Antithetic variates

In statistics, 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, a very large number of sample paths is required to obtain an accurate result. The antithetic variates method reduces the variance of the simulation results.

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

$\hat \theta = \frac{\hat \theta_1 + \hat \theta_2}{2}.$

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 independently and identically distributed, the covariance is zero 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.