Kinhom(X, lambda=NULL, ..., r = NULL, breaks = NULL,
correction=c("border", "bord.modif", "isotropic", "translate"),
renormalise=TRUE,
normpower=1,
update=TRUE,
leaveoneout=TRUE,
nlarge = 1000,
lambda2=NULL, reciplambda=NULL, reciplambda2=NULL,
sigma=NULL, varcov=NULL)"ppp"
or in a format recognised by as.ppp()X,
a pixel image (object of class "im") giving the
intensity values at all locatiolambda is present.
Passed to density.ppp if lambda is omitted."border", "bord.modif",
"isotropic", "Ripley", "translate",
"translation",
"none" or lambda is a fitted model
(class "ppm" or "kppm")
and update=TRUE (the default),
the model will first be refitted to the data X
(using density.ppp or
fitted.ppm) specifying whether to use a
leave-one-out rule when calculating the intensity.nlarge, then only the
border correction will be computed, using a fast algorithm.lambda.
Values of the estimated reciprocal $1/\lambda$
of the intensity function.
Either a vector giving the reciprocal intensity values
at the points of the pattern X,
a pixel image (olambda2.
A matrix giving values of the estimated reciprocal products
$1/\lambda(x_i)\lambda(x_j)$
of the intensities at each pair of data points
$x_i$ and $x_j$.density.ppp
to control the smoothing bandwidth, when lambda is
estimated by kernel smoothing."fv" (see fv.object).
Essentially a data frame containing at least the following columns,correction
argument. The additional columns are named
border, trans and iso
and give the estimated values of
$K_{\rm inhom}(r)$
using the border correction, translation correction,
and Ripley isotropic correction, respectively.Kest. It is defined only
for stationary point processes.
The inhomogeneous $K$ function
$K_{\rm inhom}(r)$
is a direct generalisation to nonstationary point processes.
Suppose $x$ is a point process with non-constant intensity
$\lambda(u)$ at each location $u$.
Define $K_{\rm inhom}(r)$ to be the expected
value, given that $u$ is a point of $x$,
of the sum of all terms
$1/\lambda(x_j)$
over all points $x_j$
in the process separated from $u$ by a distance less than $r$.
This reduces to the ordinary $K$ function if
$\lambda()$ is constant.
If $x$ is an inhomogeneous Poisson process with intensity
function $\lambda(u)$, then
$K_{\rm inhom}(r) = \pi r^2$.
Given a point pattern dataset, the
inhomogeneous $K$ function can be estimated
essentially by summing the values
$1/(\lambda(x_i)\lambda(x_j))$
for all pairs of points $x_i, x_j$
separated by a distance less than $r$.This allows us to inspect a point pattern for evidence of interpoint interactions after allowing for spatial inhomogeneity of the pattern. Values $K_{\rm inhom}(r) > \pi r^2$ are suggestive of clustering.
The argument lambda should supply the
(estimated) values of the intensity function $\lambda$.
It may be either
[object Object],[object Object],[object Object],[object Object],[object Object]
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, 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.
Edge corrections are used to correct bias in the estimation
of $K_{\rm inhom}$.
Each edge-corrected estimate of $K_{\rm inhom}(r)$ is
of the form
$$\widehat K_{\rm inhom}(r) = \sum_i \sum_j \frac{1{d_{ij} \le
r} e(x_i,x_j,r)}{\lambda(x_i)\lambda(x_j)}$$
where $d_{ij}$ is the distance between points
$x_i$ and $x_j$, and
$e(x_i,x_j,r)$ is
an edge correction factor. For the `border' correction,
$$e(x_i,x_j,r) =
\frac{1(b_i > r)}{\sum_j 1(b_j > r)/\lambda(x_j)}$$
where $b_i$ is the distance from $x_i$
to the boundary of the window. For the `modified border'
correction,
$$e(x_i,x_j,r) =
\frac{1(b_i > r)}{\mbox{area}(W \ominus r)}$$
where $W \ominus r$ is the eroded window obtained
by trimming a margin of width $r$ from the border of the original
window.
For the `translation' correction,
$$e(x_i,x_j,r) =
\frac 1 {\mbox{area}(W \cap (W + (x_j - x_i)))}$$
and for the `isotropic' correction,
$$e(x_i,x_j,r) =
\frac 1 {\mbox{area}(W) g(x_i,x_j)}$$
where $g(x_i,x_j)$ is the fraction of the
circumference of the circle with centre $x_i$ and radius
$||x_i - x_j||$ which lies inside the window.
If renormalise=TRUE (the default), then the estimates
are multiplied by $c^{\mbox{normpower}}$ where
$c = \mbox{area}(W)/\sum (1/\lambda(x_i)).$
This rescaling reduces the variability and bias of the estimate
in small samples and in cases of very strong inhomogeneity.
The default value of normpower is 1 (for consistency with
previous versions of lambda values so that
$\sum (1/\lambda(x_i)) = \mbox{area}(W).$
If the point pattern X contains more than about 1000 points,
the isotropic and translation edge corrections can be computationally
prohibitive. The computations for the border method are much faster,
and are statistically efficient when there are large numbers of
points. Accordingly, if the number of points in X exceeds
the threshold nlarge, then only the border correction will be
computed. Setting nlarge=Inf or correction="best"
will prevent this from happening.
Setting nlarge=0 is equivalent to selecting only the border
correction with correction="border".
The pair correlation function can also be applied to the
result of Kinhom; see pcf.
}
Kest,
pcf
# (1) intensity function estimated by model-fitting # Fit spatial trend: polynomial in x and y coordinates fit <- ppm(X, ~ polynom(x,y,2), Poisson()) # (a) predict intensity values at points themselves, # obtaining a vector of lambda values lambda <- predict(fit, locations=X, type="trend") # inhomogeneous K function Ki <- Kinhom(X, lambda) plot(Ki) # (b) predict intensity at all locations, # obtaining a pixel image lambda <- predict(fit, type="trend") Ki <- Kinhom(X, lambda) plot(Ki)
# (2) intensity function estimated by heavy smoothing Ki <- Kinhom(X, sigma=0.1) plot(Ki)
# (3) simulated data: known intensity function lamfun <- function(x,y) { 50 + 100 * x } # inhomogeneous Poisson process Y <- rpoispp(lamfun, 150, owin()) # inhomogeneous K function Ki <- Kinhom(Y, lamfun) plot(Ki)
# How to make simulation envelopes:
# Example shows method (2)
smo <- density.ppp(X, sigma=0.1)
Ken <- envelope(X, Kinhom, nsim=99,
simulate=expression(rpoispp(smo)),
sigma=0.1, correction="trans")
plot(Ken)
[object Object],[object Object],[object Object]