Jcross
Multitype J Function (i-to-j)
For a multitype point pattern, estimate the multitype $J$ function summarising the interpoint dependence between points of type $i$ and of type $j$.
- Keywords
- spatial
Usage
Jcross(X, i=1, j=2)
Jcross(X, i=1, j=2, eps, r)
Jcross(X, i=1, j=2, eps, breaks)
Arguments
- X
- The observed point pattern, from which an estimate of the multitype $J$ function $J_{ij}(r)$ 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. - 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 $J_{ij}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions o
- breaks
- An alternative to the argument
r
. Not normally invoked by the user. See the Details section.
Details
This function Jcross
and its companions
Jdot
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 spatstat
package,
a multitype pattern is represented as a single
point pattern object in which the points carry marks,
and the mark value attached to each point
determines the type of that point.
The argument 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 3rd smallest level,
not the number 3).
The ``type $i$ to type $j$'' multitype $J$ function
of a stationary multitype point process $X$
was introduced by Van lieshout and Baddeley (1999). It is defined by
$$J_{ij}(r) = \frac{1 - G_{ij}(r)}{1 -
F_{j}(r)}$$
where $G_{ij}(r)$ is the distribution function of
the distance from a type $i$ point to the nearest point of type $j$,
and $F_{j}(r)$ is the distribution
function of the distance from a fixed point in space to the nearest
point of type $j$ in the pattern.
An estimate of $J_{ij}(r)$ is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the subprocess of type $i$ points is independent of the subprocess of points of type $j$, then $J_{ij}(r) \equiv 1$. Hence deviations of the empirical estimate of $J_{ij}$ from the value 1 may suggest dependence between types.
This algorithm estimates $J_{ij}(r)$
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 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 $J_{ij}(r)$ 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 $J_{ij}(r)$, currently the Kaplan-Meier estimator. r the values of the argument $r$ at which the function $J_{ij}(r)$ has been estimated km the Kaplan-Meier estimator of $J_{ij}(r)$ rs the ``reduced sample'' or ``border correction'' estimator of $J_{ij}(r)$ un the ``uncorrected'' estimator of $J_{ij}(r)$ formed by taking the ratio of uncorrected empirical estimators of $1 - G_{ij}(r)$ and $1 - F_{j}(r)$, see Gdot
andFest
.theo the theoretical value of $J_{ij}(r)$ for a marked Poisson process, namely 1. - The result also has two attributes
"G"
and"F"
which are respectively the outputs ofGcross
andFest
for the point pattern.
synopsis
Jcross(X, i=1, j=2, eps=NULL, r=NULL, breaks=NULL)
Warnings
The argument i
is interpreted as
a level of the factor X$marks
. Beware of the usual
trap with factors: numerical values are not
interpreted in the same way as character values. See the first example.
References
Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344--361.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511--532.
See Also
Examples
# Lansing woods data: 6 types of trees
data(lansing)
<testonly>lansing <- lansing[seq(1,lansing$n, by=30), ]</testonly>
Jhm <- Jcross(lansing, "hickory", "maple")
# diagnostic plot for independence between hickories and maples
plot(Jhm)
# synthetic example with two marks "a" and "b"
pp <- runifpoispp(50)
pp <- pp %mark% sample(c("a","b"), pp$n, replace=TRUE)
J <- Jcross(pp, "a", "b")