# Gdot

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

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

- Keywords
- spatial, nonparametric

##### Usage

```
Gdot(X, i=1)
Gdot(X, i=1, r)
Gdot(X, i=1, breaks)
```

##### Arguments

- X
- The observed point pattern, from which an estimate of the distance distribution function $G_{i\bullet}(r)$ 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. - r
- numeric vector. The values of the argument $r$ at which the distribution function $G_{i\bullet}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for imp
- breaks
- An alternative to the argument
`r`

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

##### Details

This function `Gdot`

and its companions
`Gcross`

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 argument will be interpreted as a
level 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 ``dot-type'' (type $i$ to any type)
nearest neighbour distance distribution function
of a multitype point process
is the cumulative distribution function $G_{i\bullet}(r)$
of the distance from a typical random point of the process with type $i$
the nearest other point of the process, regardless of type.

An estimate of $G_{i\bullet}(r)$ is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the type $i$ points were independent of all other points, then $G_{i\bullet}(r)$ would equal $G_{ii}(r)$, the nearest neighbour distance distribution function of the type $i$ points alone. For a multitype Poisson point process with total intensity $\lambda$, we have $$G_{i\bullet}(r) = 1 - e^{ - \lambda \pi r^2}$$ Deviations between the empirical and theoretical $G_{i\bullet}$ curves may suggest dependence of the type $i$ points on the other points.

This algorithm estimates the distribution function
$G_{i\bullet}(r)$
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 $G_{i\bullet}(r)$ 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 $G_{i\bullet}(r)$. This estimate should be used with caution as $G_{i\bullet}(r)$ 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 $G_{i\bullet}$.
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 $G_{i\bullet}$ as if it were an unbiased estimator of
$G_{i\bullet}$.

##### 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 $G_{i\bullet}(r)$ has been estimated rs the ``reduced sample'' or ``border correction'' estimator of $G_{i\bullet}(r)$ km the spatial Kaplan-Meier estimator of $G_{i\bullet}(r)$ hazard the hazard rate $\lambda(r)$ of $G_{i\bullet}(r)$ by the spatial Kaplan-Meier method raw the uncorrected estimate of $G_{i\bullet}(r)$, i.e. the empirical distribution of the distances from each point of type $i$ to the nearest other point of any type. theo the theoretical value of $G_{i\bullet}(r)$ for a marked Poisson process with the same estimated intensity (see below).

##### synopsis

Gdot(X, i=1, r=NULL, breaks=NULL, ...)

##### Warnings

The argument `i`

is interpreted as a
level 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 $G_{i\bullet}$ does not necessarily have a density.

The reduced sample estimator of $G_{i\bullet}$ 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 $G_{i\bullet}$ 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
G0. <- Gdot(betacells, "off")
plot(G0.)
# synthetic example
pp <- runifpoispp(50)
pp <- pp %mark% factor(sample(0:1, pp$n, replace=TRUE))
G <- Gdot(pp, "0") # note: "0" not 0
```

*Documentation reproduced from package spatstat, version 1.9-1, License: GPL version 2 or newer*