randomFEMpts {SpatialfdaR}

Generate Random Locations in a Mesh with a Specified Density

Description

The probability that a random location is assigned to a triangle is defined by an input vector of probabilities, one per triangle. But if DensityVec = 1, any location within the mesh is accepted.

Usage

  randomFEMpts(Nsample, pts, tri, logDensityVec=rep(0,ntri))

Arguments

Nsample

The number of random locations to be generated.

pts

A two-column matrix of locations of nodes.

tri

A three-column matrix of integers specifying the vertices of triangles in anti-clockwise order.

logDensityVec

A one-column matrix of log probability density values, one for each triangle.

Details

Within triangles, the locations are uniformly distributed. If DensityVec = 1, they are uniformly distributed over the whole mesh.

Value

A two-column matrix with N rows specifying the locations of the random points.

Author(s)

Jim Ramsay

References

Sangalli, Laura M., Ramsay, James O., Ramsay, Timothy O. (2013), Spatial spline regression models, Journal of the Royal Statistical Society, Series B, 75, 681-703.

See Also

link{plotFEM.mesh}

Examples

#  ---------  density on right triangle  --------------
#  Define the locations of the vertices of the right triangle
pts  <- matrix(c(0,0, 1,0, 0,1), 3, 2, byrow=TRUE)
npts <- dim(pts)[1]
#  Specify the single triangle defined by these vertices.
#  The vertices are listed in counter-clockwise order
tri  <- matrix(c(1,2,3),1,3)
ntri <- dim(tri)[1]
#  Set up the FEM basis object for this region
edg <- NULL
RtTriBasis <- create.FEM.basis(pts, edg, tri, 1)
#  List object containing details of nodes.
nodeList <- makenodes(pts,tri,1)
#  number of random points required
Nsample <- 100
#  generate random points ... define the function
obspts <- randomFEMpts(Nsample, pts, tri)
#  plot the random points
plotFEM.mesh(pts, tri)
points(obspts[,1],obspts[,2])
#  -----------------  density on a circle  ---------------
angle = seq(0,11*pi/6,len=12)
#  define the 24 point locations
rad = 2
ctr = c(0,0)
pts  = matrix(0,25,2);
pts[ 1:12,1] = rad*cos(angle)   + ctr[1]
pts[ 1:12,2] = rad*sin(angle)   + ctr[2]
pts[13:24,1] = rad*cos(angle)/2 + ctr[1]
pts[13:24,2] = rad*sin(angle)/2 + ctr[2]
#  define the edge indices
edg <- matrix(0,12,2)
for (i in 1:11) {
  edg[i,1] <- i
  edg[i,2] <- i+1
}
edg[12,] <- c(12,1)
#  use delaunay triangulation to define the triangles
tri = delaunayn(pts)
#  plot the triangulation of the circle
plotFEM.mesh(pts, tri, xlim=c(-2,2), ylim=c(-2,2))
#  Define the true log density function
SinCosIntensFn <- function(x,y) {
  return(sin(pi*x/2)*cos(pi*y/2))
}
#  locate observation points with sin*cos log density
intDensityVec <- triDensity(pts, tri, SinCosIntensFn)
#  number of random points required
Nsample <- 100
#  generate random points ... define the function
obspts <- randomFEMpts(Nsample, pts, tri, intDensityVec)
points(obspts[,1], obspts[,2], pch="o")