pcfinhom
Inhomogeneous Pair Correlation Function
Estimates the inhomogeneous pair correlation function of a point pattern using kernel methods.
- Keywords
- spatial, nonparametric
Usage
pcfinhom(X, lambda = NULL, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("translate", "Ripley"),
renormalise = TRUE, normpower=1,
reciplambda = NULL,
sigma = NULL, varcov = NULL)Arguments
- X
- A point pattern (object of class
"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 locatio - r
- Vector of values for the argument $r$ at which $g(r)$ should be evaluated. There is a sensible default.
- kernel
- Choice of smoothing kernel, passed to
density.default. - bw
- Bandwidth for smoothing kernel, passed to
density.default. - ...
- Other arguments passed to the kernel density estimation
function
density.default. - stoyan
- Bandwidth coefficient; see Details.
- correction
- Choice of edge correction.
- renormalise
- Logical. Whether to renormalise the estimate. See Details.
- normpower
- Integer (usually either 1 or 2). Normalisation power. See Details.
- reciplambda
- Alternative to
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 patternX, a pixel image (o - sigma,varcov
- Optional arguments passed to
density.pppto control the smoothing bandwidth, whenlambdais estimated by kernel smoothing.
Details
The inhomogeneous pair correlation function $g_{\rm inhom}(r)$ is a summary of the dependence between points in a spatial point process that does not have a uniform density of points.
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 $g_{\rm inhom}(r) = 1$.
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 $K'_{\rm inhom}(r)$
is the derivative of $K_{\rm inhom}(r)$, the
inhomogeneous $K$ function. See Kinhom for information
about $K_{\rm inhom}(r)$.
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 $c^{\mbox{normpower}}$ 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
but the most sensible value is 2, which would correspond to rescaling
the lambda values so that
$\sum (1/\lambda(x_i)) = \mbox{area}(W).$
Value
- A function value table (object of class
"fv"). Essentially a data frame containing the variables r the vector of values of the argument $r$ at which the inhomogeneous pair correlation function $g_{\rm inhom}(r)$ has been estimated theo vector of values equal to 1, the theoretical value of $g_{\rm inhom}(r)$ for the Poisson process trans vector of values of $g_{\rm inhom}(r)$ estimated by translation correction iso vector of values of $g_{\rm inhom}(r)$ estimated by Ripley isotropic correction - as required.
See Also
Examples
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))