineering Mechanics 7, 123-130. 

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