Large Sample Properties of Simulations Using Latin Hypercube Sampling

Latin hypercube sampling (McKay, Conover, and Beckman 1979) is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables. The asymptotic variance of such an estimate is obtained. The estimate is also shown to be asymptotically normal. Asymptotically, the variance is less than that obtained using simple random sampling, with the degree of variance reduction depending on the degree of additivity in the function being integrated. A method for producing Latin hypercube samples when the components of the input variables are statistically dependent is also described. These techniques are applied to a simulation of the performance of a printer actuator.

Sampling efficiency in Monte Carlo based uncertainty propagation strategies: Application in seawater intrusion simulations

The convergence properties for Latin Hypercube Sampling (LHS) for reliability analyses is studied and compared with Simple Random Sampling (SRS) which has relatively wellknown error estimates. This paper summarizes the anticipated error of LHS estimates for percentile statistics of the response. For the relatively simple case, LHS yields dramatic improvements where the variance of the estimate is 1/N times that of SRS. However, for cases with many important variables, the improvement is diminished and often tends to a constant such as 1/3. In no case is the performance of LHS worse that SRS, so that one can expect LHS to continue to be a popular alternative to SRS when a direct sampling method is needed. Nomenclature fx = joint density function g = limit state function n = number of samples pf = probability of failure x = input variable(s) Z = response function

In this paper, two sampling methods which are Latin hypercube sampling (LHS) and simple random sampling were. compared to improve the modeling speed of neural network model. Sampling method was used to generate initial weights and bias set. Electrical characteristic data for <TEX>$HfO_2$</TEX> thin film was used as modeling data. 10 initial parameter sets which are initial weights and bias sets were generated using LHS and simple random sampling, respectively. Modeling was performed with generated initial parameters and measured epoch number. The other network parameters were fixed. The iterative 20 minimum epoch numbers for LHS and simple random sampling were analyzed by nonparametric method because of their nonnormality.

Uncertainty is endemic in geospatial data due to the imperfect means of recording, processing, and representing spatial information. Propagating geospatial model inputs inherent uncertainty to uncertainty in model predictions is a critical requirement in each model's impact assessment and risk-conscious policy decision-making. It is still extremely difficult, however, to perform in practice uncertainty analysis of model outputs, particularly in complex spatially distributed environmental models, partially due to computational constraints.In the field of groundwater hydrology, the "stochastic revolution" has produced an enormous number of theoretical publications and greatly influenced our perspective on uncertainty and heterogeneity; it has had relatively little impact, however, on practical modeling. Monte Carlo simulation using simple random (SR) sampling from a multivariate distribution is one of the most widely used family of methods for uncertainty propagation in hydrogeological flow and transport model predictions, the other being analytical propagation.Real-life hydrogeological problems however, consist of complex and non-linear three dimensional groundwater models with millions of nodes and irregular boundary conditions. The number of Monte-Carlo runs required in these cases, depends on the number of uncertain parameters and on the relative accuracy required for the distribution of model predictions. In the context of sensitivity studies, inverse modelling or Monte-Carlo analyses, the ensuing computational burden is usually overwhelming and computationally impractical. These tough computational constrains have to be relaxed and removed before meaningful stochastic groundwater modeling applications are possible.A computationally efficient alternative to classical Monte Carlo simulation based on SR sampling is Latin hypercube (LH) sampling, a form of stratified random sampling. The latter yields a more representative distribution of model outputs (in terms of smaller sampling variability of their statistics) for the same number of input simulated realizations. The ability to generate unbiased LH realizations becomes critical in a spatial context, where random variables are geo-referenced and exhibit spatial correlation, to ensure unbiased outputs of complex models. On this regard, this dissertation offers a detailed analysis of LH sampling and compares it with SR sampling in a hydrogeological context. Additionally, two alternative stratified sampling methods, here named stratified likelihood (SL) sampling and minimum energy (ME) sampling, are examined (proposed in a spatial context) and their efficiency is further compared to SR and LH in a hydrogeological context; also accounting for the uncertainty related to the particular model at hand via a two step sampling method. All three stratified sampling methods (accounting for model sensitivity in the second case study) were found in this work to be more efficient than simple random sampling.Additionally, this thesis proposes a novel method for the expansion of the application domain of LH sampling to very large regular grids which is the common case in environmental (hydrogeological or not) models. More specifically, a novel combination of Stein's Latin Hypercube sampling with a Monte Carlo simulation method applicable over high discretization domains is proposed, and its performance is further validated in 2D and 3D hydrogeological problems of flow and transport in a mid-heterogeneous porous media, both consisting of about $1$ million nodes. Last, an additional novel extension of the proposed LH sampling on large grids is adopted for conditional high discretized problems. In this case too, the performance of the proposed approach is evaluated in a 3D hydrogeological model of flow and transport. Results indicate that both extensions (conditional and not) of LH sampling on large grids facilitate efficient uncertainty propagation with fewer model runs due to more representative model inputs. Overall, it could be argued that all the proposed methodological approaches could reduce the time and computer resources required to perform uncertainty analysis in hydrogeological flow and transport problems. Additionally, since it is the first time that stratified sampling is performed over high discretization domains, it could be argued that the proposed extensions of LH sampling on large grids could be considered a milestone for future uncertainty analysis efforts. Moreover, all the proposed stratified methods could contribute to a wider application of uncertainty analysis endeavors in a Monte Carlo framework for any spatially distributed impact assessment study.

