pcfinhom(X, lambda = NULL, ..., r = NULL, kernel = "epanechnikov", bw = NULL, stoyan = 0.15, correction = c("translate", "Ripley"), divisor = c("r", "d"), renormalise = TRUE, normpower=1, update = TRUE, leaveoneout = TRUE, reciplambda = NULL, sigma = NULL, varcov = NULL)"ppp").
X,
a pixel image (object of class "im") giving the
intensity values at all locations, a fitted point process model
(object of class "ppm") or a function(x,y) which
can be evaluated to give the intensity value at any location.
density.default.
density.default.
density.default.
pcf.ppp.
"r" (the default) or "d".
See pcf.ppp.
lambda is a fitted model
(class "ppm", "kppm" or "dppm")
and update=TRUE (the default),
the model will first be refitted to the data X
(using update.ppm or update.kppm)
before the fitted intensity is computed.
If update=FALSE, the fitted intensity of the
model will be computed without re-fitting it to X.
density.ppp or
fitted.ppm) specifying whether to use a
leave-one-out rule when calculating the intensity.
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 (object of class "im") giving the
reciprocal intensity values at all locations,
or a function(x,y) which can be evaluated to give the
reciprocal intensity value at any location.
density.ppp
to control the smoothing bandwidth, when lambda is
estimated by kernel smoothing.
"fv").
Essentially a data frame containing the variablesas required.
The best intuitive interpretation is the following: the probability $p(r)$ of finding two points at locations $x$ and $y$ separated by a distance $r$ is equal to $$ p(r) = \lambda(x) lambda(y) g(r) \,{\rm d}x \, {\rm d}y $$ where $lambda$ is the intensity function of the point process. For a Poisson point process with intensity function $lambda$, this probability is $p(r) = lambda(x) * lambda(y)$ so $ginhom(r) = 1$.
The inhomogeneous pair correlation function
is related to the inhomogeneous $K$ function through
$$
g_{\rm inhom}(r) = \frac{K'_{\rm inhom}(r)}{2\pi r}
$$
where $Kinhom'(r)$
is the derivative of $Kinhom(r)$, the
inhomogeneous $K$ function. See Kinhom for information
about $Kinhom(r)$.
The command pcfinhom estimates the inhomogeneous
pair correlation using a modified version of
the algorithm in pcf.ppp.
If renormalise=TRUE (the default), then the estimates
are multiplied by $c^normpower$ where
$
c = area(W)/sum[i] (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
but the most sensible value is 2, which would correspond to rescaling
the lambda values so that
$
sum[i] (1/lambda(x[i])) = area(W).
$
pcf,
pcf.ppp,
Kinhom
data(residualspaper)
X <- residualspaper$Fig4b
plot(pcfinhom(X, stoyan=0.2, sigma=0.1))
fit <- ppm(X, ~polynom(x,y,2))
plot(pcfinhom(X, lambda=fit, normpower=2))
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