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As an illustration of the uniformly package, we will
show how to uniformly sample some points in a convex hull.
We give an illustration in dimension 3 (in dimension 2, use the
function runif_in_polygon).
Let’s store the vertices of an icosahedron in a matrix
vs:
vs <- t(rgl::icosahedron3d()$vb[1:3,])
head(vs)
#> [,1] [,2] [,3]
#> [1,] 0.000000 0.618034 1
#> [2,] 0.000000 0.618034 -1
#> [3,] 0.000000 -0.618034 1
#> [4,] 0.000000 -0.618034 -1
#> [5,] 0.618034 1.000000 0
#> [6,] 0.618034 -1.000000 0The icosahedron is convex, therefore its convex hull is itself.
The delaunayn function of the geometry
package calculates a “triangulation” (tetrahedralization in
dimension 3) of the convex hull of a set of points. We use it to get a
tetrahedralization of our icoshaedron:
library(geometry)
tetrahedra <- delaunayn(vs, options="Qz")
head(tetrahedra)
#> [,1] [,2] [,3] [,4]
#> [1,] 6 1 5 9
#> [2,] 6 3 1 9
#> [3,] 6 3 1 10
#> [4,] 6 12 4 2
#> [5,] 6 11 4 2
#> [6,] 6 11 5 9Each row of the tetrahedra matrix is a vector of four
identifiers of the vertices defining a tetrahedron.
Now, we calculate the volumes of each of these tetrahedra with the
volume_tetrahedron function:
library(uniformly)
volumes <-
apply(tetrahedra, 1,
function(t){
volume_tetrahedron(vs[t[1],], vs[t[2],], vs[t[3],], vs[t[4],])
})We normalize the volumes:
Now, here is the algorithm to uniformly sample a point in the icosahedron:
select a tetrahedron at random, with probability given by the normalized volumes;
uniformly sample a point in the picked tetrahedron.
That is:
i <- sample.int(nrow(tetrahedra), 1, prob=probs)
th <- tetrahedra[i,]
runif_in_tetrahedron(1, vs[th[1],], vs[th[2],], vs[th[3],], vs[th[4],])
#> [,1] [,2] [,3]
#> [1,] 0.5873218 0.7819155 -0.1332604Let’s use the algorithm to sample 100 random points:
nsims <- 100
sims <- matrix(NA_real_, nrow=nsims, ncol=3)
for(k in 1:nsims){
th <- tetrahedra[sample.int(nrow(tetrahedra), 1, prob=probs),]
sims[k,] <- runif_in_tetrahedron(1, vs[th[1],], vs[th[2],], vs[th[3],], vs[th[4],])
}library(rgl)
open3d(windowRect=c(100,100,600,600))
#> wgl
#> 1
shade3d(icosahedron3d(), color="red", alpha=0.3)
points3d(sims)
rglwidget()
We can proceed in the same way in higher dimension, using the
functions volume_simplex and runif_in_simplex
instead of the functions volume_tetrahedron and
runif_in_tetrahedron.
Note that the convexity is not the sine qua non condition to
apply the above procedure: the ingredient we need is the “triangulation”
of the object. We took a convex shape because delaunayn
provides the triangulation of a convex shape.
Let’s give an example for a 3D star. Here is the star:
vs <- rbind(
c(7.889562, 1.150329, -2.173651),
c(2.212808, 1.150329, -2.230414),
c(0.068023, 1.150328, -7.923502),
c(-2.151306, 1.150329, -2.254857),
c(-7.817406, 1.150328, -2.261558),
c(-3.523133, 1.150328, 1.888122),
c(-4.869315, 1.150328, 6.987552),
c(-0.006854, 1.150329, 4.473047),
c(4.838127, 1.150328, 7.041885),
c(3.538153, 1.150329, 1.927652),
c(0.033757, 0.000000, -0.314657),
c(0.035668, 2.269531, -0.312831)
)
faces <- rbind(
c(1, 11, 2),
c(2, 11, 3),
c(3, 11, 4),
c(4, 11, 5),
c(5, 11, 6),
c(6, 11, 7),
c(7, 11, 8),
c(8, 11, 9),
c(9, 11, 10),
c(10, 11, 1),
c(1, 12, 10),
c(10, 12, 9),
c(9, 12, 8),
c(8, 12, 7),
c(7, 12, 6),
c(6, 12, 5),
c(5, 12, 4),
c(4, 12, 3),
c(3, 12, 2),
c(2, 12, 1)
)
open3d(windowRect=c(100,100,600,600))
#> wgl
#> 2
for(i in 1:nrow(faces)){
triangles3d(rbind(
vs[faces[i,1],],
vs[faces[i,2],],
vs[faces[i,3],]),
color="red", alpha=0.4)
}
rglwidget()
This star is not convex but it is star-shaped with respect to its centroid, and its faces are triangular. Therefore we get a tetrahedralization by joining the centroid to each of the triangular faces.
Let’s calculate the volumes of these tetrahedra:
centroid <- colMeans(vs)
volumes <- apply(faces, 1,function(f){
volume_tetrahedron(vs[f[1],], vs[f[2],], vs[f[3],], centroid)
})
probs <- volumes/sum(volumes)Now we pick a face at random, with probability given by the normalized volumes, and we sample in the corresponding tetrahedron:
nsims <- 500
sims <- matrix(NA_real_, nrow=nsims, ncol=3)
for(k in 1:nsims){
f <- faces[sample.int(nrow(faces), 1, prob=probs),]
sims[k,] <- runif_in_tetrahedron(1, vs[f[1],], vs[f[2],], vs[f[3],], centroid)
}And now, let’s add the sampled points:
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