For a multitype point pattern,
estimate the inhomogeneous version of the cross
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)
The observed point pattern,
from which an estimate of the inhomogeneous cross type
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)
.
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)
.
Optional.
Values of the 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.
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.
Optional. Numeric vector giving the values of the argument
This argument is for advanced use only.
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.
Standard deviation of isotropic Gaussian smoothing kernel,
used in computing leave-one-out kernel estimates of
lambdaI
, lambdaJ
if they are omitted.
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
.
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.
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
.
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
.
Logical value (passed to density.ppp
or
fitted.ppm
) specifying whether to use a
leave-one-out rule when calculating the intensity.
An object of class "fv"
(see fv.object
).
Essentially a data frame containing numeric columns
the values of the argument
the theoretical value of
The arguments i
and j
are always interpreted as
levels of the factor X$marks
. They are converted to character
strings if they are not already character strings.
The value i=1
does not
refer to the first level of the factor.
This is a generalisation of the function Kcross
to include an adjustment for spatially inhomogeneous intensity,
in a manner similar to the function Kinhom
.
The inhomogeneous cross-type
Briefly, given a multitype point process, suppose the sub-process
of points of type
If the process of type
The argument X
must be a point pattern (object of class
"ppp"
) or any data that are acceptable to as.ppp
.
It must be a marked point pattern, and the mark vector
X$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
(object of class "im"
) which
gives the values of the type i
intensity
at all locations in the window containing X
;
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
.
which can be evaluated to give values of the intensity at any locations.
(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.)
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
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
.
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.
# NOT RUN {
# Lansing Woods data
woods <- lansing
# }
# NOT RUN {
ma <- split(woods)$maple
wh <- split(woods)$whiteoak
# method (1): estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdaW <- density.ppp(wh, sigma=0.15, at="points")
K <- Kcross.inhom(woods, "whiteoak", "maple", lambdaW, lambdaM)
# method (2): leave-one-out
K <- Kcross.inhom(woods, "whiteoak", "maple", sigma=0.15)
# method (3): fit parametric intensity model
fit <- ppm(woods ~marks * polynom(x,y,2))
# alternative (a): use fitted model as 'lambda' argument
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambdaI=fit, lambdaJ=fit, update=FALSE)
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambdaX=fit, update=FALSE)
# alternative (b): evaluate fitted intensities at data points
# (these are the intensities of the sub-processes of each type)
inten <- fitted(fit, dataonly=TRUE)
# split according to types of points
lambda <- split(inten, marks(woods))
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambda$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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