Generates a quadrature scheme (an object of class "quad")
from point patterns of data and dummy points.
quadscheme(data, dummy, method="grid", ...)An object of class "quad" describing the quadrature scheme
(data points, dummy points, and quadrature weights)
suitable as the argument Q of the function ppm() for
fitting a point process model.
The quadrature scheme can be inspected using the
print and plot methods for objects
of class "quad".
The observed data point pattern.
An object of class "ppp"
or in a format recognised by as.ppp()
The pattern of dummy points for the quadrature.
An object of class "ppp"
or in a format recognised by as.ppp()
Defaults to default.dummy(data, ...)
The name of the method for calculating quadrature weights: either
"grid" or "dirichlet".
Parameters of the weighting method (see below) and parameters for constructing the dummy points if necessary.
The following error messages need some explanation.
(See also the list of error messages in ppm.ppp).
This is not important unless the relative error is large.
In the default rule for computing the
quadrature weights, space is divided into rectangular tiles,
and the number of quadrature points (data and dummy points) in
each tile is counted. It is possible for a tile with non-zero area
to contain no quadrature points; in this case, the quadrature
scheme will contribute a bias to the model-fitting procedure.
A small relative error (less than 2 percent) is not important.
Relative errors of a few percent can occur because of the shape of
the window.
If the relative error is greater than about 5 percent, we
recommend trying different parameters for the quadrature scheme,
perhaps setting a larger value of nd to increase the number
of dummy points. A relative error greater than 10 percent
indicates a major problem with the input data. The quadrature
scheme should be inspected by plotting and printing it.
(The most likely cause of this problem is that the spatial coordinates
of the original data were not handled correctly, for example,
coordinates of the locations and the window boundary were incompatible.)
This error message is rare, and has no consequences. It is mainly of interest to programmers. It occurs when the area of a tile is calculated to be equal to zero, but a quadrature point has been placed in the tile.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner rolfturner@posteo.net
This is the primary method for producing a quadrature schemes
for use by ppm.
The function ppm fits a point process model to an
observed point pattern using
the Berman-Turner quadrature approximation (Berman and Turner, 1992;
Baddeley and Turner, 2000) to the pseudolikelihood of the model.
It requires a quadrature scheme consisting of
the original data point pattern, an additional pattern of dummy points,
and a vector of quadrature weights for all these points.
Such quadrature schemes are represented by objects of class
"quad". See quad.object for a description of this class.
Quadrature schemes are created by the function
quadscheme.
The arguments data and dummy specify the data and dummy
points, respectively. There is a sensible default for the dummy
points (provided by default.dummy).
Alternatively the dummy points
may be specified arbitrarily and given in any format recognised by
as.ppp.
There are also functions for creating dummy patterns
including corners,
gridcentres,
stratrand and
spokes.
The quadrature region is the region over which we are
integrating, and approximating integrals by finite sums.
If dummy is a point pattern object (class "ppp")
then the quadrature region is taken to be Window(dummy).
If dummy is just a list of \(x, y\) coordinates
then the quadrature region defaults to the observation window
of the data pattern, Window(data).
If dummy is missing, then a pattern of dummy points
will be generated using default.dummy, taking account
of the optional arguments ....
By default, the dummy points are arranged in a
rectangular grid; recognised arguments
include nd (the number of grid points
in the horizontal and vertical directions)
and eps (the spacing between dummy points).
If random=TRUE, a systematic random pattern
of dummy points is generated instead.
See default.dummy for details.
If method = "grid" then the optional arguments (for ...) are
(nd, ntile, eps).
The quadrature region (defined above) is divided into
an ntile[1] by ntile[2] grid of rectangular tiles.
The weight for each
quadrature point is the area of a tile divided by the number of
quadrature points in that tile.
If method="dirichlet" then the optional arguments are
(exact=TRUE, nd, eps).
The quadrature points (both data and dummy) are used to construct the
Dirichlet tessellation. The quadrature weight of each point is the
area of its Dirichlet tile inside the quadrature region.
If exact == TRUE then this area is computed exactly
using the package deldir; otherwise it is computed
approximately by discretisation.
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283--322.
Berman, M. and Turner, T.R. Approximating point process likelihoods with GLIM. Applied Statistics 41 (1992) 31--38.
ppm,
as.ppp,
quad.object,
gridweights,
dirichletWeights,
corners,
gridcentres,
stratrand,
spokes
# grid weights
Q <- quadscheme(simdat)
Q <- quadscheme(simdat, method="grid")
Q <- quadscheme(simdat, eps=0.5) # dummy point spacing 0.5 units
Q <- quadscheme(simdat, nd=50) # 1 dummy point per tile
Q <- quadscheme(simdat, ntile=25, nd=50) # 4 dummy points per tile
# Dirichlet weights
Q <- quadscheme(simdat, method="dirichlet", exact=FALSE)
# random dummy pattern
# D <- runifrect(250, Window(simdat))
# Q <- quadscheme(simdat, D, method="dirichlet", exact=FALSE)
# polygonal window
data(demopat)
X <- unmark(demopat)
Q <- quadscheme(X)
# mask window
Window(X) <- as.mask(Window(X))
Q <- quadscheme(X)
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