Description Usage Arguments Value Note References See Also Examples

The bivariate Gaussian density can be integrated based on a triangulation of
the (transformed) polygonal domain, using formulae from the
Abramowitz and Stegun (1972) handbook (Section 26.9, Example 9, pp. 956f.).
This method is quite cumbersome because the A&S formula is only for triangles
where one vertex is the origin (0,0). For each triangle of the
`tristrip`

we have to check in which of the 6 outer
regions of the triangle the origin (0,0) lies and adapt the signs in the
formula appropriately: *(AOB+BOC-AOC)* or *(AOB-AOC-BOC)* or
*(AOB+AOC-BOC)* or *(AOC+BOC-AOB)* or ....
However, the most time consuming step is the
evaluation of `pmvnorm`

.

1 |

`polyregion` |
a |

`mean, Sigma` |
mean and covariance matrix of the bivariate normal density to be integrated. |

`plot` |
logical indicating if an illustrative plot of the numerical
integration should be produced. Note that the |

The integral of the bivariate normal density over `polyregion`

.
Two attributes are appended to the integral value:

`nEval` |
number of triangles over which the standard bivariate normal density had to
be integrated, i.e. number of calls to |

`error` |
Approximate absolute integration error stemming from the error introduced by
the |

The package gpclib is required to produce the
`tristrip`

, since this is not implemented in rgeos
(as of version 0.3-25).
The restricted license of gpclib (commercial use prohibited)
has to be accepted explicitly via
`gpclibPermit()`

prior to using `polyCub.exact.Gauss`

.

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

`circleCub.Gauss`

for quasi-exact cubature of the
isotropic Gaussian density over a circular domain.

Other polyCub-methods:
`polyCub.SV()`

,
`polyCub.iso()`

,
`polyCub.midpoint()`

,
`polyCub()`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
## a function to integrate (here: isotropic zero-mean Gaussian density)
f <- function (s, sigma = 5)
exp(-rowSums(s^2)/2/sigma^2) / (2*pi*sigma^2)
## a simple polygon as integration domain
hexagon <- list(
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))
)
## quasi-exact integration based on gpclib::tristrip() and mvtnorm::pmvnorm()
if (requireNamespace("mvtnorm") && gpclibPermit()) {
hexagon.gpc <- new("gpc.poly", pts = lapply(hexagon, c, list(hole = FALSE)))
plotpolyf(hexagon.gpc, f, xlim = c(-8,8), ylim = c(-8,8))
print(polyCub.exact.Gauss(hexagon.gpc, mean = c(0,0), Sigma = 5^2*diag(2),
plot = TRUE), digits = 16)
}
``` |

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