Generating Gaussian Random Numbers

Generating Gaussian Random Numbers

http://www.taygeta.com/random/gaussian.html

This note is about the topic of generating 

Gaussia 

pseudo-random numbers given a source of 

uniform 

pseudo-random numbers. This topic comes up more frequently than I would have expected, so I decided to write this up on 

one  of the best ways to do this. At the end of this note there is a list of references

 in the literature that are relevant to this topic. You can see some 

code example

  that implement the technique, and a  

tep-by-step

  example for generating 

Weibull 

distributed random numbers.

There are many ways of solving this problem (see for example Rubinstein, 1981, for an extensive discussion of this topic) but we will only go into one important method here. If we have an equation that describes our desired distribution function, then it is possible to use some mathematical trickery based upon the fundamental transformation law of probabilities to obtain a transformation function for the distributions. This transformation takes random variables from one distribution as inputs and outputs random variables in a new distribution function. Probably the most important of these transformation functions is known as the Box-Muller (1958) transformation. It allows us to transform uniformly distributed random variables, to a new set of random variables with a Gaussian (or Normal) distribution.

The most basic form of the transformation looks like:

         y1 = sqrt( - 2 ln(x1) ) cos( 2 pi x2 )

         y2 = sqrt( - 2 ln(x1) ) sin( 2 pi x2 )

We start with 

two

 independent random numbers, x1 and x2, which come from a uniform distribution (in the range from 0 to 1). Then apply the above transformations to get two new independent random numbers which have a Gaussian distribution with zero mean and a standard deviation of one.

This particular form of the transformation has two problems with it,

1. It is slow because of many calls to the math library.
2. It can have numerical stability problems when x1 is very close to zero.
These are serious problems if you are doing 

stochastic modelling

 and generating millions of numbers.

The polar form of the Box-Muller transformation is both faster and more robust numerically. The algorithmic description of it is:

         float x1, x2, w, y1, y2;

 

         do {

                 x1 = 2.0 * ranf() - 1.0;

                 x2 = 2.0 * ranf() - 1.0;

                 w = x1 * x1 + x2 * x2;

         } while ( w >= 1.0 );



         w = sqrt( (-2.0 * log( w ) ) / w );

         y1 = x1 * w;

         y2 = x2 * w;

where 

ranf()

 is the routine to obtain a random number uniformly distributed in [0,1]. The polar form is faster because it does the equivalent of the sine and cosine geometrically without a call to the trigonometric function library. But because of the possiblity of many calls to 

ranf()

, the uniform random number generator should be fast (I generally recommend 

R250

 for most applications).

 

 

Probability transformations for Non Gaussian distributions

Finding transformations like the Box-Muller is a tedious process, and in the case of empirical distributions it is not possible. When this happens, other (often approximate) methods must be resorted to. See the reference list below (in particular 

Rubinstein, 1981

) for more information.

There are other very useful distributions for which these probability transforms have been worked out. Transformations for such distributions as the Erlang, exponential,hyperexponential, and the Weibull distribution can be found in the literature (see for example,MacDougall, 1987).

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Useful References

• Box, G.E.P, M.E. Muller 1958; A note on the generation of random normal deviates, Annals Math. Stat, V. 29, pp. 610-611

 

1. Carter, E.F, 1994; The Generation and Application of Random Numbers , Forth Dimensions Vol XVI Nos 1 & 2, Forth Interest Group, Oakland California

 

1. Knuth, D.E., 1981; The Art of Computer Programming, Volume 2 Seminumerical Algorithms, Addison-Wesley, Reading Mass., 688 pages, ISBN 0-201-03822-6

 

1. MacDougall,M.H., 1987; Simulating Computer Systems, M.I.T. Press, Cambridge, Ma., 292 pages, ISBN 0-262-13229-X

 

1. Press, W.H., B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, 1986; Numerical Recipes, The Art of Scientific Computing, Cambridge University Press, Cambridge, 818 pages, ISBN 0-512-30811-9

 

1. Rubinstein, R.Y., 1981; Simulation and the Monte Carlo method, John Wiley & Sons, ISBN 0-471-08917-6

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See Also

: A 

Reference list

 of papers on Random Number Generation.

 

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 Everett (Skip) Carter          Phone: 831-641-0645 FAX:  831-641-0647

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