Kcross.inhom: Inhomogeneous Cross K Function
In spatstat.explore: Exploratory Data Analysis for the 'spatstat' Family
Kcross.inhom R Documentation
Inhomogeneous Cross K Function
Description
For a multitype point pattern,
estimate the inhomogeneous version of the cross K function,
which counts the expected number of points of type j
within a given distance of a point of type i,
adjusted for spatially varying intensity.
Usage
Kcross.inhom(X, i, j, lambdaI=NULL, lambdaJ=NULL, ..., r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
sigma=NULL, varcov=NULL,
lambdaIJ=NULL,
lambdaX=NULL, update=TRUE, leaveoneout=TRUE)
Arguments
X
The observed point pattern,
from which an estimate of the inhomogeneous cross type K function
K_{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
The type (mark value)
of the points in X from which distances are measured.
A character string (or something that will be converted to a
character string).
Defaults to the first level of marks(X).
j
The type (mark value)
of the points in X to which distances are measured.
A character string (or something that will be
converted to a character string).
Defaults to the second level of marks(X).
lambdaI
Optional.
Values of the estimated intensity of the sub-process of
points of type i.
Either a pixel image (object of class "im"),
a numeric vector containing the intensity values
at each of the type i points in X,
a fitted point process model
(object of class "ppm" or "kppm" or "dppm"),
or a function(x,y) which
can be evaluated to give the intensity value at any location.
lambdaJ
Optional.
Values of the the estimated intensity of the sub-process of
points of type j.
Either a pixel image (object of class "im"),
a numeric vector containing the intensity values
at each of the type j points in X,
a fitted point process model
(object of class "ppm" or "kppm" or "dppm"),
or a function(x,y) which
can be evaluated to give the intensity value at any location.
r
Optional. Numeric vector giving the values of the argument r
at which the cross 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 advanced use only.
correction
A character vector containing any selection of the
options "border", "bord.modif",
"isotropic", "Ripley" ,"translate",
"translation",
"none" or "best".
It specifies the edge correction(s) to be applied.
Alternatively correction="all" selects all options.
...
Ignored.
sigma
Standard deviation of isotropic Gaussian smoothing kernel,
used in computing leave-one-out kernel estimates of
lambdaI, lambdaJ if they are omitted.
varcov
Variance-covariance matrix of anisotropic Gaussian kernel,
used in computing leave-one-out kernel estimates of
lambdaI, lambdaJ if they are omitted.
Incompatible with sigma.
lambdaIJ
Optional. A matrix containing estimates of the
product of the intensities lambdaI and lambdaJ
for each pair of points of types i and j respectively.
lambdaX
Optional. Values of the intensity for all points of X.
Either a pixel image (object of class "im"),
a numeric vector containing the intensity values
at each of the points in X,
a fitted point process model
(object of class "ppm" or "kppm" or "dppm"),
or a function(x,y) which
can be evaluated to give the intensity value at any location.
If present, this argument overrides both lambdaI and
lambdaJ.
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 refitted to the data X
(using update.ppm or update.kppm)
before the fitted intensity is computed.
If update=FALSE, the fitted intensity of the
model will be computed without re-fitting it to X.
leaveoneout
Logical value (passed to density.ppp or
fitted.ppm) specifying whether to use a
leave-one-out rule when calculating the intensity.
Details
This is a generalisation of the function Kcross
to include an adjustment for spatially inhomogeneous intensity,
in a manner similar to the function Kinhom.
The inhomogeneous cross-type K function is described by
\Moller and Waagepetersen (2003, pages 48-49 and 51-53).
Briefly, given a multitype point process, suppose the sub-process
of points of type j has intensity function
\lambda_j(u) at spatial locations u.
Suppose we place a mass of 1/\lambda_j(\zeta)
at each point \zeta of type j. Then the expected total
mass per unit area is 1. The
inhomogeneous “cross-type” K function
K_{ij}^{\mbox{inhom}}(r) equals the expected
total mass within a radius r of a point of the process
of type i.
If the process of type i points
were independent of the process of type j points,
then K_{ij}^{\mbox{inhom}}(r)
would equal \pi r^2.
Deviations between the empirical K_{ij} curve
and the theoretical curve \pi r^2
suggest dependence between the points of types i and j.
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$marks must be a factor.
The arguments i and j will be interpreted as
levels of the factor X$marks. (Warning: this means that
an integer value i=3 will be interpreted as the number 3,
not the 3rd smallest level).
If i and j are missing, they default to the first
and second level of the marks factor, respectively.
The argument lambdaI supplies the values
of the intensity of the sub-process of points of type i.
