Antithetic variates

The antithetic variates method is a variance reduction technique used in Monte Carlo methods. Considering that the error reduction in the simulated signal (using Monte Carlo methods) has a square root convergence (standard deviation of the solution), a very large number of sample paths is required to obtain an accurate result.

Underlying principle

The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path $\{\varepsilon_1,\dots,\varepsilon_M\}$ to also take $\{-\varepsilon_1,\dots,-\varepsilon_M\}$. The advantage of this technique is twofold: it reduces the number of normal samples to be taken to generate N paths, and it reduces the variance of the sample paths, improving the accuracy.

Suppose that we would like to estimate

$\theta = \mathrm{E}( h(X) ) = \mathrm{E}( Y ) \,$

For that we have generated two samples

$Y_1\text{ and }Y_2 \,$

An unbiased estimate of ${\theta}$ is given by

$\hat \theta = \frac{Y_1 + Y_2}{2}.$

And

$\text{Var}(\hat \theta) = \frac{\text{Var}(Y_1) + \text{Var}(Y_2) + 2\text{Cov}(Y_1,Y_2)}{4}$

In the case where Y₁ and Y₂ are iid, covariance self-cancels and $\text{Var}(Y_1) = \text{Var}(Y_2)$, therefore

$\text{Var}(\hat \theta) = \frac{\text{Var}(Y_1) }{2} = \frac{\text{Var}(Y_2) }{2}.$

The antithetic variates technique consists in this case of choosing the second sample in such a way that $Y_1$ and $Y_2$ are not iid anymore and $Cov(Y_1,Y_2)$ is negative. As a result, $\text{Var}(\hat \theta)$ is reduced and is smaller than the previous normal variance $\frac{\text{Var}(Y_1) }{2} = \frac{\text{Var}(Y_2) }{2}$.

Example 1

If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be $u_1, \ldots, u_n$, where, for any given i, $u_i$ is obtained from U(0, 1). The second sample is built from $u'_1, \ldots, u'_n$, where, for any given i: $u'_i = 1-u_i$. If the set $u_1$ is uniform along [0, 1], so are $u'_i$. Furthermore, covariance is negative, allowing for initial variance reduction.

Example 2: integral calculation

We would like to estimate

$I = \int_0^1 \frac{1}{1+x} \, \mathrm{d}x.$

The exact result is $I=\ln 2 \approx 0.69314718$. This integral can be seen as the expected value of $f(U)$, where

$f(x) = \frac{1}{1+x}$

And U follows a uniform distribution [0, 1].

The following table compares the classical Monte Carlo estimate (sample size: 2n, where n = 1500) to the antithetic variates estimate (sample size: n, completed with the transformed sample 1 − u_i):

	Estimate	Variance
Classical Estimate	0,69365	0,02005
Antithetic Variates	0,69399	0,00063

The use of the antithetic variates method to estimate the result shows an important variance reduction.