Emark

0th

Percentile

Diagnostics for random marking

Estimate the summary functions $E(r)$ and $V(r)$ for a marked point pattern, proposed by Schlather et al (2004) as diagnostics for dependence between the points and the marks.

Keywords
spatial, nonparametric
Usage
Emark(X, r=NULL,
correction=c("isotropic", "Ripley", "translate"),
method="density", ..., normalise=FALSE)
Vmark(X, r=NULL,
correction=c("isotropic", "Ripley", "translate"),
method="density", ..., normalise=FALSE)
Arguments
X
The observed point pattern. An object of class "ppp" or something acceptable to as.ppp. The pattern should have numeric marks.
r
Optional. Numeric vector. The values of the argument $r$ at which the function $E(r)$ or $V(r)$ should be evaluated. There is a sensible default.
correction
A character vector containing any selection of the options "isotropic", "Ripley" or "translate". It specifies the edge correction(s) to be applied.
method
A character vector indicating the user's choice of density estimation technique to be used. Options are "density", "loess", "sm" and "smrep".
...
Arguments passed to the density estimation routine (density, loess or sm.density) selected by method.
normalise
IfTRUE, normalise the estimate of $E(r)$ or $V(r)$ so that it would have value equal to 1 if the marks are independent of the points.
Details

For a marked point process, Schlather et al (2004) defined the functions $E(r)$ and $V(r)$ to be the conditional mean and conditional variance of the mark attached to a typical random point, given that there exists another random point at a distance $r$ away from it.

More formally, $$E(r) = E_{0u}[M(0)]$$ and $$V(r) = E_{0u}[(M(0)-E(u))^2]$$ where $E_{0u}$ denotes the conditional expectation given that there are points of the process at the locations $0$ and $u$ separated by a distance $r$, and where $M(0)$ denotes the mark attached to the point $0$.

These functions may serve as diagnostics for dependence between the points and the marks. If the points and marks are independent, then $E(r)$ and $V(r)$ should be constant (not depending on $r$). See Schlather et al (2004).

The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to as.ppp. It must be a marked point pattern with numeric marks.

The argument r is the vector of values for the distance $r$ at which $k_f(r)$ is estimated.

References

Schlather, M. and Ribeiro, P. and Diggle, P. (2004) Detecting dependence between marks and locations of marked point processes. Journal of the Royal Statistical Society, series B 66 (2004) 79-83.

Mark correlation markcorr, mark variogram markvario for numeric marks. Mark connection function markconnect and multitype K-functions Kcross, Kdot for factor-valued marks.

• Emark
• Vmark
Examples
data(spruces)

plot(Emark(spruces))
E <- Emark(spruces, method="density", kernel="epanechnikov")
plot(Vmark(spruces))
Documentation reproduced from package spatstat, version 1.16-2, License: GPL (>= 2)

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