Kdot.inhom {spatstat} | R Documentation |
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)
X |
The observed point pattern, from which an estimate of the inhomogeneous cross type K function Ki.(r) will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details. |
i |
Number or character string identifying the type (mark value)
of the points in |
lambdaI |
Optional.
Values of the estimated intensity of the sub-process of
points of type |
lambdadot |
Optional.
Values of the estimated intensity of the entire point process,
Either a pixel image (object of class |
... |
Ignored. |
r |
Optional. Numeric vector giving the values of the argument r at which the cross K function Kij(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. |
breaks |
Optional. An alternative to the argument |
correction |
A character vector containing any selection of the
options |
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
|
lambdaIdot |
Optional. A matrix containing estimates of the
product of the intensities |
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(z) at each point z of the process. Then the expected total mass per unit area is 1. The inhomogeneous “dot-type” K function K[i.]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.]inhom(r) would equal pi * r^2. Deviations between the empirical Ki. curve and the theoretical curve pi * r^2 suggest dependence between the points of types i and j for j != 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 3rd smallest level,
not the number 3).
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.
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.
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 Ki.(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
.
An object of class "fv"
(see fv.object
).
Essentially a data frame containing numeric columns
r |
the values of the argument r at which the function Ki.(r) has been estimated |
theo |
the theoretical value of Ki.(r) for a marked Poisson process, namely pi * r^2 |
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 Ki.(r)
obtained by the edge corrections named.
The argument i
is interpreted as a
level of the factor X$marks
. Beware of the usual
trap with factors: numerical values are not
interpreted in the same way as character values.
Adrian Baddeley Adrian.Baddeley@csiro.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner r.turner@auckland.ac.nz
Moller, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes Chapman and Hall/CRC Boca Raton, 2003.
Kdot
,
Kinhom
,
Kcross.inhom
,
pcf
# Lansing Woods data data(lansing) lansing <- lansing[seq(1,lansing$n, by=10)] ma <- split(lansing)$maple lg <- unmark(lansing) # 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(lansing, "maple", lambdaI=lambdaM, lambdadot=lambdadot) # Equivalent K <- Kdot.inhom(lansing, "maple", sigma=0.15) # 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)