Latin hypercube sampling and geostatistical modeling of spatial uncertainty in a spatially explicit forest landscape model simulation

This chapter discusses the use of computer models for such diverse applications as safety assessments for geologic isolation of radioactive waste and for nuclear power plants; loss cost projections for hurricanes; reliability analyses for manufacturing equipment; transmission of HIV; and subsurface storm flow modelling. Such models are usually characterized by a large number of input variables (perhaps as many as a few hundred), and usually, only a handful of these inputs are important for a given response. In addition, the model response is frequently multivariate and time dependent. Latin hypercube sampling (LHS) uses a stratified sampling scheme to improve on the coverage of thek‐dimensional input space for such computer models. This means that a single sample will provide useful information when some input variable(s) dominate certain responses (or certain time intervals), while other input variables dominate other responses (or time intervals). By sampling over the entire range, each variable has the opportunity to show up as important, if it indeed is important. If an input variable is not important, then the method of sampling is of little or no concern. The values of the stratified sampling scheme can be paired to ensure a desired correlation structure among thekinput variables. LHS is more efficient than simple random sampling in a large range of conditions.

This entry discusses the use of computer models for such diverse applications as safety assessments for geologic isolation of radioactive waste and for nuclear power plants; loss cost projections for hurricanes; reliability analyses for manufacturing equipment; transmission of HIV; and subsurface storm flow modelling. Such models are usually characterized by a large number of input variables (perhaps as many as a few hundred), and usually, only a handful of these inputs are important for a given response. In addition, the model response is frequently multivariate and time dependent. Latin hypercube sampling (LHS) uses a stratified sampling scheme to improve on the coverage of thek‐dimensional input space for such computer models. This means that a single sample will provide useful information when some input variable(s) dominate certain responses (or certain time intervals), while other input variables dominate other responses (or time intervals). By sampling over the entire range, each variable has the opportunity to show up as important, if it indeed is important. If an input variable is not important, then the method of sampling is of little or no concern. The values of the stratified sampling scheme can be paired to ensure a desired correlation structure among thekinput variables. LHS is more efficient than simple random sampling in a large range of conditions.

&lt;p&gt;Reliability of pipe structure is one aspect to be considered in reactor safety analysis. MSC NASTRAN is a computer code that can be used to calculate pipe deflection for reliability evaluation. MSC PATRAN can be used to generate input for this code. Uncertainty evaluation needs to be done in the input variable to understand uncertainty range in the analysis results. A computer code for evaluating structure reliability has been developed in our previous study. The code has implemented latin hypercube sampling (LHS) to assess uncertainty in the input variable, such as load and modulus of elasticity. In this study, comparison of two uncertainty methods, i.e. simple random sampling (SRS) and LHS, was carried out for the developed software. The comparison was subjected to pipe deflection calculation using 100 samples. Comparison analysis shows that LHS method produces a robust mean of variance for all sample size. The results also confirm that variance of pipe deflection using LHS is smaller by 3% than SRS one. It can be concluded that LHS is appropriate to be implemented for uncertainty analysis in the developed code.&lt;/p&gt;&lt;p&gt;&lt;strong&gt; &lt;/strong&gt;&lt;/p&gt;

