Voronoi weighting of samples in Monte Carlo intergration

conference paper

English

The standard way to numerically calculate integrals such as the ones featured in estimation of statistical moments of functions of random variables using Monte Carlo procedure is to: (i) perform selection of samples from the random vector, (ii) approximate the integrals using averages of the functions evaluated at the sampling points. If the N sim points are selected with an equal probability (with respect to the joint distribution function) such as in Monte Carlo sampling, the averages use equal weights 1/N sim . The problem with Monte Carlo sampling is that the estimated values exhibit a large variance due to the fact that the sampling points are usually not spread uniformly over the domain of sampling probabilities. One way to improve the accuracy would be to perform a more advanced sampling. The paper explores another way to improve the Monte Carlo integration approach: by considering unequal weights. These weights are obtained by transforming the sampling points into sampling probabilities (points within a unit hypercube), and subsequently by associating the sampling points with weights obtained as volumes of regions/cells around the sampling points within a unit hypercube. These cells are constructed by the Voronoi tessellation around each point. Supposedly, this approach could have been considered superior over the naive one because it can suppress inaccuracies stemming from clusters of sampling points. The paper also explores utilization of the Voronoi diagram for identification of optimal locations for sample size extension.

Voronoi tessellation, Reweighting, Monte Carlo integration, Weighted statistical estimators, Sampling plan

VOŘECHOVSKÝ, M.; SADÍLEK, V.; ELIÁŠ, J.

15. 6. 2017

978-618-82844-4-9

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering

478

491

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