# polyCub.exact.Gauss: Quasi-Exact Cubature of the Bivariate Normal Density In polyCub: Cubature over Polygonal Domains

## Description

The bivariate Gaussian density can be integrated based on a triangulation of the (transformed) polygonal domain, using formulae from the Abramowitz and Stegun (1972) handbook (Section 26.9, Example 9, pp. 956f.). This method is quite cumbersome because the A&S formula is only for triangles where one vertex is the origin (0,0). For each triangle of the `tristrip` we have to check in which of the 6 outer regions of the triangle the origin (0,0) lies and adapt the signs in the formula appropriately: (AOB+BOC-AOC) or (AOB-AOC-BOC) or (AOB+AOC-BOC) or (AOC+BOC-AOB) or .... However, the most time consuming step is the evaluation of `pmvnorm`.

## Usage

 `1` ```polyCub.exact.Gauss(polyregion, mean = c(0, 0), Sigma = diag(2), plot = FALSE) ```

## Arguments

 `polyregion` a `"gpc.poly"` polygon or something that can be coerced to this class, e.g., an `"owin"` polygon (via `owin2gpc`), an `"sfg"` polygon (via `sfg2gpc`), or – given rgeos is available – a `"SpatialPolygons"` object. `mean, Sigma` mean and covariance matrix of the bivariate normal density to be integrated. `plot` logical indicating if an illustrative plot of the numerical integration should be produced. Note that the `polyregion` will be transformed (shifted and scaled).

## Value

The integral of the bivariate normal density over `polyregion`. Two attributes are appended to the integral value:

 `nEval` number of triangles over which the standard bivariate normal density had to be integrated, i.e. number of calls to `pmvnorm` and `pnorm`, the former of which being the most time-consuming operation. `error` Approximate absolute integration error stemming from the error introduced by the `nEval` `pmvnorm` evaluations. For this reason, the cubature method is in fact only quasi-exact (as is the `pmvnorm` function).

## Note

The package gpclib is required to produce the `tristrip`, since this is not implemented in rgeos (as of version 0.3-25). The restricted license of gpclib (commercial use prohibited) has to be accepted explicitly via `gpclibPermit()` prior to using `polyCub.exact.Gauss`.

## References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications.

`circleCub.Gauss` for quasi-exact cubature of the isotropic Gaussian density over a circular domain.
Other polyCub-methods: `polyCub.SV()`, `polyCub.iso()`, `polyCub.midpoint()`, `polyCub()`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```## a function to integrate (here: isotropic zero-mean Gaussian density) f <- function (s, sigma = 5) exp(-rowSums(s^2)/2/sigma^2) / (2*pi*sigma^2) ## a simple polygon as integration domain hexagon <- list( list(x = c(7.33, 7.33, 3, -1.33, -1.33, 3), y = c(-0.5, 4.5, 7, 4.5, -0.5, -3)) ) ## quasi-exact integration based on gpclib::tristrip() and mvtnorm::pmvnorm() if (requireNamespace("mvtnorm") && gpclibPermit()) { hexagon.gpc <- new("gpc.poly", pts = lapply(hexagon, c, list(hole = FALSE))) plotpolyf(hexagon.gpc, f, xlim = c(-8,8), ylim = c(-8,8)) print(polyCub.exact.Gauss(hexagon.gpc, mean = c(0,0), Sigma = 5^2*diag(2), plot = TRUE), digits = 16) } ```