Variance reduction

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The variance of randomly generated points within a unit square can be reduced through a stratification process.

In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates that can be obtained for a given number of iterations. Every output random variable from the simulation is associated with a variance which limits the precision of the simulation results. In order to make a simulation statistically efficient, i.e., to obtain a greater precision and smaller confidence intervals for the output random variable of interest, variance reduction techniques can be used. The main ones are: Common random numbers, antithetic variates, control variates, importance sampling and stratified sampling. Under these headings are a variety of specialized techniques; for example, particle transport simulations make extensive use of "weight windows" and "splitting/Russian roulette" techniques, which are a form of importance sampling.

Variation reduction also plays an important role in commerce as quality and safety ever increase in importance. The methods that Walter A. Shewhart established at the Hawthorne Plant remain today a fundamental tool to those involved in quality and continuous improvement. Shewhart's delineation between Assignable and Common causes of variation and the control chart to make this delineation operational have the unusual characteristic of applicability across a wide spectrum of life. The methods are as easily applied to manufacturing of materials in telephone manufacturing as they are to the making hamburgers at McDonald's or being a Barista at Starbucks. Wherever we are experiencing service or piece parts we want for goodness. We want for reduced variation, for everything to be great all the time, but we are confronted with variation.

The quest to improve quality has been going strong since at least the 1980s, and we have improved. In some industries improved tremendously.

Common Random Numbers (CRN)[edit]

The common random numbers variance reduction technique is a popular and useful variance reduction technique which applies when we are comparing two or more alternative configurations (of a system) instead of investigating a single configuration. CRN has also been called Correlated sampling, Matched streams or Matched pairs.

CRN requires synchronization of the random number streams, which ensures that in addition to using the same random numbers to simulate all configurations, a specific random number used for a specific purpose in one configuration is used for exactly the same purpose in all other configurations. For example, in queueing theory, if we are comparing two different configurations of tellers in a bank, we would want the (random) time of arrival of the Nth customer to be generated using the same draw from a random number stream for both configurations.

Underlying principle of the CRN technique[edit]

Suppose X_{1j} and X_{2j} are the observations from the first and second configurations on the jth independent replication.

We want to estimate

\xi= E(X_{1j})-E(X_{2j})=\mu_1-\mu_2. \,

If we perform n replications of each configuration and let

Z_j=X_{1j}-X_{2j} \quad\mbox{for } j=1,2,\ldots, n,

then E(Z_j)=\xi and Z(n) = \frac{\sum_{j=1,\ldots,n} Z_j}{n} is an unbiased estimator of \xi.

And since the Z_j's are independent identically distributed random variables,

\operatorname{Var}[Z(n)] = \frac{\operatorname{Var}(Z_j)}{n}.

In case of independent sampling, i.e., no common random numbers used then Cov(X1j, X2j) = 0. But if we succeed to induce an element of positive correlation between X1 and X2 such that Cov(X1j, X2j) > 0, it can be seen from the equation above that the variance is reduced.

It can also be observed that if the CRN induces a negative correlation, i.e., Cov(X1j, X2j) < 0, this technique can actually backfire, where the variance is increased and not decreased (as intended).

See also[edit]