Kcross.inhom {spatstat.explore} | R Documentation |
Inhomogeneous Cross K Function
Description
For a multitype point pattern,
estimate the inhomogeneous version of the cross function,
which counts the expected number of points of type
within a given distance of a point of type
,
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 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 has intensity function
at spatial locations
.
Suppose we place a mass of
at each point
of type
. Then the expected total
mass per unit area is 1. The
inhomogeneous “cross-type”
function
equals the expected
total mass within a radius
of a point of the process
of type
.
If the process of type points
were independent of the process of type
points,
then
would equal
.
Deviations between the empirical
curve
and the theoretical curve
suggest dependence between the points of types
and
.
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 typei
intensity at all locations in the window containingX
;- a numeric vector
containing the values of the type
i
intensity evaluated only at the data points of typei
. The length of this vector must equal the number of typei
points 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=TRUE
the model will first be refitted to the dataX
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 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.
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
obtained by the edge corrections named.
Warnings
The arguments i
and j
are always interpreted as
levels of the factor X$marks
. They are converted to character
strings if they are not already character strings.
The value i=1
does not
refer to the first level of the factor.
Author(s)
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner rolfturner@posteo.net and Ege Rubak rubak@math.aau.dk.
References
Baddeley, A., Moller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
Moller, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes Chapman and Hall/CRC Boca Raton, 2003.
See Also
Kcross
,
Kinhom
,
Kdot.inhom
,
Kmulti.inhom
,
pcf
Examples
# Lansing Woods data
woods <- lansing
ma <- split(woods)$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)