Van der Corput

Let’s generate a 10 point van der Corput sequence.

npoints = 10
qmc <- data.frame(j = 1:npoints, x = halton(npoints))
ggplot(qmc,aes(x=x,y=j)) + geom_point(size=4) + scale_y_continuous('Sample Number')

ggplot(qmc,aes(x=j,y=x)) + geom_line() + geom_point(size=4) + scale_x_continuous('Sample Number')

Halton Sequence

Now a Halton Sequence in 2-D

npoints = 50
xy = halton(npoints,dim=2)
qmc <- data.frame(j = 1:npoints)
qmc[,c("x","y")] = xy
ggplot(qmc,aes(x=x,y=y)) + geom_point(size=4)

The warts begin to appear in a Halton Sequence in 6-D

npoints = 50
xy = halton(npoints,dim=6)
qmc <- data.frame(j = 1:npoints)
qmc[,2:7] = xy
ggpairs(qmc,columns=2:7)

15-D is just plain bad.

npoints = 50
xy = halton(npoints,dim=15)
qmc <- data.frame(j = 1:npoints)
qmc[,2:16] = xy
ggpairs(qmc,columns=10:15)

Sobol Sequnce

In 2-D the Sobol sequence is similar to Halton.

npoints = 50
xy = sobol(npoints,dim=2)
qmc <- data.frame(j = 1:npoints)
qmc[,c("x","y")] = xy
ggplot(qmc,aes(x=x,y=y)) + geom_point(size=4)

In 6-D the Sobol appears to be somewhat better than Halton, but still imperfect.

npoints = 50
xy = sobol(npoints,dim=6)
qmc <- data.frame(j = 1:npoints)
qmc[,2:7] = xy
ggpairs(qmc,columns=2:7)

15-D is much better than Halton.

npoints = 50
xy = sobol(npoints,dim=15)
qmc <- data.frame(j = 1:npoints)
qmc[,2:16] = xy
ggpairs(qmc,columns=10:15)

Torus

In 1-D the torus is the fractional part of \(\sqrt{2}\):

npoints = 10
qmc <- data.frame(j = 1:npoints, x = torus(npoints), y = (1:npoints)*sqrt(2))
ggplot(qmc,aes(x=x,y=y,color=j)) + geom_point(size=4) + scale_y_continuous('j times root 2') + scale_colour_gradient(low="blue",high="red")

In 2-D the torus doesn’t appear too random:

npoints = 50
xy = torus(npoints,dim=2)
qmc <- data.frame(j = 1:npoints)
qmc[,c("x","y")] = xy
ggplot(qmc,aes(x=x,y=y)) + geom_point(size=4)

In 6-D torus is probably the worst

npoints = 50
xy = torus(npoints,dim=6)
qmc <- data.frame(j = 1:npoints)
qmc[,2:7] = xy
ggpairs(qmc,columns=2:7)

15-D

npoints = 50
xy = torus(npoints,dim=15)
qmc <- data.frame(j = 1:npoints)
qmc[,2:16] = xy
ggpairs(qmc,columns=10:15)