rect.integrate: Density integration on the two-dimensional simplex In lbelzile/BMAmevt: Multivariate Extremes: Bayesian Estimation of the Spectral Measure

Description

The integral is approximated by a rectangular method, using the values stored in matrix `density`.

Usage

 `1` ```rect.integrate(density, npoints, eps) ```

Arguments

 `density` A `npoints*npoints` matrix containing the density's values scattered on the discretization grid defined by `npoints, equi, eps` (see `discretize`). `npoints` The number of grid nodes on the squared grid containing the desired triangle. `eps` Positive number: minimum distance from any node inside the simplex to the simplex boundary

Details

Integration is made with respect to the Lebesgue measure on the projection of the simplex onto the plane (x,y): x > 0, y > 0, x+y < 1. It is assumed that `density` has been constructed on a grid obtained via function `discretize`, with argument `equi` set to `FALSE` and `npoints` and `eps` equal to those passed to `rect.integrate`.

Value

The value of the estimated integral of `density`.

Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```wrapper <- function(x, y, my.fun,...) { sapply(seq_along(x), FUN = function(i) my.fun(x[i], y[i],...)) } grid <- discretize(npoints=40,eps=1e-3,equi=FALSE) Density <- outer(grid\$X,grid\$Y,FUN=wrapper, my.fun=function(x,y){10*((x/2)^2+y^2)}) rect.integrate(Density,npoints=40,eps=1e-3) ```

lbelzile/BMAmevt documentation built on June 13, 2019, 12:43 p.m.