polyCub (version 0.4-3)

polyCub.iso: Cubature of Isotropic Functions over Polygonal Domains

Description

Conducts numerical integration of a two-dimensional isotropic function $f(x,y) = f_r(||(x,y)-\boldsymbol{\mu}||)$, with $\mu$ being the center of isotropy, over a polygonal domain. It internally solves a line integral along the polygon boundary using integrate where the integrand requires the antiderivative of $r f_r(r)$), which ideally is analytically available and supplied to the function as argument intrfr. The two-dimensional integration problem thereby reduces to an efficient adaptive quadrature in one dimension. See Meyer and Held (2014, Section 2.4 of Supplement B) for mathematical details.

.polyCub.iso is a bare-bone version of polyCub.iso.

Usage

polyCub.iso(polyregion, f, intrfr, ..., center, control = list(),
  check.intrfr = FALSE, plot = FALSE)
.polyCub.iso(polys, intrfr, ..., center, control = list(),
  .witherror = FALSE)

Arguments

intrfr

analytical antiderivative of $r f_r(r)$ from 0 to R (first argument, not necessarily named "R", must be vectorized). If missing, intrfr is approximated numerically using

...

further arguments for f or intrfr.

center

numeric vector of length 2, the center of isotropy.

control

list of arguments passed to integrate, the quadrature rule used for the line integral along the polygon boundary.

check.intrfr

logical (or numeric vector) indicating if (for which r's) the supplied intrfr function should be checked against a numeric approximation. This check requires f to be specified. If TRUE, the set o

polys

something like owin$bdry, but see xylist.

.witherror

logical indicating if an upperbound for the absolute integration error should be attached as an attribute to the result?

polyregion

a polygonal integration domain. The following classes are supported: "owin", "gpc.poly", "SpatialPolygon

f

two-dimensional function. As its first argument it must take a coordinate matrix, i.e. a numeric matrix with two columns.

plot

logical indicating if an illustrative plot of the numerical integration should be produced.

Value

• The approximate integral of the isotropic function f over polyregion. If the intrfr function is provided (which is assumed to be exact), an upperbound for the absolute integration error is appended to the result as attribute "abs.error". It equals the sum of the absolute errors reported by all integrate calls (there is one for each edge of polyregion).

References

Hedevang, E. (2013). Personal communication at the Summer School on Topics in Space-Time Modeling and Inference (May 2013, Aalborg, Denmark).

Meyer, S. and Held, L. (2014). Power-law models for infectious disease spread. Revised for the Annals of Applied Statistics. http://arxiv.org/abs/1308.5115{arXiv:1308.5115}. Supplements are available from http://www.biostat.uzh.ch/research/manuscripts/powerlaw.html.

See Also

Other polyCub.methods: polyCub.SV; polyCub.exact.Gauss; polyCub.midpoint; polyCub

Examples

## we use the example polygon and f (exponential decay) from
example(plotpolyf)

## numerical approximation of 'intrfr'
(intISOnum <- polyCub.iso(letterR, f, center=fcenter))

## analytical 'intrfr' (recall: f_r(r)=dexp(r), we need int_0^R r*f(r) dr)
intrfr <- function (R, rate=1) pgamma(R, 2, rate) / rate
(intISOana <- polyCub.iso(letterR, intrfr=intrfr, center=fcenter))

stopifnot(all.equal(intISOana, intISOnum, check.attributes=FALSE))


### polygon area: f(r) = 1, f(x,y) = 1, center does not really matter

intrfr.const <- function (R) R^2/2
(area.ISO <- polyCub.iso(letterR, intrfr=intrfr.const, center=c(0,0)))

stopifnot(all.equal(spatstat::area.owin(letterR),
                    area.ISO,
                    check.attributes=FALSE))
## the hole is subtracted correctly