For a multitype point pattern, estimate the inhomogeneous version of the dot \(K\) function, which counts the expected number of points of any type within a given distance of a point of type \(i\), adjusted for spatially varying intensity.
Kdot.inhom(X, i, lambdaI=NULL, lambdadot=NULL, …, r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
sigma=NULL, varcov=NULL, lambdaIdot=NULL,
lambdaX=NULL, update=TRUE, leaveoneout=TRUE)The observed point pattern, from which an estimate of the inhomogeneous cross type \(K\) function \(K_{i\bullet}(r)\) will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.
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).
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.
Optional.
Values of the estimated intensity of the entire point process,
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.
Ignored.
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 \(r\).
This argument is for internal 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.
Standard deviation of isotropic Gaussian smoothing kernel,
used in computing leave-one-out kernel estimates of
lambdaI, lambdadot if they are omitted.
Variance-covariance matrix of anisotropic Gaussian kernel,
used in computing leave-one-out kernel estimates of
lambdaI, lambdadot if they are omitted.
Incompatible with sigma.
Optional. A matrix containing estimates of the
product of the intensities lambdaI and lambdadot
for each pair of points, the first point of type i and
the second of any type.
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
lambdadot.
Logical value indicating what to do when
lambdaI, lambdadot 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 \(r\) at which the function \(K_{i\bullet}(r)\) has been estimated
the theoretical value of \(K_{i\bullet}(r)\) for a marked Poisson process, namely \(\pi r^2\)
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.
This is a generalisation of the function Kdot
to include an adjustment for spatially inhomogeneous intensity,
in a manner similar to the function Kinhom.
Briefly, given a multitype point process, consider the points without their types, and suppose this unmarked point process has intensity function \(\lambda(u)\) at spatial locations \(u\). Suppose we place a mass of \(1/\lambda(\zeta)\) at each point \(\zeta\) of the process. Then the expected total mass per unit area is 1. The inhomogeneous ``dot-type'' \(K\) function \(K_{i\bullet}^{\mbox{inhom}}(r)\) equals the expected total mass within a radius \(r\) of a point of the process of type \(i\), discounting this point itself.
If the process of type \(i\) points were independent of the points of other types, then \(K_{i\bullet}^{\mbox{inhom}}(r)\) would equal \(\pi r^2\). Deviations between the empirical \(K_{i\bullet}\) curve and the theoretical curve \(\pi r^2\) suggest dependence between the points of types \(i\) and \(j\) for \(j\neq i\).
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).
If i is missing, it defaults to the first
level of the marks factor, i = levels(X$marks)[1].
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.
of the form function(x,y)
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 the argument lambdadot should contain
estimated values of the intensity of the entire point process.
It may be either a pixel image, a numeric vector of length equal
to the number of points in X, a function, or omitted.
Alternatively if the argument lambdaX is given, then it specifies
the intensity values for all points of X, and the
arguments lambdaI, lambdadot will be ignored.
(The two arguments lambdaI, lambdadot allow the user
to specify two different methods for calculating the intensities of
the two kinds of points, while lambdaX ensures that the same
method is used for both kinds of points.)
For advanced use only, the optional argument lambdaIdot
is a matrix containing estimated
values of the products of these two intensities for each pair of
points, the first point of type i and the second of any type.
The argument r is the vector of values for the
distance \(r\) at which \(K_{i\bullet}(r)\) should be evaluated.
The values of \(r\) must be increasing nonnegative numbers
and the maximum \(r\) value must 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.
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
woods <- woods[seq(1,npoints(woods), by=10)]
ma <- split(woods)$maple
lg <- unmark(woods)
# Estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdadot <- density.ppp(lg, sigma=0.15, at="points")
K <- Kdot.inhom(woods, "maple", lambdaI=lambdaM,
lambdadot=lambdadot)
# Equivalent
K <- Kdot.inhom(woods, "maple", sigma=0.15)
# Fit model
fit <- ppm(woods ~ marks * polynom(x,y,2))
K <- Kdot.inhom(woods, "maple", lambdaX=fit, update=FALSE)
# synthetic example: type A points have intensity 50,
# type B points have intensity 50 + 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
lamdot <- as.im(function(x,y) { 100 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
K <- Kdot.inhom(X, "B", lambdaI=lamB, lambdadot=lamdot)
# }
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