Posting about measuring clock drift got me in the mood of poking random numbers a bit more. Well, about an year ago I decided I needed normally distributed prng output, but never really used it for anything. I am sharing a solution in my best hope that someone will spare themselves the headbanging part.
The normal distribution is one of those things one should be acquainted with if about to deal with statistics, simulations or even procedural art. Anyhow, the main problem with normal pseudo-random numbers in flash is obtaining them, because the usual output of every prng follows the uniform distribution and one has to map that output to the distribution of interest. The reasonable solutions I have explored are three; one is using lookup tables, which is self-explanatory and very inexpensive, but unfortunately also painfully boring.
Another is making use of the central limit theorem, which in real life boils down to this. One can take, say, 10 numbers extracted from a uniformly distributed population (read Math.random) and take their average, whose sampling distribution will approximate to some extent the normal distribution.
I fancy using the Box-Muller transform or, more specifically, the Marsaglia polar method, which is a less computationally expensive variation of the former. Both take a pair of uniformly distributed inputs and map them to two (independent and uncorrelated) standard normal outputs. Effectively, this means that one could take any prng, be it Math.random, a Park-Miller lcg (again, check out the one on Polygonal labs), a Mersenne twister or whatever, and map its output to the standard normal distribution.
For better performance, the mapping algorithm could be built into the prng so that all operations are performed within one call. This is even more valid for the Marsaglia method, because it relies on rejective sampling, which means that it rejects a pair of random inputs every now and then and requests another. The code snippets below demonstrate the Marsaglia transform built into a Park-Miller prng, whose core logic consists in just a couple of lines:
public class ParkMiller
{
/**
* Seeds the prng.
*/
private var s : int;
public function seed ( seed : uint ) : void
{
s = seed > 1 ? seed % 2147483647 : 1;
}
/**
* Returns a Number ~ U(0,1)
*/
public function uniform () : Number
{
return ( ( s = ( s * 16807 ) % 2147483647 ) / 2147483647 );
}
When you call uniform(), the seed value is multiplied by 16807 (a primitive root modulo) and set to the remainder of the product divided by 2147483647 (a Mersenne prime, 2^31-1, or the int.MAX_VALUE). This new value is returned as a Number in the range (0,1).
The histogram below illustrates the uniform distribution of the prng output:
The uniform pseudo-random values will be fed to the Marsaglia transform, whose simple algorithm is well described in its Wikipedia article. An as3 implementation with inlined uniform() getters could look like this:
/**
* Returns a Number ~ N(0,1);
*/
private var ready : Boolean;
private var cache : Number;
public function standardNormal () : Number
{
if ( ready )
{ // Return a cached result
ready = false; // from a previous call
return cache; // if available.
}
var x : Number, // Repeat extracting uniform values
y : Number, // in the range ( -1,1 ) until
w : Number; // 0 < w = x*x + y*y < 1
do
{
x = ( s = ( s * 16807 ) % 2147483647 ) / 1073741823.5 - 1;
y = ( s = ( s * 16807 ) % 2147483647 ) / 1073741823.5 - 1;
w = x * x + y * y;
}
while ( w >= 1 || !w );
w = Math.sqrt ( -2 * Math.log ( w ) / w );
ready = true;
cache = x * w; // Cache one of the outputs
return y * w; // and return the other.
}
}
The following histogram displays the distribution of this new getter:
Performance-wise, there is some ground for reducing the algorithm’s overhead. One could, for ranges of values, approximate the square root with, for example, an inlined Babylonian calculation. The natural logarithm can also be easily approximated for values not too close to zero. Still, every approximation comes at the cost of some precision, and the above implementation will be fast enough for most uses; on my notebook I get a million numbers for about 350 milliseconds.
Finally, the Ziggurat algorithm is an alternative to the Marsaglia transform, and has the promise of better perfomance if well optimised. Still, I personally haven’t managed to make it work all that great in as3.
Source: ParkMiller.as
