About the Monte Carlo Sampling Techniques

Sampling Techniques Overview

Simple Random Sampling is the basic, most commonly used Monte Carlo simulation technique; however, the number of simulations necessary for the Simple Random Sampling technique to converge on a solution is usually more than desirable and often more than practical.

Other sampling techniques, called variance reduction techniques, have been developed to reduce the sample size (number of simulations) without sacrificing the quality of the statistical description of the behavior of the system. These techniques reduce the variance of the statistical estimates derived from the Monte Carlo simulation data. As a result, the error in estimates is reduced (estimates from multiple simulations are more consistent). Alternatively, fewer points are needed with variance reduction techniques to obtain error or confidence levels similar to those obtained through simple random sampling. Both Descriptive Sampling and Sobol Sampling are variance reduction techniques. Sobol sampling is best suited when the number of samples needed to estimate a statistic is not known a priori and convergence is expected with fewer samples than provided. Descriptive sampling is best suited when the exact number of samples is known and a convergence check is not performed.

The figure below shows two random variables in standard space (U-space). Each row and column in the discretized two-variable space is sampled only once in random order. The cloud of points generated using Simple Random Sampling is shown for comparison.

The difference between simple random sampling and descriptive sampling or Sobol sampling cannot necessarily be observed with a single simulation using each technique. The actual estimates may be similar. However, through repeated simulation (with different randomization), it is observed that the variance of the set of estimates from descriptive sampling or Sobol sampling will be less than that from simple random sampling. The range of values observed for the estimates from descriptive sampling or Sobol sampling will be small (tighter range); therefore, the confidence in the estimates is increased. Given this property of descriptive sampling or Sobol sampling, and variance reduction techniques in general, descriptive sampling or Sobol sampling will, on average, provide better estimates for the same number of sampling points as simple random sampling, but different comparable estimates can be obtained with fewer sampling points.

Simple Random Sampling

The Simple Random Sampling technique generates sample points by generating N uniformly distributed random numbers between 0 and 1 for each random variable and obtaining corresponding values from each random variable distribution. The Simple Random Sampling technique is the easiest to understand and is the default selection. However, if the Monte Carlo Simulation is computationally expensive, you may want to try the Descriptive Sampling technique.

Descriptive Sampling

The Descriptive Sampling technique generates sample points by dividing each random variable distribution into N intervals of equal probability and randomly combining samples from these intervals for each random variable to produce design points.

The Descriptive Sampling technique divides the space defined by each random variable into subsets of equal probability, and the analysis is performed with each subset of each random variable only once (each subset of one random variable is combined with only one subset of each other random variable).

Sobol Sampling

Sobol Sampling generates a quasi-random sequence of numbers that are more uniformly distributed than both simple random sampling and descriptive sampling. Samples obtained using Sobol's sequences exhibit a probability density function that is closer to the true density function. The Sobol Sampling technique generates numbers as binary fractions of appropriate length from a set of special binary fractions.