circleCub.Gauss {polyCub}

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 (Abramowitz and 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

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

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
## Not run: ## (this example requires gpclib)
disc.poly <- spatstat.geom::disc(radius=3, centre=c(1,2), npoly=32)
polyCub.exact.Gauss(disc.poly, mean=c(4,5), Sigma=6^2*diag(2))

## End(Not run)