It may be either
- a pixel image
(object of class "im") which
gives the values of the type i intensity
at all locations in the window containing X;
- a numeric vector
containing the values of the
type i intensity evaluated only
at the data points of type i. The length of this vector
must equal the number of type i points in X.
- a function
-
which can be evaluated to give values of the intensity at
any locations.
- a fitted point process model
-
(object of class "ppm", "kppm" or "dppm")
whose fitted trend can be used as the fitted intensity.
(If update=TRUE the model will first be refitted to the
data X before the trend is 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 should contain
estimated values of the intensity of the sub-process of points of
type j. It may be either a pixel image, a function,
a numeric vector, or omitted.
Alternatively if the argument lambdaX is given, then it specifies
the intensity values for all points of X, and the
arguments lambdaI, lambdaJ will be ignored.
The optional argument lambdaIJ is for advanced use only.
It is a matrix containing estimated
values of the products of these two intensities for each pair of
data points of types i and j respectively.
The argument r is the vector of values for the
distance r at which K_{ij}(r) should be evaluated.
The values of r must be increasing nonnegative numbers
and the maximum r value must not exceed the radius of the
largest disc contained in the window.
The argument correction chooses the edge correction
as explained e.g. in Kest.
The pair correlation function can also be applied to the
result of Kcross.inhom; see pcf.
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.
Warnings
The arguments i and j are always interpreted as
levels of the factor X$marks. They are converted to character
strings if they are not already character strings.
The value i=1 does not
refer to the first level of the factor.
Author(s)
\spatstatAuthors.
References
Baddeley, A., \Moller, J. and Waagepetersen, R. (2000)
Non- and semiparametric estimation of interaction in
inhomogeneous point patterns.
Statistica Neerlandica 54, 329–350.
\Moller, J. and Waagepetersen, R.
Statistical Inference and Simulation for Spatial Point Processes
Chapman and Hall/CRC
Boca Raton, 2003.
See Also
Kcross,
Kinhom,
Kdot.inhom,
Kmulti.inhom,
pcf
Examples
# Lansing Woods data
woods <- lansing
ma <- split(woods)$maple
wh <- split(woods)$whiteoak
# method (1): estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdaW <- density.ppp(wh, sigma=0.15, at="points")
K <- Kcross.inhom(woods, "whiteoak", "maple", lambdaW, lambdaM)
# method (2): leave-one-out
K <- Kcross.inhom(woods, "whiteoak", "maple", sigma=0.15)
# method (3): fit parametric intensity model
if(require("spatstat.model")) {
fit <- ppm(woods ~marks * polynom(x,y,2))
# alternative (a): use fitted model as 'lambda' argument
online <- interactive()
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambdaI=fit, lambdaJ=fit,
update=online, leaveoneout=online)
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambdaX=fit,
update=online, leaveoneout=online)
# alternative (b): evaluate fitted intensities at data points
# (these are the intensities of the sub-processes of each type)
inten <- fitted(fit, dataonly=TRUE, leaveoneout=FALSE)
# split according to types of points
lambda <- split(inten, marks(woods))
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambda$whiteoak, lambda$maple)
}
# synthetic example: type A points have intensity 50,
# type B points have intensity 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
K <- Kcross.inhom(X, "A", "B",
lambdaI=as.im(50, Window(X)), lambdaJ=lamB)
spatstat.explore documentation built on Oct. 22, 2024, 9:07 a.m.
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| Kcross.inhom | R Documentation |
Inhomogeneous Cross K Function
Description
For a multitype point pattern,
estimate the inhomogeneous version of the cross K function,
which counts the expected number of points of type j
within a given distance of a point of type i,
adjusted for spatially varying intensity.
Usage
Kcross.inhom(X, i, j, lambdaI=NULL, lambdaJ=NULL, ..., r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
sigma=NULL, varcov=NULL,
lambdaIJ=NULL,
lambdaX=NULL, update=TRUE, leaveoneout=TRUE)
Arguments
X |
The observed point pattern,
from which an estimate of the inhomogeneous cross type |
i |
The type (mark value)
of the points in |
j |
The type (mark value)
of the points in |
lambdaI |
Optional.
Values of the estimated intensity of the sub-process of
points of type |
lambdaJ |
Optional.
