Why does my quasi-random numbers, generated with the HALTONSET, not fill each of the dimensions?

I am generating some quasi-random numbers using the HALTONSET class. However, the points returned are not always uniformly distributed over each of the dimensions.

For example, the following code generates quasi-random points that should fill the 30 dimensional unit hypercube:

 

p = haltonset(30,'Skip',1e7,'Leap',1e2);
X = net(p,1000);
figure, boxplot(X)

However, you can see that values for dimension 26 are not evenly distributed over the unit interval.

Answer 1

[MathWorks Support Team]

This change has been incorporated into the documentation in Release 2010a (R2010a). For previous releases, read below for any additional information:

Many Leap values create sequences that fail to touch on large sub-hyper-rectangles of the unit hypercube, and so fail to be a uniform quasi-random point set.

A known rule for choosing Leap values for Halton sets is to set it to (P-1) where P is a prime number that has not been used to generate one of the dimensions, i.e. for a k-dimensional point set P would be the (k+1)th or greater prime.

So in this 30 dimensional example, a suitable Leap value would be the 31st prime - 1, i.e., 127-1 = 126:

 

p = haltonset(30,'Skip',1e7,'Leap',126);
X = net(p,1000);
boxplot(X)