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.

For the special case of a rectangular domain along the axes, the
package **cubature**
is more appropriate (cf. `CRAN Task View: Numerical Mathematics`

).

The integration domain is described by a polygonal boundary (or multiple polygons, including holes). Various R packages for spatial data analysis provide classes for polygons. The implementations differ in vertex order (which direction represents a hole) and if the first vertex is repeated.

All of **polyCub**’s cubature methods understand

`"owin"`

from package**spatstat.geom**,`"gpc.poly"`

from**gpclib**,`"SpatialPolygons"`

from package**sp**, and`"(MULTI)POLYGON"`

from package**sf**.

Internally, **polyCub** uses its auxiliary
`xylist()`

function to extract a plain list of lists of
vertex coordinates from these classes, such that vertices are ordered
anticlockwise (for exterior boundaries) and the first vertex is not
repeated (i.e., the `"owin"`

convention).

The following cubature methods are implemented in
**polyCub**:

`polyCub.SV()`

: Product Gauss cubature`polyCub.midpoint()`

: Two-dimensional midpoint rule`polyCub.iso()`

: Adaptive cubature for radially symmetric functions \(f(x,y) = f_r(\lVert(x-x_0,y-y_0)\rVert)\)`polyCub.exact.Gauss()`

(*currently disabled*): Accurate integration of the bivariate Gaussian density

The following section details and illustrates the different cubature methods.

`library("polyCub")`

We consider the integration of a function f(x,y) which all of the
above cubature methods can handle: an isotropic zero-mean Gaussian
density. **polyCub** expects the integrand f to take a
two-column coordinate matrix as its first argument (as opposed to
separate arguments for the x and y coordinates), so:

```
<- function (s, sigma = 5)
f
{exp(-rowSums(s^2)/2/sigma^2) / (2*pi*sigma^2)
}
```

We use a simple hexagon as polygonal integration domain, here
specified via an `"xylist"`

of vertex coordinates:

```
<- list(
hexagon list(x = c(7.33, 7.33, 3, -1.33, -1.33, 3),
y = c(-0.5, 4.5, 7, 4.5, -0.5, -3))
)
```

An image of the function and the integration domain can be produced
using **polyCub**’s rudimentary (but convenient) plotting
utility:

`plotpolyf(hexagon, f, xlim = c(-8,8), ylim = c(-8,8))`

`polyCub.SV()`

The **polyCub** package provides an R-interfaced
C-translation of “polygauss: Matlab code for Gauss-like cubature over
polygons” (Sommariva and Vianello, 2013, https://www.math.unipd.it/~alvise/software.html), an
algorithm described in Sommariva and Vianello (2007, *BIT Numerical
Mathematics*, https://doi.org/10.1007/s10543-007-0131-2). The cubature
rule is based on Green’s integration formula and incorporates
appropriately transformed weights and nodes of one-dimensional
Gauss-Legendre quadrature in both dimensions, thus the name “product
Gauss cubature”. It is exact for all bivariate polynomials if the number
of cubature nodes is sufficiently large (depending on the degree of the
polynomial).

For the above example, a reasonable approximation is already obtained
with degree `nGQ = 3`

of the one-dimensional Gauss-Legendre
quadrature:

```
polyCub.SV(hexagon, f, nGQ = 3, plot = TRUE)
#> [1] 0.2741456
```

The involved nodes (displayed in the figure above) and weights can be
extracted by calling `polyCub.SV()`

with
`f = NULL`

, e.g., to determine the number of nodes:

```
nrow(polyCub.SV(hexagon, f = NULL, nGQ = 3)[[1]]$nodes)
#> [1] 72
```

For illustration, we create a variant of `polyCub.SV()`

,
which returns the number of function evaluations as an attribute:

```
<- function (polyregion, f, ..., nGQ = 20) {
polyCub.SVn <- polyCub.SV(polyregion, f = NULL, ..., nGQ = nGQ)
nw ## nw is a list with one element per polygon of 'polyregion'
<- sapply(nw, function (x)
res c(result = sum(x$weights * f(x$nodes, ...)), nEval = nrow(x$nodes)))
structure(sum(res["result",]), nEval = sum(res["nEval",]))
}polyCub.SVn(hexagon, f, nGQ = 3)
#> [1] 0.2741456
#> attr(,"nEval")
#> [1] 72
```

We can use this function to investigate how the accuracy of the
approximation depends on the degree `nGQ`

and the associated
number of cubature nodes:

```
for (nGQ in c(1:5, 10, 20)) {
<- polyCub.SVn(hexagon, f, nGQ = nGQ)
result cat(sprintf("nGQ = %2i: %.12f (n=%i)\n", nGQ, result, attr(result, "nEval")))
}#> nGQ = 1: 0.285265369245 (n=12)
#> nGQ = 2: 0.274027610314 (n=36)
#> nGQ = 3: 0.274145638288 (n=72)
#> nGQ = 4: 0.274144768964 (n=120)
#> nGQ = 5: 0.274144773834 (n=180)
#> nGQ = 10: 0.274144773813 (n=660)
#> nGQ = 20: 0.274144773813 (n=2520)
```

`polyCub.midpoint()`

The two-dimensional midpoint rule in **polyCub** is a
simple wrapper around `as.im.function()`

and
`integral.im()`

from package **spatstat.geom**.
In other words, the polygon is represented by a binary pixel image and
the integral is approximated as the sum of (pixel area * f(pixel
midpoint)) over all pixels whose midpoint is part of the polygon.

