R: Quasi-Exact Cubature of the Bivariate Normal Density

polyCub.exact.Gauss {polyCub} | R Documentation

## Quasi-Exact Cubature of the Bivariate Normal Density

### 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

    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 (converted via `owin2gpc` and – given rgeos is available – `"SpatialPolygons"` also work.
`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.

### See Also

`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`

### Examples

    ## 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)
    }

[Package polyCub version 0.7.1 Index]