Estimates the locally-rescaled \(K\)-function of a point process.
Kscaled(X, lambda=NULL, ..., r = NULL, breaks = NULL,
rmax = 2.5,
correction=c("border", "isotropic", "translate"),
renormalise=FALSE, normpower=1,
sigma=NULL, varcov=NULL) Lscaled(...)
An object of class "fv" (see fv.object).
Essentially a data frame containing at least the following columns,
the vector of values of the argument \(r\) at which the pair correlation function \(g(r)\) has been estimated
vector of values of \(\pi r^2\), the theoretical value of \(K_{\rm scaled}(r)\) for an inhomogeneous Poisson process
and containing additional columns
according to the choice specified in the correction
argument. The additional columns are named
border, trans and iso
and give the estimated values of
\(K_{\rm scaled}(r)\)
using the border correction, translation correction, and Ripley isotropic correction, respectively.
The observed data point pattern,
from which an estimate of the locally scaled \(K\) function
will be computed.
An object of class "ppp"
or in a format recognised by as.ppp().
Optional.
Values of the estimated intensity function.
Either a vector giving the intensity values
at the points of the pattern X,
a pixel image (object of class "im") giving the
intensity values at all locations, a function(x,y) which
can be evaluated to give the intensity value at any location,
or a fitted point process model (object of class "ppm").
Arguments passed from Lscaled to Kscaled
and from Kscaled to density.ppp
if lambda is omitted.
vector of values for the argument \(r\) at which the locally scaled \(K\) function should be evaluated. (These are rescaled distances.) Not normally given by the user; there is a sensible default.
This argument is for internal use only.
maximum value of the argument \(r\) that should be used. (This is the rescaled distance).
A character vector containing any selection of the
options "border",
"isotropic", "Ripley",
"translate", "translation",
"none" or "best".
It specifies the edge correction(s) to be applied.
Alternatively correction="all" selects all options.
Logical. Whether to renormalise the estimate. See Details.
Integer (usually either 1 or 2). Normalisation power. See Details.
Optional arguments passed to density.ppp
to control the smoothing bandwidth, when lambda is
estimated by kernel smoothing.
Ute Hahn, Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz
Kscaled computes an estimate of the \(K\) function
for a locally scaled point process.
Lscaled computes the corresponding \(L\) function
\(L(r) = \sqrt{K(r)/\pi}\).
Locally scaled point processes are a class of models for inhomogeneous point patterns, introduced by Hahn et al (2003). They include inhomogeneous Poisson processes, and many other models.
The template \(K\) function of a locally-scaled process is a counterpart of the ``ordinary'' Ripley \(K\) function, in which the distances between points of the process are measured on a spatially-varying scale (such that the locally rescaled process has unit intensity).
The template \(K\) function is an indicator of interaction between the points. For an inhomogeneous Poisson process, the theoretical template \(K\) function is approximately equal to \(K(r) = \pi r^2\). Values \(K_{\rm scaled}(r) > \pi r^2\) are suggestive of clustering.
Kscaled computes an estimate of the template \(K\) function
and Lscaled computes the corresponding \(L\) function
\(L(r) = \sqrt{K(r)/\pi}\).
The locally scaled interpoint distances are computed using an approximation proposed by Hahn (2007). The Euclidean distance between two points is multiplied by the average of the square roots of the intensity values at the two points.
The argument lambda should supply the
(estimated) values of the intensity function \(\lambda\).
It may be either
containing the values
of the intensity function at the points of the pattern X.
(object of class "im")
assumed to contain the values of the intensity function
at all locations in the window.
which can be evaluated to give values of the intensity at any locations.
if lambda is omitted, then it will be estimated using
a `leave-one-out' kernel smoother.
If lambda is a numeric vector, then its length should
be equal to the number of points in the pattern X.
The value lambda[i] is assumed to be the
the (estimated) value of the intensity
\(\lambda(x_i)\) for
the point \(x_i\) of the pattern \(X\).
Each value must be a positive number; NA's are not allowed.
If lambda is a pixel image, the domain of the image should
cover the entire window of the point pattern. If it does not (which
may occur near the boundary because of discretisation error),
then the missing pixel values
will be obtained by applying a Gaussian blur to lambda using
blur, then looking up the values of this blurred image
for the missing locations.
(A warning will be issued in this case.)
If lambda is a function, then it will be evaluated in the
form lambda(x,y) where x and y are vectors
of coordinates of the points of X. It should return a numeric
vector with length equal to the number of points in X.
If lambda is omitted, then it will be estimated using
a `leave-one-out' kernel smoother,
as described in Baddeley, Moller
and Waagepetersen (2000). The estimate lambda[i] for the
point X[i] is computed by removing X[i] from the
point pattern, applying kernel smoothing to the remaining points using
density.ppp, and evaluating the smoothed intensity
at the point X[i]. The smoothing kernel bandwidth is controlled
by the arguments sigma and varcov, which are passed to
density.ppp along with any extra arguments.
If renormalise=TRUE, the estimated intensity lambda
is multiplied by \(c^(normpower/2)\) before performing other calculations,
where \(c = area(W)/sum[i] (1/lambda(x[i]))\). This
renormalisation has about the same effect as in Kinhom,
reducing the variability and bias of the estimate
in small samples and in cases of very strong inhomogeneity.
Edge corrections are used to correct bias in the estimation
of \(K_{\rm scaled}\). First the interpoint distances are
rescaled, and then edge corrections are applied as in Kest.
See Kest for details of the edge corrections
and the options for the argument correction.
The pair correlation function can also be applied to the
result of Kscaled; see pcf and pcf.fv.
Baddeley, A., Moller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329--350.
Hahn, U. (2007) Global and Local Scaling in the Statistics of Spatial Point Processes. Habilitationsschrift, Universitaet Augsburg.
Hahn, U., Jensen, E.B.V., van Lieshout, M.N.M. and Nielsen, L.S. (2003) Inhomogeneous spatial point processes by location-dependent scaling. Advances in Applied Probability 35, 319--336.
Prokesova, M., Hahn, U. and Vedel Jensen, E.B. (2006) Statistics for locally scaled point patterns. In A. Baddeley, P. Gregori, J. Mateu, R. Stoica and D. Stoyan (eds.) Case Studies in Spatial Point Pattern Modelling. Lecture Notes in Statistics 185. New York: Springer Verlag. Pages 99--123.
Kest,
pcf
X <- unmark(bronzefilter)
K <- Kscaled(X)
if(require("spatstat.model")) {
fit <- ppm(X, ~x)
lam <- predict(fit)
K <- Kscaled(X, lam)
}
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