pcfinhom: Inhomogeneous Pair Correlation Function

Description

Estimates the inhomogeneous pair correlation function of a point pattern using kernel methods.

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

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

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.

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.

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.

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 $\sum (1/\lambda(x_i)) = \mbox{area}(W).$

Examples

Run this code

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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