Thursday, January 24, 2013

Sampling a Normal

The normal distribution is one of the most commonly used probability distributions because it turns out that many probabilities in the real world are approximately normal as a result of the central limit theorem. The standard normal distribution has a probability density function of

$$f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}$$
For something so common, its density function is not particularly nice to deal with; its cumulative distribution cannot be expressed in terms of elementary functions. This makes it a bit tricky to sample from as well, since the easiest way to sample a random variable is to sample a value from the uniform distribution on $[0, 1]$ and then invert the cumulative distribution function. Fortunately, people have come up with very clever ways of sampling a normal, and I'll provide a derivation of one, called the Box-Muller transform, here.

Consider the joint distribution of two independent normal variables:

$$f(x, y) = f(x)f(y) = \frac{1}{2\pi} e^{-\frac{1}{2}(x^2+y^2)}$$
This is a distribution across all points in the two-dimensional plane given by their $(x, y)$ coordinates. If we instead consider the polar representation of points, we can transform the given joint distribution via the change of variables formula. Using $x = r\cos{\theta}$ and $y = r\sin{\theta}$ gives (most calculations omitted for brevity)

$$f(r, \theta) = f(x, y) \left| \begin{array}{cc} \cos{\theta} & -r\sin{\theta} \\ \sin{\theta} & r\cos{\theta} \end{array} \right| = f(x, y) \cdot r = \frac{1}{2\pi} r e^{-\frac{1}{2} r^2}$$
Integrating out $\theta$ over the range $[0, 2\pi)$ leaves us with something that we can obtain the cumulative distribution for:

$$f(r) = r e^{-\frac{1}{2} r^2} \Rightarrow F(r) = \int_0^r f(t) dt = \int_0^r t e^{-\frac{1}{2} t^2} dt = 1 - e^{-\frac{1}{2} r^2}$$
Now we can apply the inversion technique to sample $r$; let $U_1$ be a random variable drawn from a uniform distribution on $[0, 1]$. We can compute $r = F^{-1}(1-U_1) = \sqrt{-2\ln{U_1}}$ (we decide to swap $U_1$ with $1-U_1$ to make the result nicer since the two have the same distribution). Notice that we can sample $\theta$ as well because it's actually uniformly distributed on $[0, 2\pi)$, so if $U_2$ is another uniform random variable on $[0, 1]$ then we can take $\theta = 2 \pi U_2$. Putting it all together, we get two (independent) sampled normals

$$x = r\cos{\theta} = \sqrt{-2\ln{U_1}} \cos{(2 \pi U_2)} \\ y = r\sin{\theta} = \sqrt{-2\ln{U_1}} \sin{(2 \pi U_2)}$$
To wrap things up and make it a bit more practical, here's a Java snippet of this in action.

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