Monday, January 14, 2013

Box-Muller Transformation

The Box-Muller transform is a method for generating normally distributed random numbers from uniformly distributed random numbers. The Box-Muller transformation can be summarized as follows, suppose u1 and u2 are independent random variables that are uniformly distributed between 0 and 1 and let


then z1 and z2 are independent random variables with a standard normal distribution. Intuitively, the transformation maps each circle of points around the origin to another circle of points around the origin where larger outer circles are mapped to closely-spaced inner circles and inner circles to outer circles.
Let's see a Python snippet that implements the transformation:
from numpy import random, sqrt, log, sin, cos, pi
from pylab import show,hist,subplot,figure

# transformation function
def gaussian(u1,u2):
  z1 = sqrt(-2*log(u1))*cos(2*pi*u2)
  z2 = sqrt(-2*log(u1))*sin(2*pi*u2)
  return z1,z2

# uniformly distributed values between 0 and 1
u1 = random.rand(1000)
u2 = random.rand(1000)

# run the transformation
z1,z2 = gaussian(u1,u2)

# plotting the values before and after the transformation
figure()
subplot(221) # the first row of graphs
hist(u1)     # contains the histograms of u1 and u2 
subplot(222)
hist(u2)
subplot(223) # the second contains
hist(z1)     # the histograms of z1 and z2
subplot(224)
hist(z2)
show()
The result should be similar to the following:


In the first row of the graph we can see, respectively, the histograms of u1 and u2 before the transformation and in the second row we can see the values after the transformation, respectively z1 and z2. We can observe that the values before the transformation are distributed uniformly while the histograms of the values after the transformation have the typical Gaussian shape.

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