Kmulti.inhom: Inhomogeneous Marked K-Function
In spatstat.explore: Exploratory Data Analysis for the 'spatstat' Family
Kmulti.inhom R Documentation
Inhomogeneous Marked K-Function
Description
For a marked point pattern,
estimate the inhomogeneous version of the multitype K function
which counts the expected number of points of subset J
within a given distance from a typical point in subset I,
adjusted for spatially varying intensity.
Usage
Kmulti.inhom(X, I, J, lambdaI=NULL, lambdaJ=NULL,
...,
r=NULL, breaks=NULL, rmax=NULL,
correction=c("border", "isotropic", "Ripley", "translate"),
lambdaIJ=NULL,
sigma=NULL, varcov=NULL,
lambdaX=NULL, update=TRUE, leaveoneout=TRUE,
Ilevels=NULL, Jlevels=NULL)
Arguments
X
The observed point pattern,
from which an estimate of the inhomogeneous multitype K function
K_{IJ}(r) will be computed.
It must be a marked point pattern.
See under Details.
I
Subset index specifying the points of X
from which distances are measured. See Details.
J
Subset index specifying the points in X to which
distances are measured. See Details.
lambdaI
Optional.
Values of the estimated intensity function of the points
belonging to subset I.
(Ignored if lambdaX is given.)
Either a numeric vector giving the intensity values
at the data points of subset I,
a pixel image (object of class "im") giving the
intensity values of points of subset I at all locations,
a function(x,y) which
can be evaluated to give the intensity value of points of subset I
at any location, or a fitted point process model
(class "ppm", "kppm", "dppm" or "slrm")
which could be used to predict the
intensity values of points of subset J at all locations.
If lambdaI is an unmarked point process model, it is assumed
to have been fitted to X[I]; if it is a multitype point
process model, it is assumed to have been fitted to X,
and the argument Ilevels is required.
lambdaJ
Optional.
Values of the estimated intensity function of the points
belonging to subset J.
A numeric vector, pixel image, function(x,y),
or fitted point process model.
Ignored if lambdaX is given.
...
Ignored.
r
Optional. Numeric vector. The values of the argument r
at which the multitype K function
K_{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 on r.
breaks
This argument is for internal use only.
rmax
Optional. Maximum desired value of the argument r.
A single numeric value. There is a sensible default.
correction
A character vector containing any selection of the
options "border", "bord.modif",
"isotropic", "Ripley", "translate",
"none" or "best".
It specifies the edge correction(s) to be applied.
Alternatively correction="all" selects all options.
lambdaIJ
Optional. A matrix containing estimates of
the product of the intensities lambdaI and lambdaJ
for each pair of points, the first point belonging to subset
I and the second point to subset J.
sigma, varcov
Optional arguments passed to density.ppp
to control the smoothing bandwidth, when lambda is
estimated by kernel smoothing.
lambdaX
Optional.
Alternative to lambdaI and lambdaJ.
Data which can be used to calculate the intensity functions
of both the subsets I and J.
A list of pixel images (one image for each possible type of point),
a function(x,y,mark) giving the intensity at each location
for each possible type of point, a numeric matrix with one row for
each point in X and one column for each possible type of
point, or a fitted multitype point process model (class
"ppm"). The arguments Ilevels and Jlevels are
required in this case.
update
Logical value indicating what to do when
lambdaI, lambdaJ or lambdaX
is a fitted point process model
(class "ppm", "kppm" or "dppm").
If update=TRUE (the default),
the model will first be updated (refitted to the data)
before the fitted intensity lambdaI or lambdaJ
at the data points is computed.
If update=FALSE, the fitted intensity of the
model will be computed without re-fitting the model.
leaveoneout
Logical value (passed to density.ppp or
fitted.ppm) specifying whether to use a
leave-one-out rule when calculating the intensity.
Ilevels
Character vector containing the types of points
(levels of marks(X)) which
comprise the subset I.
Required when lambdaX is given
or when lambdaI is a multitype point process model.
Jlevels
Character vector containing the types of points
which comprise the subset J.
Required when lambdaX is given
or when lambdaJ is a multitype point process model.
