pcfinhom

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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 pattern X, a pixel image (o
sigma,varcov
Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is 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
  • rthe vector of values of the argument $r$ at which the inhomogeneous pair correlation function $g_{\rm inhom}(r)$ has been estimated
  • theovector of values equal to 1, the theoretical value of $g_{\rm inhom}(r)$ for the Poisson process
  • transvector of values of $g_{\rm inhom}(r)$ estimated by translation correction
  • isovector of values of $g_{\rm inhom}(r)$ estimated by Ripley isotropic correction
  • as required.

See Also

pcf, pcf.ppp, Kinhom

Aliases
  • pcfinhom
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))
Documentation reproduced from package spatstat, version 1.29-0, License: GPL (>= 2)

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