It shouldn't matter if there are repeats or not if there is a uniform bias in the random number generator. However most pseudo-random number generators fail this criteria (ones in practice, in theory they often pass :) ).
What I was proposing you use are sometimes called quasi-random numbers. This is because they are not really too random at all. They have a heavy bias against being chosen twice. The advantage here is that you converge faster. Additionally in the real world they will be more likely to be correct (since any unatural skew in the bias would have to be tempered with the non-repeating bias).
The reason it converges faster is because the numbers chosen are forced to not repeat, but still spread themselves evenly through a given space. I hope you can see why this would converge more quickly -- it's a tad hard to draw an ascii picture to represent this.
Basically it's like using an iteratively increasing grid (like my last post), but then you do not have to commit to any particular number of samples (with the grid, you have to do 4 the first iteration, then 5 more the next, then 10 the next, etc).
I have to be off to work, but as an example here is a base two Hamilton sequence: 0.0, 0.1, 0.01, 0.11, 0.001, 0.101, 0.011, and 0.111 . You may want to convert these numbers into decimal so you can see the sequence: 0.000, 0.500, 0.250, 0.750, 0.125, 0.625, 0.375, 0.875 . Normally, you would use a higher base (which works best when the base is also a prime number), but using 2 lets you see how it is slowly filling up a grid for you. Sobol sequences produce more random results than Hamilton, so using them is more ideal. But anyway, you get the picture.
Ciao,
Gryn | [reply] |