Details
The function Kmulti.inhom
is the counterpart, for spatially-inhomogeneous marked point patterns,
of the multitype K function Kmulti.
Suppose X is a marked point process, with marks of any kind.
Suppose X_I, X_J are two sub-processes, possibly
overlapping. Typically X_I would consist of those points
of X whose marks lie in a specified range of mark values,
and similarly for X_J. Suppose that
\lambda_I(u), \lambda_J(u) are the
spatially-varying intensity functions of X_I and
X_J respectively. Consider all the pairs of points
(u,v) in the point process X such that the first point
u belongs to X_I, the second point v
belongs to X_J, and the distance between u and v
is less than a specified distance r. Give this pair (u,v)
the numerical weight
1/(\lambda_I(u)\lambda_J(u)).
Calculate the sum of these weights over all pairs of points as
described. This sum (after appropriate edge-correction and
normalisation) is the estimated inhomogeneous multitype K function.
The argument X must be a point pattern (object of class
"ppp") or any data that are acceptable to as.ppp.
The arguments I and J specify two subsets of the
point pattern. They may be any type of subset indices, for example,
logical vectors of length equal to npoints(X),
or integer vectors with entries in the range 1 to
npoints(X), or negative integer vectors.
Alternatively, I and J may be functions
that will be applied to the point pattern X to obtain
index vectors. If I is a function, then evaluating
I(X) should yield a valid subset index. This option
is useful when generating simulation envelopes using
envelope.
The argument lambdaI supplies the values
of the intensity of the sub-process identified by index I.
It may be either
- a pixel image
(object of class "im") which
gives the values of the intensity of X[I]
at all locations in the window containing X;
- a numeric vector
containing the values of the
intensity of X[I] evaluated only
at the data points of X[I]. The length of this vector
must equal the number of points in X[I].
- a function
-
of the form function(x,y)
which can be evaluated to give values of the intensity at
any locations.
- A fitted multitype point process model
-
(class "lppm" or "ppm")
that was fitted to a multitype point pattern. If
update=TRUE, this model will be re-fitted to the data
pattern X. Then the intensity of this model for
the subset I will be computed (using the information
Ilevels).
- A fitted unmarked point process model
-
(class "lppm", "ppm", "kppm", "dppm" or "slrm")
that was fitted to an unmarked point pattern. If
update=TRUE, this model will be re-fitted to the subset of
the data, unmark(X[I]). Then the intensity of this fitted model
will be computed.
- omitted:
-
if lambdaI is omitted then it will be estimated
using a leave-one-out kernel smoother.
If lambdaI is omitted, then it will be estimated using
a ‘leave-one-out’ kernel smoother, as described in Baddeley,
\Moller and Waagepetersen (2000). The estimate of lambdaI for a given
point is computed by removing the point from the
point pattern, applying kernel smoothing to the remaining points using
density.ppp, and evaluating the smoothed intensity
at the point in question. The smoothing kernel bandwidth is controlled
by the arguments sigma and varcov, which are passed to
density.ppp along with any extra arguments.
Similarly lambdaJ supplies the values
of the intensity of the sub-process identified by index J.
Alternatively if the argument lambdaX is given,
then it should
provide data giving the intensities of both subsets I and
J. It may be
A list of pixel images, one for each possible type of point
A function(x,y,mark) which can be evaluated to
give the intensity of each type of point at any location
A matrix with one row for each data point in X
and one column for each possible type of point, giving the estimated
intensity of each type of point at each data location
A fitted point process model (class "lppm" or "ppm")
that was fitted to a multitype point pattern. If
update=TRUE, this model will be re-fitted to the data
pattern X. Then the fitted intensities for subsets I and
J will be computed (using the information Ilevels
and Jlevels).
The arguments lambdaI, lambdaJ will be ignored
if lambdaX is given.
The argument r is the vector of values for the
distance r at which K_{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.
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.
Biases due to edge effects are
treated in the same manner as in Kinhom.
The edge corrections implemented here are
- border
the border method or
“reduced sample” estimator (see Ripley, 1988). This is
the least efficient (statistically) and the fastest to compute.
It can be computed for a window of arbitrary shape.
- isotropic/Ripley
Ripley's isotropic correction
(see Ripley, 1988; Ohser, 1983).
