For a marked point pattern,
estimate the distribution of the distance
from a typical point in subset `I`

to the nearest point of subset \(J\).

```
Gmulti(X, I, J, r=NULL, breaks=NULL, ...,
disjoint=NULL, correction=c("rs", "km", "han"))
```

An object of class `"fv"`

(see `fv.object`

).

Essentially a data frame containing six numeric columns

- r
the values of the argument \(r\) at which the function \(G_{IJ}(r)\) has been estimated

- rs
the ``reduced sample'' or ``border correction'' estimator of \(G_{IJ}(r)\)

- han
the Hanisch-style estimator of \(G_{IJ}(r)\)

- km
the spatial Kaplan-Meier estimator of \(G_{IJ}(r)\)

- hazard
the hazard rate \(\lambda(r)\) of \(G_{IJ}(r)\) by the spatial Kaplan-Meier method

- raw
the uncorrected estimate of \(G_{IJ}(r)\), i.e. the empirical distribution of the distances from each point of type \(i\) to the nearest point of type \(j\)

- theo
the theoretical value of \(G_{IJ}(r)\) for a marked Poisson process with the same estimated intensity

- X
The observed point pattern, from which an estimate of the multitype distance distribution function \(G_{IJ}(r)\) will be computed. It must be a marked point pattern. See under Details.

- I
Subset of points of

`X`

from which distances are measured.- J
Subset of points in

`X`

to which distances are measured.- r
Optional. Numeric vector. The values of the argument \(r\) at which the distribution function \(G_{IJ}(r)\) should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions on \(r\).

- breaks
This argument is for internal use only.

- ...
Ignored.

- disjoint
Optional flag indicating whether the subsets

`I`

and`J`

are disjoint. If missing, this value will be computed by inspecting the vectors`I`

and`J`

.- correction
Optional. Character string specifying the edge correction(s) to be used. Options are

`"none"`

,`"rs"`

,`"km"`

,`"hanisch"`

and`"best"`

. Alternatively`correction="all"`

selects all options.

The function \(G_{IJ}\) does not necessarily have a density.

The reduced sample estimator of \(G_{IJ}\) is pointwise approximately unbiased, but need not be a valid distribution function; it may not be a nondecreasing function of \(r\). Its range is always within \([0,1]\).

The spatial Kaplan-Meier estimator of \(G_{IJ}\) is always nondecreasing but its maximum value may be less than \(1\).

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

The function `Gmulti`

generalises `Gest`

(for unmarked point
patterns) and `Gdot`

and `Gcross`

(for
multitype point patterns) to arbitrary marked point patterns.

Suppose \(X_I\), \(X_J\) are subsets, possibly overlapping, of a marked point process. This function computes an estimate of the cumulative distribution function \(G_{IJ}(r)\) of the distance from a typical point of \(X_I\) to the nearest distinct point of \(X_J\).

The argument `X`

must be a point pattern (object of class
`"ppp"`

) or any data that are acceptable to `as.ppp`

.

The arguments `I`

and `J`

specify two subsets of the
point pattern. They may be any type of subset indices, for example,
logical vectors of length equal to `npoints(X)`

,
or integer vectors with entries in the range 1 to
`npoints(X)`

, or negative integer vectors.

Alternatively, `I`

and `J`

may be **functions**
that will be applied to the point pattern `X`

to obtain
index vectors. If `I`

is a function, then evaluating
`I(X)`

should yield a valid subset index. This option
is useful when generating simulation envelopes using
`envelope`

.

This algorithm estimates the distribution function \(G_{IJ}(r)\)
from the point pattern `X`

. It assumes that `X`

can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in `X`

as `Window(X)`

)
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in `Gest`

.

The argument `r`

is the vector of values for the
distance \(r\) at which \(G_{IJ}(r)\) should be evaluated.
It is also used to determine the breakpoints
(in the sense of `hist`

)
for the computation of histograms of distances. The reduced-sample and
Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation
error which is controlled by the fineness of the breakpoints.

First-time users would be strongly advised not to specify `r`

.
However, if it is specified, `r`

must satisfy `r[1] = 0`

,
and `max(r)`

must be larger than the radius of the largest disc
contained in the window. Furthermore, the successive entries of `r`

must be finely spaced.

The algorithm also returns an estimate of the hazard rate function, \(\lambda(r)\), of \(G_{IJ}(r)\). This estimate should be used with caution as \(G_{IJ}(r)\) is not necessarily differentiable.

The naive empirical distribution of distances from each point of
the pattern `X`

to the nearest other point of the pattern,
is a biased estimate of \(G_{IJ}\).
However this is also returned by the algorithm, as it is sometimes
useful in other contexts. Care should be taken not to use the uncorrected
empirical \(G_{IJ}\) as if it were an unbiased estimator of
\(G_{IJ}\).

Cressie, N.A.C. *Statistics for spatial data*.
John Wiley and Sons, 1991.

Diggle, P.J. *Statistical analysis of spatial point patterns*.
Academic Press, 1983.

Diggle, P. J. (1986).
Displaced amacrine cells in the retina of a
rabbit : analysis of a bivariate spatial point pattern.
*J. Neurosci. Meth.* **18**, 115--125.

Harkness, R.D and Isham, V. (1983)
A bivariate spatial point pattern of ants' nests.
*Applied Statistics* **32**, 293--303

Lotwick, H. W. and Silverman, B. W. (1982).
Methods for analysing spatial processes of several types of points.
*J. Royal Statist. Soc. Ser. B* **44**, 406--413.

Ripley, B.D. *Statistical inference for spatial processes*.
Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J.
*Stochastic geometry and its applications*.
2nd edition. Springer Verlag, 1995.

Van Lieshout, M.N.M. and Baddeley, A.J. (1999)
Indices of dependence between types in multivariate point patterns.
*Scandinavian Journal of Statistics* **26**, 511--532.

`Gcross`

,
`Gdot`

,
`Gest`

```
trees <- longleaf
# Longleaf Pine data: marks represent diameter
# \testonly{
trees <- trees[seq(1, npoints(trees), by=50), ]
# }
Gm <- Gmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
plot(Gm)
```

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