Re: Query: Spatially uncorrelated reproducible random values

[Paul E Johnson]

Nigel Gilbert wrote:
> 
>  Do you know that there is no correlation between a seed
> and the (next) generated value, or is this just an assumption you've
> made?

NO, I was just shooting from the hip. Jan B wrote to me with the same
point.  I think I would have to experiment with it.  Since you have only
the goal of a number uncorrelated with the x,y position, I suppose I'd
test it after taking a couple of precautions.  Instead of using the
first draw after reseeding at a point, I'd use the 10th number drawn for
each position.  I think I saw a note from Ted Belding a while back on
this question. Perhaps Ted or Sven Thomessen is reading this thread and
can give us a definitive answer.

I suppose the meaning of "correlated" is open to debate here. Obviously,
using my suggestion, the values will be related to x,y because a known
process is used to generate the value from the coordinate. But it could
be uncorrelated, in a second sense that, to the observer, it is not
possible to predict the random value without knowing the formula. I
think this is the meaning of pseudo-randomness (or is it quasi
randomness).

I'm looking at the code for Swarm's MT19937 generator (thanks, Sven) and
when it draws a number on the basis of the state (or, at first, the
seed), it does the following.  I had the bit math on the top of my
tongue last month when I was doing the ArtificialStockMarket revision,
but the details have escaped me now, but it appears to me that the
numbers drawn after seeding are uncorrelated in the second sense above: 

if (index == N)
    {
      /* generate N words at one time */
      unsigned kk;

      for (kk = 0; kk < N - M; kk++)
        {
          y = (state[kk]&UPPER_MASK)|(state[kk+1]&LOWER_MASK);
          state[kk] = state[kk+M] ^ (y >> 1) ^ a[y & 0x1];
        }

      for (; kk < N - 1; kk++)
        {
          y = (state[kk]&UPPER_MASK)|(state[kk+1]&LOWER_MASK);
          state[kk] = state[kk+(M-N)] ^ (y >> 1) ^ a[y & 0x1];
        }

      y = (state[N-1] & UPPER_MASK) | (state[0] & LOWER_MASK);
      state[N - 1] = state[M - 1] ^ (y >> 1) ^ a[y & 0x1];

      index = 0;                // on subsequent rounds we start at 0
    }

  y = state[index++];

  y ^= ( y >> TEMPERING_SHIFT_U );
  y ^= ( ( y << TEMPERING_SHIFT_S ) & TEMPERING_MASK_B );
  y ^= ( ( y << TEMPERING_SHIFT_T ) & TEMPERING_MASK_C );
  y &= 0xffffffff; /* you may delete this line if word size = 32 */
  y ^= ( y >> TEMPERING_SHIFT_L );
 
  // -----------

  if (antiThetic)
    return (unsignedMax - y);
  else
    return y;


-- 
Paul E. Johnson                       email: address@hidden
Dept. of Political Science            http://lark.cc.ukans.edu/~pauljohn
University of Kansas                  Office: (785) 864-9086
Lawrence, Kansas 66045                FAX: (785) 864-5700


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