Latin hypercube

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

The Systems Module constructs and displays fault trees using EASYFLOW which aic read automatically to generate minimal cutsets that can be transferred, for solution, to SETS. CAFT A. or IRRAS and then transferred to RISKMAN for point estimates and uncertainty analysi,s using Monte Carlo simulations or Latin hypercube. Uncertainty analysis is performed on the systems lev el using a probability quantification model and using Monte Carlo simulations from unavailability distributions. 

Iman, R. L. and M. J. Shortencarrier, 1984, A FORTRAN 77 Program and User s Guide for the Generation of Latin Hypercube and Random Samples for Use with Computer Model, NUREG/CR-3624, March. 

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 ... 

Sequences. See also Hammersley sequence sampling (HSS) Latin hypercube entries Sobol sequence quasi-Monte Carlo, 26 1011, 1013 quasirandom, 26 1016, 1036, 1048 Sequence tagged sites (STS), 12 513, 515 Sequencing batch reactor (SBR)... 

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

Tang B. 1993. Orthogonal array-based Latin hypercubes. J Am Stat Assoc 88 1392-1397. 

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). 

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. 

A spacefilling Latin Hypercube Design was used to generate model control variable input sets that were spread throughout factor space. Statistical tools were used to develop models from the results of the simulation experiments" and to suggest additional simulation points to explore regions of interest. Over 130 different sets of control variables were eventually simulated. 

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... 

Iman RL, Shortencarier MJ (1984) A FORTRAN 77 program and user s guide for the generation of Latin hypercube and random samples for use with computer models. Albuquerque, NM, Sandia National Laboratories (Report Nos. SAND83-2365 and NUREG/CR-3624). 

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. 

Figure 1 shows scatter plots of the raw data from a Latin hypercube experimental design (see McKay et al., 1979) with 500 runs of the Wonderland code. The output variable HDI is plotted against two of the input variables shown in Section 6 to be important economic innovation in the north (e.inov.n) and sustainable pollution in the south (v.spoll.s). 

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