Re: Stochastic Positioning of A Point in A 3D Gaussian Distribution

On 1 Sep 2006 09:56:43 -0700, hoffmann@xxxxxxxxxxxx wrote:
1. Why is the Box-Muller Transform, which converts two PRNG
(pseudo random number generator) numbers u1,u2 with
uniform distributions into a geometrical pair x,y with Gaussian
distributions, better than a transform which converts simply
one PRNG number with uniform distribution into one number
with Gaussian distribution ?
http://en.wikipedia.org/wiki/Box-Muller_transform
The OP needs obviously THREE numbers for x,y,z.

2. Numbers by a PRNG with uniform distribution can be trans-
formed into numbers with Gaussian distribution either by adding
N numbers or by averaging N numbers.
E.g. one can find that N=6 is reasonable and N=12 is near to
perfect.
IMO this is perhaps faster than using Box-Muller.
Is any comparison of quality available ?

There is no reason anyone should need to write their own pseudo-random
number generator, whether for uniform or Gaussian distributions. It's
really easy to create a bad uniform generator, and even the venerable
Box-Muller generator for Gaussians has two major variations, one of
which is troublesome.

Four possibilities are:

* Box-Muller: this gives two independent variates for each call, and
is still fast even if one is discarded.
* Ratio: This is a method using rejection, and the ellipse regions of
Leva make it fast as well as brief.
* Averaging uniform: This is both slow and inaccurate, it is not to
be recommended even with some known adjustments.
* Ziggurat: A variation of the "rectangle-wedge-tail" approach, this
tends to be the fastest, but requires a large table.

Two good sources to consult are Luc Devroye,

<http://cg.scs.carleton.ca/~luc/rng.html>

(including his book, a free download),

<http://cg.scs.carleton.ca/~luc/books-luc.html>

and Knuth,

Knuth, D. The Art of Computer Programming, Vol. II., 3/e.

Or just grab an implementation, such as winrand,

<http://crypto.mat.sbg.ac.at/ftp/pub/data/winrand.zip>

or the GNU Scientific Library (GSL).

<http://www.gnu.org/software/gsl/>
.

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