Jdot {spatstat.explore} | R Documentation |
Multitype J Function (i-to-any)
Description
For a multitype point pattern,
estimate the multitype function
summarising the interpoint dependence between
the type
points and the points of any type.
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 |
i |
The type (mark value)
of the points in |
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 |
breaks |
This argument is for internal use only. |
... |
Ignored. |
correction |
Optional. Character string specifying the edge correction(s)
to be used. Options are |
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 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 number 3,
not the 3rd smallest level.)
The “type to any type” multitype
function
of a stationary multitype point process
was introduced by Van lieshout and Baddeley (1999). It is defined by
where is the distribution function of
the distance from a type
point to the nearest other point
of the pattern, and
is the distribution
function of the distance from a fixed point in space to the nearest
point of the pattern.
An estimate of
is a useful summary statistic in exploratory data analysis
of a multitype point pattern. If the pattern is
a marked Poisson point process, then
.
If the subprocess of type
points is independent
of the subprocess of points of all types not equal to
,
then
equals
, the ordinary
function
(see
Jest
and Van Lieshout and Baddeley (1996))
of the points of type .
Hence deviations from zero of the empirical estimate of
may suggest dependence between types.
This algorithm estimates
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 Window(X)
)
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 at which
should be evaluated.
The values of
must be increasing nonnegative numbers
and the maximum
value must not exceed the radius of the
largest disc contained in the window.
Value
An object of class "fv"
(see fv.object
).
Essentially a data frame containing six numeric columns
J |
the recommended
estimator of |
r |
the values of the argument |
km |
the Kaplan-Meier
estimator of |
rs |
the “reduced sample” or “border correction”
estimator of |
han |
the Hanisch-style
estimator of |
un |
the “uncorrected”
estimator of |
theo |
the theoretical value of |
The result also has two attributes "G"
and "F"
which are respectively the outputs of Gdot
and Fest
for the point pattern.
Warnings
The argument i
is interpreted as
a level of the factor X$marks
. It is converted to a character
string if it is not already a character string.
The value i=1
does not
refer to the first level of the factor.
Author(s)
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner rolfturner@posteo.net.
References
Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344–361.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
See Also
Examples
# Lansing woods data: 6 types of trees
woods <- lansing
Jh. <- Jdot(woods, "hickory")
plot(Jh.)
# diagnostic plot for independence between hickories and other trees
Jhh <- Jest(split(woods)$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")