Box–Muller Gaussian Transform

📐 Definition

The Box–Muller method maps two independent uniform samples on ((0,1)) to a pair of independent standard normal variables using logarithm, square root, and trigonometric transforms.

Z_1 = \sqrt{-2\ln U_1}\,\cos(2\pi U_2),\qquad Z_2 = \sqrt{-2\ln U_1}\,\sin(2\pi U_2)

Domain and Codomain

Accepts uniform random inputs; outputs real-valued Gaussian samples with zero mean and unit variance.

⚙️ Key Properties

The transform yields independent $N(0,1)$ variables. The Jacobian ensures the correct density under ideal arithmetic; extremely small $U_1$ values are often clamped to avoid overflow in $\ln U_1$ .

Fixing $U_2$ to a constant collapses the phase and destroys the rotational symmetry; the outputs are no longer Gaussian (e.g. $U_2=0$ gives $Z_2=0$ and $Z_1=\sqrt{-2\ln U_1}$ , which is Rayleigh-distributed). Restricting to one output simply discards half of each uniform pair; each component is still $N(0,1)$ .

🎯 Special Cases and Limits

Using only $Z_1$ discards half of each uniform pair.
Extremely small $U_1$ requires care due to $\ln U_1$ .

Gaussian white noise uses the generated samples; inverse-CDF Gaussian sampling provides an alternative; the Marsaglia polar method avoids trigonometric evaluations while producing the same distribution.

Usage in Oakfield

Oakfield uses Box–Muller-style transforms internally when it needs Gaussian deviates from uniform RNGs:

White-noise stimuli (e.g. stimulus_white_noise) use Gaussian sampling to populate noise fields deterministically from a seed.
Stochastic operators (e.g. stochastic_noise) generate Gaussian sources for OU-style updates.
Integrator stochastic hooks use an internal RNG to generate Gaussian noise by default (with uniform/Laplace alternatives available).

Historical Foundations

📜 Transform Sampling

The Box–Muller transform is a classic method for turning uniform random variables into Gaussian samples using an analytic change of variables.

🌍 Modern Perspective

While often superseded by faster methods in high-performance settings, it remains a standard reference implementation due to simplicity and exactness.

📚 References

Box & Muller, “A Note on the Generation of Random Normal Deviates” (1958)
Devroye, Non-Uniform Random Variate Generation