# Gmulti

##### Marked Nearest Neighbour Distance Function

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

to the nearest point of subset $J$.

- Keywords
- spatial

##### Usage

```
Gmulti(X, I, J)
Gmulti(X, I, J, r)
Gmulti(X, I, J, breaks)
```

##### Arguments

- 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
- 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
- breaks
- An alternative to the argument
`r`

. Not normally invoked by the user. See the**Details**section.

##### Details

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 logical vectors of length equal to
`X$n`

, or integer vectors with entries in the range 1 to
`X$n`

, etc.

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 `X$window`

)
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
$G_{IJ}$.

##### Value

- 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 $G_{IJ}(r)$ 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

##### synopsis

Gmulti(X, I, J, r=NULL, breaks=NULL, ...)

##### Warnings

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 $1$.

##### References

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.

##### See Also

##### Examples

```
data(longleaf)
# Longleaf Pine data: marks represent diameter
<testonly>longleaf <- longleaf[seq(1,longleaf$n, by=50), ]</testonly>
Gm <- Gmulti(longleaf, longleaf$marks <= 15, longleaf$marks >= 25)
plot(Gm)
```

*Documentation reproduced from package spatstat, version 1.5-9, License: GPL version 2 or newer*