Monte Carlo simulation method combined with simple random sampling (SRS) suffers from long computation time and heavy computer storage requirement when used in probabilistic load flow (PLF) evaluation and other power system probabilistic analyses. This paper proposes the use of an efficient sampling method, Latin hypercube sampling (LHS) combined with Cholesky decomposition method (LHS-CD), into Monte Carlo simulation for solving the PLF problems. The LHS-CD sampling method is investigated using IEEE 14-bus and 118-bus systems. The method is compared with SRS and LHS only with random permutation (LHS-RP). LHS-CD is found to be robust and flexible and has the potential to be applied in many power system probabilistic problems.

The failure of stress corrosion fraction (SCF) will lead to the ejection of nuclear power plant related equipment, which will affect the safety and economy of nuclear power plant. In this paper, a SCF failure analysis method for primary coolant circuits materials was established based on Paris model, and the uncertainty of fracture toughness was transformed into an integral form to improve the calculation efficiency. Taking the weld of thermowell as an example, the probability of SCF failure was calculated, and the uncertainty was analyzed by Wilks’ method. The influences of simple random sampling (SRS), Latin hypercube sampling (LHS) and Halton low discrepancy sequence on the uncertainty quantification of the calculated results were studied. At the same time, a surrogate model was established based on polynomial chaos expansion (PCE) method to study whether this method was suitable for SCF probability calculation. The results show that Halton sequence with 1000 samples can make the mean and variance convergence of failure probability better than SRS and LHS. When calculating the upper limit of tolerance interval, the mean and median of results corresponding to LHS are similar to those of SRS, but the dispersion degree of LHS is lower than SRS, while the results corresponding to Halton sequence are smaller than those corresponding to SRS and LHS. The increase of Wilks’ order can reduce the conservatism. When the order is 4, both the computational efficiency and the computational accuracy are considered. The results of the surrogate model based on PCE are basically consistent with those of the original program, but the amount of calculation is greatly reduced. This method is suitable for SCF probability analysis.

Two methods for generating representative realizations from Gaussian and lognormal random field models are studied in this paper, with term representative implying realizations efficiently spanning the range of possible attribute values corresponding to the multivariate (log)normal probability distribution. The first method, already established in the geostatistical literature, is multivariate Latin hypercube sampling, a form of stratified random sampling aiming at marginal stratification of simulated values for each variable involved under the constraint of reproducing a known covariance matrix. The second method, scarcely known in the geostatistical literature, is stratified likelihood sampling, in which representative realizations are generated by exploring in a systematic way the structure of the multivariate distribution function itself. The two sampling methods are employed for generating unconditional realizations of saturated hydraulic conductivity in a hydrogeological context via a synthetic case study involving physically-based simulation of flow and transport in a heterogeneous porous medium; their performance is evaluated for different sample sizes (number of realizations) in terms of the reproduction of ensemble statistics of hydraulic conductivity and solute concentration computed from a very large ensemble set generated via simple random sampling. The results show that both Latin hypercube and stratified likelihood sampling are more efficient than simple random sampling, in that overall they can reproduce to a similar extent statistics of the conductivity and concentration fields, yet with smaller sampling variability than the simple random sampling.

Approximation of the exponential integral (well function) using sampling methods

A recently developed centroidal Voronoi tessellation (CVT) unstructured sampling method is investigated here to assess its suitability for use in statistical sampling and function integration. CVT efficiently generates a highly uniform distribution of sample points over arbitrarily shaped M-dimensional parameter spaces. It has recently been shown on several 2D test problems to provide superior point distributions for generating locally conforming response surfaces. Its performance as a statistical sampling and function integration method is compared to that of latin-hypercube sampling (LHS) and simple random sampling (SRS) Monte Carlo methods, and Halton and Hammersley quasiMonte-Carlo sequence methods. Specifically, sampling efficiencies are compared for function integration and for resolving various statistics of response in a 2D test problem. It is found that on balance CVT performs best of all these sampling methods on our test problems

Latin hypercube sampling and the sensitivity analysis of a Monte Carlo epidemic model