This is currently implemented only for rectangular windows.
- translate
Translation correction (Ohser, 1983).
Implemented for all window geometries.
The pair correlation function pcf can also be applied to the
result of Kmulti.inhom.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r
the values of the argument r
at which the function K_{IJ}(r) has been estimated
theo
the theoretical value of K_{IJ}(r)
for a marked Poisson process, namely \pi r^2
together with a column or columns named
"border", "bord.modif",
"iso" and/or "trans",
according to the selected edge corrections. These columns contain
estimates of the function K_{IJ}(r)
obtained by the edge corrections named.
Author(s)
\adrian
and \rolf
References
Baddeley, A., \Moller, J. and Waagepetersen, R. (2000)
Non- and semiparametric estimation of interaction in
inhomogeneous point patterns.
Statistica Neerlandica 54, 329–350.
See Also
Kmulti,
Kdot.inhom,
Kcross.inhom,
pcf
Examples
# Finnish Pines data: marked by diameter and height
plot(finpines, which.marks="height")
II <- (marks(finpines)$height <= 2)
JJ <- (marks(finpines)$height > 3)
K <- Kmulti.inhom(finpines, II, JJ)
plot(K)
# functions determining subsets
f1 <- function(X) { marks(X)$height <= 2 }
f2 <- function(X) { marks(X)$height > 3 }
K <- Kmulti.inhom(finpines, f1, f2)
spatstat.explore documentation built on March 22, 2026, 5:06 p.m.
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| Kmulti.inhom | R Documentation |
Inhomogeneous Marked K-Function
Description
For a marked point pattern,
estimate the inhomogeneous version of the multitype K function
which counts the expected number of points of subset J
within a given distance from a typical point in subset I,
adjusted for spatially varying intensity.
Usage
Kmulti.inhom(X, I, J, lambdaI=NULL, lambdaJ=NULL,
...,
r=NULL, breaks=NULL, rmax=NULL,
correction=c("border", "isotropic", "Ripley", "translate"),
lambdaIJ=NULL,
sigma=NULL, varcov=NULL,
lambdaX=NULL, update=TRUE, leaveoneout=TRUE,
Ilevels=NULL, Jlevels=NULL)
Arguments
X |
The observed point pattern,
from which an estimate of the inhomogeneous multitype |
I |
Subset index specifying the points of |
J |
Subset index specifying the points in |
lambdaI |
Optional.
Values of the estimated intensity function of the points
belonging to subset |
lambdaJ |
Optional.
Values of the estimated intensity function of the points
belonging to subset |
... |
Ignored. |
r |
Optional. Numeric vector. The values of the argument |
breaks |
This argument is for internal use only. |
rmax |
Optional. Maximum desired value of the argument |
correction |
A character vector containing any selection of the
options |
lambdaIJ |
Optional. A matrix containing estimates of
the product of the intensities |
sigma, varcov |
Optional arguments passed to |
lambdaX |
Optional.
Alternative to |
update |
Logical value indicating what to do when
|
leaveoneout |
Logical value (passed to |
Ilevels |
Character vector containing the types of points
(levels of |
Jlevels |
Character vector containing the types of points
which comprise the subset |
Details
The function Kmulti.inhom
is the counterpart, for spatially-inhomogeneous marked point patterns,
of the multitype K function Kmulti.
Suppose X is a marked point process, with marks of any kind.
Suppose X_I, X_J are two sub-processes, possibly
overlapping. Typically X_I would consist of those points
of X whose marks lie in a specified range of mark values,
and similarly for X_J. Suppose that
\lambda_I(u), \lambda_J(u) are the
spatially-varying intensity functions of X_I and
X_J respectively. Consider all the pairs of points
(u,v) in the point process X such that the first point
u belongs to X_I, the second point v
belongs to X_J, and the distance between u and v
is less than a specified distance r. Give this pair (u,v)
the numerical weight
1/(\lambda_I(u)\lambda_J(u)).
Calculate the sum of these weights over all pairs of points as
described. This sum (after appropriate edge-correction and
normalisation) is the estimated inhomogeneous multitype K function.
The argument X must be a point pattern (object of class
"ppp") or any data that are acceptable to as.ppp.