To use `polyCub.midpoint()`

, we need to convert our
polygon to **spatstat.geom**’s “owin” class:

```
library("spatstat.geom")
<- owin(poly = hexagon) hexagon.owin
```

Using a pixel size of `eps = 0.5`

(here yielding 270
pixels), we obtain:

```
polyCub.midpoint(hexagon.owin, f, eps = 0.5, plot = TRUE)
#> [1] 0.2736067
```

`polyCub.iso()`

A radially symmetric function can be expressed in terms of the
distance r from its point of symmetry: f(r). If the antiderivative of r
times f(r), called `intrfr()`

, is analytically available,
Green’s theorem leads us to a cubature rule which only needs
*one-dimensional* numerical integration. More specifically,
`intrfr()`

will be `integrate()`

d along the edges
of the polygon. The mathematical details are given in Meyer and Held
(2014, *The Annals of Applied Statistics*, https://doi.org/10.1214/14-AOAS743, Supplement B,
Section 2.4).

For the bivariate Gaussian density `f`

defined above, the
integral from 0 to R of `r*f(r)`

is analytically available
as:

```
<- function (R, sigma = 5)
intrfr
{1 - exp(-R^2/2/sigma^2))/2/pi
( }
```

With this information, we can apply the cubature rule as follows:

```
polyCub.iso(hexagon, intrfr = intrfr, center = c(0,0))
#> [1] 0.2741448
#> attr(,"abs.error")
#> [1] 3.043618e-15
```

Note that we do not even need the original function
`f`

.

If `intrfr()`

is missing, it can be approximated
numerically using `integrate()`

for `r*f(r)`

as
well, but the overall integration will then be much less efficient than
product Gauss cubature.

Package **polyCub** exposes a C-version of
`polyCub.iso()`

for use by other R packages (notably **surveillance**)
via `LinkingTo: polyCub`

and
`#include <polyCubAPI.h>`

. This requires the
`intrfr()`

function to be implemented in C as well. See https://github.com/bastistician/polyCub/blob/master/tests/polyiso_powerlaw.c
for an example.

`polyCub.exact.Gauss()`

*This cubature method is currently disabled* (#2). It
requires polygon triangulation originally performed using
`tristrip()`

from the **gpclib**
package; unfortunately, it has become unavailable from mainstream
repositories.

Abramowitz and Stegun (1972, Section 26.9, Example 9) offer a formula
for the integral of the bivariate Gaussian density over a triangle with
one vertex at the origin. This formula can be used after triangulation
of the polygonal domain. The core of the formula is an integral of the
bivariate Gaussian density with zero mean, unit variance and some
correlation over an infinite rectangle [h, Inf] x [0, Inf], which can be
computed accurately using `pmvnorm()`

from the **mvtnorm**
package.

For the above example, we obtained:

```
polyCub.exact.Gauss(hexagon.owin, mean = c(0,0), Sigma = 5^2*diag(2))
#> [1] 0.2741448
#> attr(,"nEval")
#> [1] 48
#> attr(,"error")
#> [1] 4.6e-14
```

The required triangulation as well as the numerous calls of
`pmvnorm()`

make this integration algorithm quite cumbersome.
For large-scale integration tasks, it is thus advisable to resort to the
general-purpose product Gauss cubature rule
`polyCub.SV()`

.

Note: **polyCub** provides an auxiliary function
`circleCub.Gauss()`

to calculate the integral of an
*isotropic* Gaussian density over a *circular* domain
(which requires nothing more than a single call of
`pchisq()`

).

We use the last result from `polyCub.exact.Gauss()`

as a
reference value and tune the number of cubature nodes in
`polyCub.SV()`

and `polyCub.midpoint()`

until the
absolute error is below 10^-8. This leads to `nGQ = 4`

for
product Gauss cubature and a 1200 x 1200 pixel image for the midpoint
rule. For `polyCub.iso()`

, we keep the default tolerance
levels of `integrate()`

. For comparison, we also run
`polyCub.iso()`

without the analytically derived
`intrfr`

function, which leads to a
double-`integrate`

approximation.

The median runtimes [ms] of the different cubature methods are given below.

```
<- microbenchmark::microbenchmark(
benchmark SV = polyCub.SV(hexagon.owin, f, nGQ = 4),
midpoint = polyCub.midpoint(hexagon.owin, f, dimyx = 1200),
iso = polyCub.iso(hexagon.owin, intrfr = intrfr, center = c(0,0)),
iso_double_approx = polyCub.iso(hexagon.owin, f, center = c(0,0)),
exact = polyCub.exact.Gauss(hexagon.owin, mean = c(0,0), Sigma = 5^2*diag(2)),
times = 9,
check = function (values) all(abs(unlist(values) - 0.274144773813434) < 1e-8))
```

`summary(benchmark, unit = "ms")[c("expr", "median")]`

expr | median |
---|---|

SV | 0.07 |

midpoint | 179.58 |

iso | 0.28 |

iso_double_approx | 3.97 |

exact | 7.05 |

The general-purpose SV-method is the clear winner of this small
competition. A disadvantage of that method is that the number of
cubature nodes needs to be tuned manually. This also holds for the
midpoint rule, which is by far the slowest option. In contrast, the
“iso”-method for radially symmetric functions is based on R’s
`integrate()`

function, which implements automatic tolerance
levels. Furthermore, the “iso”-method can also be used with “spiky”
integrands, such as a heavy-tailed power-law kernel \(f(r) = (r+1)^{-2}\).