# Compute the probabilities that a random location will be within one of the triangles of an FEM mesh. 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. ```r 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. ## Returns 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. ## 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. ## Author(s) Jim Ramsay ## See Also `smooth.FEM.density` ## Examples ```r # --------- 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))) ```

triDensity function