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##### Generate a Logistic Regression Quadrature Scheme from a Point Pattern

Generates a logistic regression quadrature scheme (an object of class "logiquad" inheriting from "quad") from point patterns of data and dummy points.

Keywords
spatial, datagen
##### Usage
quadscheme.logi(data, dummy, dummytype = "stratrand",
nd = NULL, mark.repeat = FALSE, …)
##### Arguments
data

The observed data point pattern. An object of class "ppp" or in a format recognised by as.ppp()

dummy

The pattern of dummy points for the quadrature. An object of class "ppp" or in a format recognised by as.ppp(). If missing a sensible default is generated.

dummytype

The name of the type of dummy points to use when "dummy" is missing. Currently available options are: "stratrand" (default), "binomial", "poisson", "grid" and "transgrid".

nd

Integer, or integer vector of length 2 controlling the intensity of dummy points when "dummy" is missing.

mark.repeat

Repeating the dummy points for each level of a marked data pattern when "dummy" is missing. (See details.)

Ignored.

##### Details

This is the primary method for producing a quadrature schemes for use by ppm when the logistic regression approximation (Baddeley et al. 2013) to the pseudolikelihood of the model is applied (i.e. when method="logi" in ppm).

The function ppm fits a point process model to an observed point pattern. When used with the option method="logi" it requires a quadrature scheme consisting of the original data point pattern and an additional pattern of dummy points. Such quadrature schemes are represented by objects of class "logiquad".

Quadrature schemes are created by the function quadscheme.logi. The arguments data and dummy specify the data and dummy points, respectively. There is a sensible default for the dummy points. Alternatively the dummy points may be specified arbitrarily and given in any format recognised by as.ppp.

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, taking account of the optional arguments dummytype, nd, and mark.repeat.

The currently accepted values for dummytype are:

• "grid" where the frame of the window is divided into a nd * nd or nd[1] * nd[2] regular grid of tiles and the centers constitutes the dummy points.

• "transgrid" where a regular grid as above is translated by a random vector.

• "stratrand" where each point of a regular grid as above is randomly translated within its tile.

• "binomial" where nd * nd or nd[1] * nd[2] points are generated uniformly in the frame of the window. "poisson" where a homogeneous Poisson point process with intensity nd * nd or nd[1] * nd[2] is generated within the frame of observation window.

Then if the window is not rectangular, any dummy points lying outside it are deleted.

If data is a multitype point pattern the dummy points should also be marked (with the same levels of the marks as data). If dummy is missing and the dummy pattern is generated by quadscheme.logi the default behaviour is to attach a uniformly distributed mark (from the levels of the marks) to each dummy point. Alternatively, if mark.repeat=TRUE each dummy point is repeated as many times as there are levels of the marks with a distinct mark value attached to it.

Finally, each point (data and dummy) is assigned the weight 1. The weights are never used and only appear to be compatible with the class "quad" from which the "logiquad" object inherits.

##### Value

An object of class "logiquad" inheriting from "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".

##### References

Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for spatial Gibbs point processes. Biometrika 101 (2) 377--392.

ppm, as.ppp

##### Aliases
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