Pseudo-random number sampling
Pseudo-random number sampling or non-uniform pseudo-random variate generation is the numerical practice of generating pseudo-random numbers that are distributed according to a given probability distribution.
Methods of sampling a non-uniform distribution are typically based on the availability of a pseudo-random number generator producing numbers X that are uniformly distributed. 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.
Historically, basic methods of pseudo-random number sampling were developed for Monte-Carlo simulations in the Manhattan project; they were first published by John von Neumann in the early 1950s.
Finite discrete distributions
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
- Slice sampling
- Ziggurat algorithm, for monotonously 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.
Generic methods for generating correlated samples (often necessary for unusually-shaped or high-dimensional distributions):
- Markov chain Monte Carlo, the general principle
- Metropolis–Hastings algorithm
- Gibbs sampling
- Slice sampling
- Reversible-jump Markov chain Monte Carlo, when the number of dimensions is not fixed (e.g. when estimating a mixture model and simultaneously estimating the number of mixture components)
- Particle filters, when the observed data is connected in a Markov chain and should be processed sequentially
For generating a normal distribution:
For generating a Poisson distribution:
GNU Scientific Library has a section entitled "Random Number Distributions" with routines for sampling under more than twenty different distributions.
- Devroye, L. (1986) Non-Uniform Random Variate Generation. New York: Springer
- Fishman, G.S. (1996) Monte Carlo. Concepts, Algorithms, and Applications. New York: Springer
- Hörmann, W.; J Leydold, G Derflinger (2004,2011) Automatic Nonuniform Random Variate Generation. Berlin: Springer.
- Knuth, D.E. (1997) The Art of Computer Programming, Vol. 2 Seminumerical Algorithms, Chapter 3.4.1 (3rd edition).
- Ripley, B.D. (1987) Stochastic Simulation. Wiley.