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Concept
Non-uniform random variate generation
Summary
Non-uniform random variate generation or pseudo-random number sampling is the numerical practice of generating pseudo-random numbers (PRN) that follow a given probability distribution.
Methods are typically based on the availability of a uniformly distributed PRN generator. Computational algorithms are then used to manipulate a single random variate, X, or often several such variates, into a new random variate Y such that these values have the required distribution.
The first methods were developed for Monte-Carlo simulations in the Manhattan project, published by John von Neumann in the early 1950s.
For a discrete probability distribution with a finite number n of indices at which the probability mass function f takes non-zero values, the basic sampling algorithm is straightforward. The interval [0, 1) is divided in n intervals [0, f(1)), [f(1), f(1) + f(2)), ... The width of interval i equals the probability f(i).
One draws a uniformly distributed pseudo-random number X, and searches for the index i of the corresponding interval. The so determined i will have the distribution f(i).
Formalizing this idea becomes easier by using the cumulative distribution function
It is convenient to set F(0) = 0. The n intervals are then simply [F(0), F(1)), [F(1), F(2)), ..., [F(n − 1), F(n)). The main computational task is then to determine i for which F(i − 1) ≤ X < F(i).
This can be done by different algorithms:
Linear search, computational time linear in n.
Binary search, computational time goes with log n.
Indexed search, also called the cutpoint method.
Alias method, computational time is constant, using some pre-computed tables.
There are other methods that cost constant time.
Generic methods for generating independent samples:
Rejection sampling for arbitrary density functions
Inverse transform sampling for distributions whose CDF is known
Ratio of uniforms, combining a change of variables and rejection sampling
Slice sampling
Ziggurat algorithm, for monotonically decreasing density functions as well as symmetric unimodal distributions
Convolution random number generator, not a sampling method in itself: it describes the use of arithmetics on top of one or more existing sampling methods to generate more involved distributions.
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