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'''Pseudo-random number sampling''' or '''non-uniform pseudo-random variate generation''' is the [[Numerical analysis|numerical]] practice of generating [[pseudo-random number]]s that are distributed according to a given [[probability distribution]].


Methods of sampling a non-[[Uniform distribution (continuous)|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 method|Monte-Carlo simulations]] in the [[Manhattan project]];{{Citation needed|date=June 2011}} they were first published by [[John von Neumann]] in the early 1950s.{{Citation needed|date=June 2011}}
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== 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 <nowiki>[</nowiki>0,&nbsp;1<nowiki>)</nowiki> is divided in ''n'' intervals [0,&nbsp;''f''(1)), [''f''(1),&nbsp;''f''(1)&nbsp;+&nbsp;''f''(2)),&nbsp;... The width of interval ''i'' equals the probability&nbsp;''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&nbsp;''f''(''i'').
 
Formalizing this idea becomes easier by using the cumulative distribution function
:<math>F(i)=\sum_{j=1}^i f(j).</math>
It is convenient to set ''F''(0)&nbsp;=&nbsp;0. The ''n'' intervals are then simply [''F''(0),&nbsp;''F''(1)), [''F''(1),&nbsp;''F''(2)), ..., [''F''(''n''&nbsp;&minus;&nbsp;1),&nbsp;''F''(''n'')). The main computational task is then to determine ''i'' for which ''F''(''i''&nbsp;&minus;&nbsp;1)&nbsp;≤&nbsp;''X''&nbsp;<&nbsp;''F''(''i'').
 
This can be done by different algorithms:
* [[Linear search]], computational time linear in&nbsp;''n''.
* [[Binary search]], computational time goes with&nbsp;log&nbsp;''n''.
* [[Indexed search]],<ref>Ripley (1987) {{Page needed|date=June 2011}}</ref> also called the ''cutpoint method''.<ref>Fishman (1996) {{Page needed|date=June 2011}}</ref>
* [[Alias method]], computational time is constant, using some pre-computed tables.
* There are other methods that cost constant time.<ref>Fishman (1996) {{Page needed|date=June 2011}}</ref>
 
== Continuous distributions ==
 
Generic methods for generating [[statistical independence|independent]] samples:
* [[Rejection sampling]]
* [[Inverse transform sampling]]
* [[Slice sampling]]
* [[Ziggurat algorithm]], for monotonously decreasing density functions
* [[Convolution random number generator]], not a sampling method in itself: it describes the use of arithmetics on top of one ore 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 filter]]s, when the observed data is connected in a [[Markov chain]] and should be processed sequentially
 
For generating a [[normal distribution]]:
* [[Box–Muller transform]]
* [[Marsaglia polar method]]
 
For generating a [[Poisson distribution]]:
* See [[Poisson distribution#Generating Poisson-distributed random variables]]
 
== Software Libraries ==
 
[[GNU Scientific Library]] has a section entitled "Random Number Distributions" with routines for sampling under more than twenty different distributions.
 
== Footnotes ==
 
{{reflist}}
 
== Literature ==
 
* 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.
* [[Donald Knuth|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.
 
{{DEFAULTSORT:Pseudo-Random Number Sampling}}
[[Category:Pseudorandom number generators]]
[[Category:Non-uniform random numbers]]

Latest revision as of 05:33, 2 December 2014


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