The Marsaglia polar method is a pseudo-random number sampling method for generating a pair of independent standard normal random variables.
Standard normal random variables are frequently used in computer science, computational statistics, and in particular, in applications of the Monte Carlo method.
The polar method works by choosing random points (x, y) in the square −1 < x < 1, −1 < y < 1 until
and then returning the required pair of normal random variables as
or, equivalently,
where and represent the cosine and sine of the angle that the vector (x, y) makes with x axis.
The underlying theory may be summarized as follows:
If u is uniformly distributed in the interval 0 ⤠u < 1, then the point (cos(2ÃÂu), sin(2ÃÂu)) is uniformly distributed on the unit circumference x<sup>2</sup> + y<sup>2</sup> = 1, and multiplying that point by an independent random variable àwhose distribution is
will produce a point
whose coordinates are jointly distributed as two independent standard normal random variables.
This idea dates back to Laplace, whom Gauss credits with finding the above
by taking the square root of
The transformation to polar coordinates makes evident that ø is uniformly distributed (constant density) from 0 to 2ÃÂ, and that the radial distance r has density
(r<sup>2</sup> has the appropriate chi square distribution.)
This method of producing a pair of independent standard normal variates by radially projecting a random point on the unit circumference to a distance given by the square root of a chi-square-2 variate is called the polar method for generating a pair of normal random variables,
A direct application of this idea,
is called the BoxâÂÂMuller transform, in which the chi variate is usually generated as
but that transform requires logarithm, square root, sine and cosine functions. On some processors, the cosine and sine of the same argument can be calculated in parallel using a single instruction. Notably for Intel-based machines, one can use fsincos assembler instruction or the expi instruction (available e.g. in D), to calculate complex
and just separate the real and imaginary parts.
Note: To explicitly calculate the complex-polar form use the following substitutions in the general form,
Let and Then
In contrast, the polar method here removes the need to calculate a cosine and sine. Instead, by solving for a point on the unit circle, these two functions can be replaced with the x and y coordinates normalized to the radius. In particular, a random point (x, y) inside the unit circle is projected onto the unit circumference by setting and forming the point
which is a faster procedure than calculating the cosine and sine. Some researchers argue that the conditional if instruction (for rejecting a point outside of the unit circle), can make programs slower on modern processors equipped with pipelining and branch prediction. Also this procedure requires about 27% more evaluations of the underlying random number generator (only of generated points lie inside of unit circle).
That random point on the circumference is then radially projected the required random distance by means of
using the same s because that s is independent of the random point on the circumference and is itself uniformly distributed from 0 to 1.
A simple implementation in Python:
Simple implementation in Java using the mean and standard deviation:
A non-thread safe implementation in C++ using the mean and standard deviation:
C++11 GNU GCC libstdc++'s implementation of std::normal_distribution uses the Marsaglia polar method, as quoted from herein.
A simple Julia implementation:The for loop can be parallelized by using the <code>Threads.@threads</code> macro.