The arguments I and J specify two subsets of the
point pattern. They may be any type of subset indices, for example,
logical vectors of length equal to npoints(X),
or integer vectors with entries in the range 1 to
npoints(X), or negative integer vectors.
Alternatively, I and J may be functions
that will be applied to the point pattern X to obtain
index vectors. If I is a function, then evaluating
I(X) should yield a valid subset index. This option
is useful when generating simulation envelopes using
envelope.
The argument lambdaI supplies the values
of the intensity of the sub-process identified by index I.
It may be either
- a pixel image
(object of class
"im") which gives the values of the intensity ofX[I]at all locations in the window containingX;- a numeric vector
containing the values of the intensity of
X[I]evaluated only at the data points ofX[I]. The length of this vector must equal the number of points inX[I].- a function
-
of the form
function(x,y)which can be evaluated to give values of the intensity at any locations. - A fitted multitype point process model
-
(class
"lppm"or"ppm") that was fitted to a multitype point pattern. Ifupdate=TRUE, this model will be re-fitted to the data patternX. Then the intensity of this model for the subsetIwill be computed (using the informationIlevels). - A fitted unmarked point process model
-
(class
"lppm","ppm","kppm","dppm"or"slrm") that was fitted to an unmarked point pattern. Ifupdate=TRUE, this model will be re-fitted to the subset of the data,unmark(X[I]). Then the intensity of this fitted model will be computed. - omitted:
-
if
lambdaIis omitted then it will be estimated using a leave-one-out kernel smoother.
If lambdaI is omitted, then it will be estimated using
a ‘leave-one-out’ kernel smoother, as described in Baddeley,
\Moller and Waagepetersen (2000). The estimate of lambdaI for a given
point is computed by removing the point from the
point pattern, applying kernel smoothing to the remaining points using
density.ppp, and evaluating the smoothed intensity
at the point in question. The smoothing kernel bandwidth is controlled
by the arguments sigma and varcov, which are passed to
density.ppp along with any extra arguments.
Similarly lambdaJ supplies the values
of the intensity of the sub-process identified by index J.
Alternatively if the argument lambdaX is given,
then it should
provide data giving the intensities of both subsets I and
J. It may be
A list of pixel images, one for each possible type of point
A
function(x,y,mark)which can be evaluated to give the intensity of each type of point at any locationA matrix with one row for each data point in
Xand one column for each possible type of point, giving the estimated intensity of each type of point at each data locationA fitted point process model (class
"lppm"or"ppm") that was fitted to a multitype point pattern. Ifupdate=TRUE, this model will be re-fitted to the data patternX. Then the fitted intensities for subsetsIandJwill be computed (using the informationIlevelsandJlevels).
The arguments lambdaI, lambdaJ will be ignored
if lambdaX is given.
The argument r is the vector of values for the
distance r at which K_{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.
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.
Biases due to edge effects are
treated in the same manner as in Kinhom.
The edge corrections implemented here are
- border
the border method or “reduced sample” estimator (see Ripley, 1988). This is the least efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary shape.
- isotropic/Ripley
Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is currently implemented only for rectangular windows.
- translate
Translation correction (Ohser, 1983). Implemented for all window geometries.
The pair correlation function pcf can also be applied to the
result of Kmulti.inhom.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r |
the values of the argument |
theo |
the theoretical value of |
together with a column or columns named
"border", "bord.modif",
"iso" and/or "trans",
according to the selected edge corrections. These columns contain
estimates of the function K_{IJ}(r)
obtained by the edge corrections named.
Author(s)
\adrianand \rolf
References
Baddeley, A., \Moller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
See Also
Kmulti,
Kdot.inhom,
Kcross.inhom,
pcf
Examples
# Finnish Pines data: marked by diameter and height
plot(finpines, which.marks="height")
II <- (marks(finpines)$height <= 2)
JJ <- (marks(finpines)$height > 3)
K <- Kmulti.inhom(finpines, II, JJ)
plot(K)
# functions determining subsets
f1 <- function(X) { marks(X)$height <= 2 }
f2 <- function(X) { marks(X)$height > 3 }
K <- Kmulti.inhom(finpines, f1, f2)
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