Illustration

To illustrate the method, we choose $N=5$, and $D=3$ so that $N-D=2$ so that the data will always appear uniformly distributed when projected on a plane. Figure 5.8 shows the data as seen from the $x_2,x_5$ plane after 4 and 20 iterations of the systematic method. The five facets of the valid region are caused by the manifold reaching the positivity limit of each of the five dimensions. Non-uniformity can be clearly seen for 4, but not for 20 iterations.

Figure: Visual uniformity test for $N=5$, $D=3$. The 2-D manifold as seen from the two dimensions $x_2,x_5$ after 4 iteration (left) and 20 iterations (right).

Later, we will discuss the difference between the systematic and the random-directions algorithm. But first, we must solve for the asymptotic mean.