For a multitype point pattern on a linear network, estimate the inhomogeneous multitype \(K\) function which counts the expected number of points of type \(j\) within a given distance of a point of type \(i\).
linearKcross.inhom(X, i, j, lambdaI=NULL, lambdaJ=NULL,
r=NULL, ..., correction="Ang", normalise=TRUE,
sigma=NULL)An object of class "fv" (see fv.object).
The observed point pattern,
from which an estimate of the cross type \(K\) function
\(K_{ij}(r)\) will be computed.
An object of class "lpp" which
must be a multitype point pattern (a marked point pattern
whose marks are a factor).
Number or character string identifying the type (mark value)
of the points in X from which distances are measured.
Defaults to the first level of marks(X).
Number or character string identifying the type (mark value)
of the points in X to which distances are measured.
Defaults to the second level of marks(X).
Intensity values for the points of type i. Either a numeric vector,
a function, a pixel image
(object of class "im" or "linim") or
a fitted point process model (object of class "ppm"
or "lppm") or NULL.
Intensity values for the points of type j. Either a numeric vector,
a function, a pixel image
(object of class "im" or "linim") or
a fitted point process model (object of class "ppm"
or "lppm") or NULL.
numeric vector. The values of the argument \(r\) at which the \(K\)-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 \(r\).
Geometry correction.
Either "none" or "Ang". See Details.
Arguments passed to lambdaI and lambdaJ if
they are functions.
Logical. If TRUE (the default), the denominator of the estimator is
data-dependent (equal to the sum of the reciprocal intensities at
the points of type i), which reduces the sampling variability.
If FALSE, the denominator is the length of the network.
Smoothing bandwidth passed to density.lpp
for estimation of intensities when either lambdaI or
lambdaJ is NULL.
The arguments i and j are interpreted as
levels of the factor marks(X). Beware of the usual
trap with factors: numerical values are not
interpreted in the same way as character values.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au.
This is a counterpart of the function Kcross.inhom
for a point pattern on a linear network (object of class "lpp").
The arguments i and j will be interpreted as
levels of the factor marks(X).
If i and j are missing, they default to the first
and second level of the marks factor, respectively.
The argument r is the vector of values for the
distance \(r\) at which \(K_{ij}(r)\) should be evaluated.
The values of \(r\) must be increasing nonnegative numbers
and the maximum \(r\) value must not exceed the radius of the
largest disc contained in the window.
If lambdaI or lambdaJ is missing or NULL, it will
be estimated by kernel smoothing using density.lpp.
If lambdaI or lambdaJ is a fitted point process model,
the default behaviour is to update the model by re-fitting it to
the data, before computing the fitted intensity.
This can be disabled by setting update=FALSE.
Baddeley, A, Jammalamadaka, A. and Nair, G. (2014) Multitype point process analysis of spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical Society, Series C), 63, 673--694.
linearKdot,
linearK.
lam <- table(marks(chicago))/(summary(chicago)$totlength)
lamI <- function(x,y,const=lam[["assault"]]){ rep(const, length(x)) }
lamJ <- function(x,y,const=lam[["robbery"]]){ rep(const, length(x)) }
K <- linearKcross.inhom(chicago, "assault", "robbery", lamI, lamJ)
# using fitted models for the intensity
# fit <- lppm(chicago ~marks + x)
# K <- linearKcross.inhom(chicago, "assault", "robbery", fit, fit)
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