Values of the the estimated intensity of the sub-process of
points of type |
r |
Optional. Numeric vector giving the values of the argument |
breaks |
This argument is for advanced use only. |
correction |
A character vector containing any selection of the
options |
... |
Ignored. |
sigma |
Standard deviation of isotropic Gaussian smoothing kernel,
used in computing leave-one-out kernel estimates of
|
varcov |
Variance-covariance matrix of anisotropic Gaussian kernel,
used in computing leave-one-out kernel estimates of
|
lambdaIJ |
Optional. A matrix containing estimates of the
product of the intensities |
lambdaX |
Optional. Values of the intensity for all points of |
update |
Logical value indicating what to do when
|
leaveoneout |
Logical value (passed to |
Details
This is a generalisation of the function Kcross
to include an adjustment for spatially inhomogeneous intensity,
in a manner similar to the function Kinhom.
The inhomogeneous cross-type K function is described by
\Moller and Waagepetersen (2003, pages 48-49 and 51-53).
Briefly, given a multitype point process, suppose the sub-process
of points of type j has intensity function
\lambda_j(u) at spatial locations u.
Suppose we place a mass of 1/\lambda_j(\zeta)
at each point \zeta of type j. Then the expected total
mass per unit area is 1. The
inhomogeneous “cross-type” K function
K_{ij}^{\mbox{inhom}}(r) equals the expected
total mass within a radius r of a point of the process
of type i.
If the process of type i points
were independent of the process of type j points,
then K_{ij}^{\mbox{inhom}}(r)
would equal \pi r^2.
Deviations between the empirical K_{ij} curve
and the theoretical curve \pi r^2
suggest dependence between the points of types i and j.
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 arguments i and j will be interpreted as
levels 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).
If i and j are missing, they default to the first
and second level of the marks factor, respectively.
The argument lambdaI supplies the values
of the intensity of the sub-process of points of type i.
It may be either
- a pixel image
(object of class
"im") which gives the values of the typeiintensity at all locations in the window containingX;- a numeric vector
containing the values of the type
iintensity evaluated only at the data points of typei. The length of this vector must equal the number of typeipoints inX.- a function
-
which can be evaluated to give values of the intensity at any locations.
- a fitted point process model
-
(object of class
"ppm","kppm"or"dppm") whose fitted trend can be used as the fitted intensity. (Ifupdate=TRUEthe model will first be refitted to the dataXbefore the trend is 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 should contain
estimated values of the intensity of the sub-process of points of
type j. It may be either a pixel image, a function,
a numeric vector, or omitted.
Alternatively if the argument lambdaX is given, then it specifies
the intensity values for all points of X, and the
arguments lambdaI, lambdaJ will be ignored.
The optional argument lambdaIJ is for advanced use only.
It is a matrix containing estimated
values of the products of these two intensities for each pair of
data points of types i and j respectively.
The argument r is the vector of values for the
distance r at which K_{ij}(r) should be evaluated.
The values of r must be increasing nonnegative numbers
and the maximum r value must not exceed the radius of the
largest disc contained in the window.
The argument correction chooses the edge correction
as explained e.g. in Kest.
The pair correlation function can also be applied to the
result of Kcross.inhom; see pcf.
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.
Warnings
The arguments i and j are always interpreted as
levels of the factor X$marks. They are converted to character
strings if they are not already character strings.
The value i=1 does not
refer to the first level of the factor.
Author(s)
\spatstatAuthors.
References
Baddeley, A., \Moller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
\Moller, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes Chapman and Hall/CRC Boca Raton, 2003.
See Also
Kcross,
Kinhom,
Kdot.inhom,
Kmulti.inhom,
pcf
Examples
# Lansing Woods data
woods <- lansing
ma <- split(woods)$maple
wh <- split(woods)$whiteoak
# method (1): estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdaW <- density.ppp(wh, sigma=0.15, at="points")
K <- Kcross.inhom(woods, "whiteoak", "maple", lambdaW, lambdaM)
# method (2): leave-one-out
K <- Kcross.inhom(woods, "whiteoak", "maple", sigma=0.15)
# method (3): fit parametric intensity model
if(require("spatstat.model")) {
fit <- ppm(woods ~marks * polynom(x,y,2))
# alternative (a): use fitted model as 'lambda' argument
online <- interactive()
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambdaI=fit, lambdaJ=fit,
update=online, leaveoneout=online)
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambdaX=fit,
update=online, leaveoneout=online)
# alternative (b): evaluate fitted intensities at data points
# (these are the intensities of the sub-processes of each type)
inten <- fitted(fit, dataonly=TRUE, leaveoneout=FALSE)
# split according to types of points
lambda <- split(inten, marks(woods))
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambda$whiteoak, lambda$maple)
}
# synthetic example: type A points have intensity 50,
# type B points have intensity 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
K <- Kcross.inhom(X, "A", "B",
lambdaI=as.im(50, Window(X)), lambdaJ=lamB)
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