polyCub.exact.Gauss: Quasi-Exact Cubature of the Bivariate Normal Density

Description Usage Arguments Value Note References See Also Examples

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

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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.

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

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## 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)
}

polyCub documentation built on Jan. 27, 2021, 1:05 a.m.