Kscaled: Locally Scaled K-function
In spatstat.explore: Exploratory Data Analysis for the 'spatstat' Family
Kscaled R Documentation
Locally Scaled K-function
Description
Estimates the locally-rescaled K-function of
a point process.
Usage
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(...)
Arguments
X
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().
lambda
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.
r
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.
breaks
This argument is for internal use only.
rmax
maximum value of the argument r that should be used.
(This is the rescaled distance).
correction
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.
renormalise
Logical. Whether to renormalise the estimate. See Details.
normpower
Integer (usually either 1 or 2).
Normalisation power. See Details.
sigma, varcov
Optional arguments passed to density.ppp
to control the smoothing bandwidth, when lambda is
estimated by kernel smoothing.
Details
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
- a numeric vector
-
containing the values
of the intensity function at the points of the pattern X.
- a pixel image
-
(object of class "im")
assumed to contain the values of the intensity function
at all locations in the window.
- a function
-
which can be evaluated to give values of the intensity at
any locations.
- omitted:
-
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.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing at least the following columns,
r
the vector of values of the argument r
at which the pair correlation function g(r) has been estimated
theo
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.
Author(s)
Ute Hahn,
\adrian
and \rolf
References
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.
See Also
Kest,
pcf
Examples
X <- unmark(bronzefilter)
K <- Kscaled(X)
if(require("spatstat.model")) {
fit <- ppm(X, ~x)
lam <- predict(fit)
K <- Kscaled(X, lam)
}
spatstat.explore documentation built on Oct. 22, 2024, 9:07 a.m.
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| Kscaled | R Documentation |
Locally Scaled K-function
Description
Estimates the locally-rescaled K-function of
a point process.
Usage
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(...)
Arguments
X |
The observed data point pattern,
from which an estimate of the locally scaled |
lambda |
Optional.
Values of the estimated intensity function.
Either a vector giving the intensity values
at the points of the pattern |
... |
Arguments passed from |
r |
vector of values for the argument |
breaks |
This argument is for internal use only. |
rmax |
maximum value of the argument |
correction |
A character vector containing any selection of the
options |
renormalise |
Logical. Whether to renormalise the estimate. See Details. |
normpower |
Integer (usually either 1 or 2). Normalisation power. See Details. |
sigma, varcov |
Optional arguments passed to |
Details
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
- a numeric vector
-
containing the values of the intensity function at the points of the pattern
X. - a pixel image
-
(object of class
"im") assumed to contain the values of the intensity function at all locations in the window. - a function
-
which can be evaluated to give values of the intensity at any locations.
- omitted:
-
if
lambdais 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.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing at least the following columns,
r |
the vector of values of the argument |
theo |
vector of values of |
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.
Author(s)
Ute Hahn, \adrian and \rolf
References
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.
See Also
Kest,
pcf
Examples
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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