# Gcross

##### Multitype Nearest Neighbour Distance Function (i-to-j)

For a multitype point pattern, estimate the distribution of the distance from a point of type $i$ to the nearest point of type $j$.

- Keywords
- spatial

##### Usage

```
Gcross(X, i=1, j=2)
Gcross(X, i=1, j=2, r)
Gcross(X, i=1, j=2, breaks)
```

##### Arguments

- X
- The observed point pattern, from which an estimate of the cross type distance distribution function $G_{ij}(r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.
- i
- Number or character string identifying the type (mark value)
of the points in
`X`

from which distances are measured. - j
- Number or character string identifying the type (mark value)
of the 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

This function `Gcross`

and its companions
`Gdot`

and `Gmulti`

are generalisations of the function `Gest`

to multitype point patterns.

A multitype point pattern is a spatial pattern of
points classified into a finite number of possible
``colours'' or ``types''. In the `X`

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

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

.
It must be a marked point pattern, and the mark vector
`X$marks`

must be a factor.
The arguments `i`

and `j`

will be interpreted as
levels of the factor `X$marks`

. (Warning: this means that
an integer value `i=3`

will be interpreted as the 3rd smallest level,
not the number 3).
The ``cross-type'' (type $i$ to type $j$)
nearest neighbour distance distribution function
of a multitype point process
is the cumulative distribution function $G_{ij}(r)$
of the distance from a typical random point of the process with type $i$
the nearest point of type $j$.

An estimate of $G_{ij}(r)$ is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the process of type $i$ points were independent of the process of type $j$ points, then $G_{ij}(r)$ would equal $F_j(r)$, the empty space function of the type $j$ points. For a multitype Poisson point process where the type $i$ points have intensity $\lambda_i$, we have $$G_{ij}(r) = 1 - e^{ - \lambda_j \pi r^2}$$ Deviations between the empirical and theoretical $G_{ij}$ curves may suggest dependence between the points of types $i$ and $j$.

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 (see below).

##### synopsis

Gcross(X, i=1, j=2, r=NULL, breaks=NULL, ...)

##### Warnings

The arguments `i`

and `j`

are interpreted as
levels of the factor `X$marks`

. Beware of the usual
trap with factors: numerical values are not
interpreted in the same way as character values. See the first example.

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(betacells)
# cat retina data
G01 <- Gcross(betacells, "off", "on")
plot(G01)
# empty space function of `on' points
F1 <- Fest(betacells[betacells == "on"], r = G01$r, eps=10.0)
lines(F1$r, F1$km, lty=3)
# synthetic example
pp <- runifpoispp(50)
pp <- pp %mark% factor(sample(0:1, pp$n, replace=TRUE))
G <- Gcross(pp, "0", "1") # note: "0" not 0
```

*Documentation reproduced from package spatstat, version 1.9-0, License: GPL version 2 or newer*