ExampleĪverage Joe has been offered free wood scraps from the local lumber yard. Independant uniform selection is done on each of the variable’s generated values. Once each variable has been sampled using this method, a random grouping of variables is selected for each Monte Carlo calculation. How this is done is different for every distribution, but it’s generally just a matter of reversing the process of the probability function. P(X <= x) = n is solved for x, where n is the random point in the segment. This number would be used to calculate the actual variable value based upon its distribution. For the second segment, a number would be chosen between 0.2% and 0.4%. For the first segment, a number would be chosen between 0.0% and 0.2%. A probability is randomly picked within each segment using a uniform distribution, and then mapped to the correct representative value in of the variable’s actual distribution.Ī simulation with 500 iterations would split the probability into 500 segments, each representing 0.2% of the total distribution. To perform the stratified sampling, the cumulative probability (100%) is divided into segments, one for each iteration of the Monte Carlo simulation. A cumulative frequency plot of “recovery factor”, which was log-normally distributed with a mean of 60% and a standard deviation of 5%. 500 samples were taken using the stratified sampling method described here, which generated a very smooth curve.įigure 2. Differences within the plot, such as the left axis label and the black lines, are due to ongoing development of the software application and are not related to the issue being demonstrated.)įigure 1. (These figures were generated using different versions of the same software. Figure 1 and figure 2 demonstrate the difference between a pure random sampling and a stratified sampling of a log-normal distribution. The sampling algorithm ensures that the distribution function is sampled evenly, but still with the same probability trend. Variables are sampled using a even sampling method, and then randomly combined sets of those variables are used for one calculation of the target function. The concept behind LHS is not overly complex. There are many resources available describing Monte-Carlo ( history, examples, software). Monte-Carlo simulations provide statistical answers to problems by performing many calculations with randomized variables, and analyzing the trends in the output data. LHS can be incorporated into an existing Monte Carlo model fairly easily, and work with variables following any analytical probability distribution. The method commonly used to reduce the number or runs necessary for a Monte Carlo simulation to achieve a reasonably accurate random distribution. Les trois réplicats peuvent tous reproduire précisément les cdfs RS-MC correspondantes pour toutes les concentrations de BTEX dans les trois phases.Latin hypercube sampling (LHS) is a form of stratified sampling that can be applied to multiple variables. La stabilité du LHS-MC est également évaluée en comparant trois réplicats d’un LHS-MC. Les cdfs LHS-MC pour les trois différentes tailles d’échantillons peuvent reproduire avec précision les cdfs RS-MC correspondantes pour les concentrations de benzène, de toluène, d’éthylbenzène et de xylène (BTEX) dans les phases liquides, gazeuses et solides. Pour évaluer la capacité du LHS-MC à produire des fonctions de distribution cumulative (cdfs) qui reproduit les cdfs de l’échantillonnage aléatoire Monte Carlo (RS-MC), les cdfs de sortie obtenues avec des tailles d’échantillons LHS-MC de 100, 300 et 500 ainsi qu’un échantillon RS-MC de taille 10,000 sont comparées en utilisant les tests de Kolmogoroff-Smirnoff à deux échantillons. Dans le but de faciliter l’analyse d’incertitude d’un modèle MOFAT de transport à éléments finis polyphasé et à multicomposantes, cet article se veut un guide pour la sélection de la dimension de l’échantillonnage latin hypercube Monte Carlo (LHS-MC).
0 Comments
Leave a Reply. |