R: Cubature over Polygonal Domains

polyCub-package {polyCub} | R Documentation

## Cubature over Polygonal Domains

### Description

The R package polyCub implements cubature (numerical integration) over polygonal domains. It solves the problem of integrating a continuously differentiable function f(x,y) over simple closed polygons.

### Details

polyCub provides the following cubature methods, which can either be called explicitly or via the generic `polyCub` function:

`polyCub.SV`:

  General-purpose product Gauss cubature (Sommariva and Vianello, 2007)

`polyCub.midpoint`:

  Simple two-dimensional midpoint rule based on `as.im.function` from spatstat (Baddeley and Turner, 2005)

`polyCub.iso`:

  Adaptive cubature for radially symmetric functions via line `integrate()` along the polygon boundary (Meyer and Held, 2014, Supplement B, Section 2.4).

`polyCub.exact.Gauss`:

  Accurate (but slow) integration of the bivariate Gaussian density based on polygon triangulation (via `tristrip` from gpclib) and (numerous) evaluations of cumulative densities (via `pmvnorm` from package mvtnorm). Note that there is also a function `circleCub.Gauss` to integrate the isotropic Gaussian density over a circular domain.

A more detailed description and benchmark experiment of the above cubature methods can be found in the `vignette("polyCub")` and in Meyer (2010, Section 3.2).

### References

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

Baddeley, A. and Turner, R. (2005). spatstat: an R package for analyzing spatial point patterns. Journal of Statistical Software, 12 (6), 1-42.

Meyer, S. (2010). Spatio-Temporal Infectious Disease Epidemiology based on Point Processes. Master's Thesis, LMU Munich. Available as http://epub.ub.uni-muenchen.de/11703/.

Meyer, S. and Held, L. (2014). Power-law models for infectious disease spread. The Annals of Applied Statistics, 8 (3), 1612-1639.
DOI-Link: https://doi.org/10.1214/14-AOAS743, arXiv:1308.5115

Sommariva, A. and Vianello, M. (2007). Product Gauss cubature over polygons based on Green's integration formula. BIT Numerical Mathematics, 47 (2), 441-453.
DOI-Link: https://doi.org/10.1007/s10543-007-0131-2

### See Also

`vignette("polyCub")`

For the special case of a rectangular domain along the axes (e.g., a bounding box), the cubature package is more appropriate.

[Package polyCub version 0.7.1 Index]