triDensity {SpatialfdaR}

Compute the probabilities that a random location will be within one of the triangles of an FEM mesh.

Description

Given a function defining a log-density surface for an FEM density, compute the probabilities that a random location will be within each of the triangles of an FEM mesh. This involves numerical quadrature over each rectangle, and nquad is the order of approximation.

Usage

triDensity(pts, tri, logIntensFn, nquad=5)

Arguments

pts

A two-column matrix with each row corresponding to a location within a two-dimensional region bounded by line segments and containing a triangular mesh.

tri

A three-column matrix indexing for each triangle its vertices.

logIntensFn

A function of x and y giving the surface height for an unnormalized log density function. It must return a square matrix of order equal to nquad. This implies that any multiplications must be pointwise.

nquad

The order of the quadrature over a triangle. The default order of five is good for most applications.

Value

A vector of length equal to the number of triangles in a mesh containing the probabilities that a random observation will fall within the triangles.

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

smooth.FEM.density

Examples

#  ---------  right triangle mesh  with zero log density --------------
#  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)
#  set up function to provide log intensity zero
ZeroFn <- function(x,y) {
  xdim <- dim(x)
  ZeroIntens <- matrix(0,xdim[1],xdim[1])
  return(ZeroIntens)
}
#  Define the true log density, which is flat with height 0
intDensityVec <- triDensity(pts, tri, ZeroFn)
#  ---------  square mesh with sin*cos log density  --------------
nsquare <- 3
result <- squareMesh(nsquare)
pts <- result$p
tri <- result$t
pts <- pts/nsquare
#  Set up and plot an FEM basis object
edge <- NULL
SquareBasis3 <- create.FEM.basis(pts, edg, tri)
plotFEM.mesh(pts, tri)
#  set up function to provide log intensity sine*cosine
SinCosIntensFn <- function(x,y) {
  scale <- 1
  SinCosIntens <- scale*sin(2*pi*x)*cos(2*pi*y)
  return(SinCosIntens)
}
#  Computation of probabilities for each triangle of having a
#  location within it.
intDensityVec <- triDensity(pts, tri, SinCosIntensFn)
#  Display triangle numbers with probabilities
print(cbind(matrix(1:ntri,ntri,1), round(intDensityVec,3)))