Jdot
Multitype J Function (i-to-any)
For a multitype point pattern, estimate the multitype $J$ function summarising the interpoint dependence between the type $i$ points and the points of any type.
- Keywords
- spatial, nonparametric
Usage
Jdot(X, i, eps=NULL, r=NULL, breaks=NULL, ..., correction=NULL)Arguments
- X
- The observed point pattern, from which an estimate of the multitype $J$ function $J_{i\bullet}(r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.
- i
- The type (mark value)
of the points in
Xfrom which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level ofmarks(X). - eps
- A positive number. The resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.
- r
- numeric vector. The values of the argument $r$ at which the function $J_{i\bullet}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important condit
- breaks
- This argument is for internal use only.
- ...
- Ignored.
- correction
- Optional. Character string specifying the edge correction(s)
to be used. Options are
"none","rs","km","Hanisch"and"best".
Details
This function Jdot and its companions
Jcross and Jmulti
are generalisations of the function Jest
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$marks must be a factor.
The argument i will be interpreted as a
level of the factor X$marks. (Warning: this means that
an integer value i=3 will be interpreted as the number 3,
not the 3rd smallest level.)
The ``type $i$ to any type'' multitype $J$ function
of a stationary multitype point process $X$
was introduced by Van lieshout and Baddeley (1999). It is defined by
$$J_{i\bullet}(r) = \frac{1 - G_{i\bullet}(r)}{1 -
F_{\bullet}(r)}$$
where $G_{i\bullet}(r)$ is the distribution function of
the distance from a type $i$ point to the nearest other point
of the pattern, and $F_{\bullet}(r)$ is the distribution
function of the distance from a fixed point in space to the nearest
point of the pattern.
An estimate of $J_{i\bullet}(r)$
is a useful summary statistic in exploratory data analysis
of a multitype point pattern. If the pattern is
a marked Poisson point process, then
$J_{i\bullet}(r) \equiv 1$.
If the subprocess of type $i$ points is independent
of the subprocess of points of all types not equal to $i$,
then $J_{i\bullet}(r)$ equals
$J_{ii}(r)$, the ordinary $J$ function
(see Jest and Van Lieshout and Baddeley (1996))
of the points of type $i$.
Hence deviations from zero of the empirical estimate of
$J_{i\bullet} - J_{ii}$
may suggest dependence between types.
This algorithm estimates $J_{i\bullet}(r)$
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 Jest,
using the Kaplan-Meier and border corrections.
The main work is done by Gmulti and Fest.
The argument r is the vector of values for the
distance $r$ at which $J_{i\bullet}(r)$ should be evaluated.
The values of $r$ must be increasing nonnegative numbers
and the maximum $r$ value must exceed the radius of the
largest disc contained in the window.
Value
- An object of class
"fv"(seefv.object).Essentially a data frame containing six numeric columns
J the recommended estimator of $J_{i\bullet}(r)$, currently the Kaplan-Meier estimator. r the values of the argument $r$ at which the function $J_{i\bullet}(r)$ has been estimated km the Kaplan-Meier estimator of $J_{i\bullet}(r)$ rs the ``reduced sample'' or ``border correction'' estimator of $J_{i\bullet}(r)$ han the Hanisch-style estimator of $J_{i\bullet}(r)$ un the ``uncorrected'' estimator of $J_{i\bullet}(r)$ formed by taking the ratio of uncorrected empirical estimators of $1 - G_{i\bullet}(r)$ and $1 - F_{\bullet}(r)$, see GdotandFest.theo the theoretical value of $J_{i\bullet}(r)$ for a marked Poisson process, namely 1. - The result also has two attributes
"G"and"F"which are respectively the outputs ofGdotandFestfor the point pattern.
Warnings
The argument i is interpreted as
a level of the factor X$marks. It is converted to a character
string if it is not already a character string.
The value i=1 does not
refer to the first level of the factor.
References
Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344--361.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511--532.
See Also
Examples
# Lansing woods data: 6 types of trees
data(lansing)
<testonly>lansing <- lansing[seq(1,lansing$n, by=30), ]</testonly>
Jh. <- Jdot(lansing, "hickory")
plot(Jh.)
# diagnostic plot for independence between hickories and other trees
Jhh <- Jest(lansing[lansing$marks == "hickory", ])
plot(Jhh, add=TRUE, legendpos="bottom")
# synthetic example with two marks "a" and "b"
pp <- runifpoint(30) %mark% factor(sample(c("a","b"), 30, replace=TRUE))
J <- Jdot(pp, "a")