circleCub.Gauss: Integration of the Isotropic Gaussian Density over Circular...

In polyCub: Cubature over Polygonal Domains

Integration of the Isotropic Gaussian Density over Circular Domains

Description

This function calculates the integral of the bivariate, isotropic Gaussian density (i.e., \Sigma = sd^2*diag(2)) over a circular domain via the cumulative distribution function pchisq of the (non-central) Chi-Squared distribution \bibcitep|R:Abramowitz+Stegun:1972|Formula 26.3.24.

Usage

circleCub.Gauss(center, r, mean, sd)

Arguments

center	numeric vector of length 2 (center of the circle).
r	numeric (radius of the circle). Several radii may be supplied.
mean	numeric vector of length 2 (mean of the bivariate Gaussian density).
sd	numeric (common standard deviation of the isotropic Gaussian density in both dimensions).

Value

The integral value (one for each supplied radius).

Note

The non-centrality parameter of the evaluated chi-squared distribution equals the squared distance between the mean and the center. If this becomes too large, the result becomes inaccurate, see pchisq.

References

\bibshow

*

Examples

circleCub.Gauss(center=c(1,2), r=3, mean=c(4,5), sd=6)

## compare with cubature over a polygonal approximation of a circle
d2norm <- function (s, mean, sd)
  dnorm(s[,1], mean=mean[1], sd=sd) * dnorm(s[,2], mean=mean[2], sd=sd)
if (requireNamespace("spatstat.geom")) { # for the disc()
  npoly <- 32  # increase this for a closer match
  disc.poly <- spatstat.geom::disc(radius=3, centre=c(1,2), npoly=npoly)
  polyCub.iso(disc.poly, d2norm, mean=c(4,5), sd=6, center=c(4,5))
}

polyCub documentation built on April 24, 2026, 5:07 p.m.