polyCub (version 0.4-2)

circleCub.Gauss: 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 circular domains via the cumulative distribution function of the (non-central) Chi-Squared distribution (pchisq), cp. Formula 26.3.24 in Abramowitz and Stegun (1970).

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).

References

M. Abramowitz and I. A. Stegun (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (9th ed.). New York: Dover Publications.

Examples

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

if (gpclibPermit()) {
  ## compare with cubature over a polygonal approximation of a circle
  disc.poly <- spatstat::disc(radius=3, centre=c(1,2), npoly=32)
  polyCub.exact.Gauss(disc.poly, mean=c(4,5), Sigma=6^2*diag(2))
}