Kmm: Mark-weighted K-function

In ecespa: Functions for Spatial Point Pattern Analysis

Mark-weighted K-function

Description

This is a functional data summary for marked point patterns that measures the joint pattern of points and marks at different scales determined by r.

Usage

Kmm(mippp, r = 1:10, nsim=NULL)

## S3 method for ploting objects of class 'ecespa.kmm':
## S3 method for class 'ecespa.kmm'
plot(x, type="Kmm.n", q=0.025, 
            xlime=NULL, ylime=NULL,  maine=NULL, add=FALSE, kmean=TRUE,
            ylabe=NULL, xlabe=NULL, lty=c(1,2,3), col=c(1,2,3), lwd=c(1,1,1),
             ...)

Arguments

mippp	A marked point pattern. An object with the ppp format of spatstat.
r	Sequence of distances at which Kmm is estimated.
nsim	Number of simulated point patterns to be generated when computing the envelopes.
x	An object of class 'ecespa.kmm'. The result of applying Kmm to a marked point pattern.
type	Type of mark-weighted K-function to plot. One of "Kmm" ("plain" mark-weighted K-function) or "Kmm.n" (normalized mark-weighted K-function).
q	Quantile for selecting the simulation envelopes.
xlime	Max and min coordinates for the x-axis.
ylime	Max and min coordinates for the y-axis.
maine	Title to add to the plot.
add	Logical. Should the kmm.object be added to a previous plot?
kmean	Logical. Should the mean of the simulated Kmm envelopes be ploted?
ylabe	Text or expression to label the y-axis.
xlabe	Text or expression to label the x-axis.
lty	Vector with the line type for the estimated Kmm function, the simulated envelopes and the mean of the simulated envelopes.
col	Vector with the color for the estimated Kmm function, the simulated envelopes and the mean of the simulated envelopes.
lwd	Vector with the line width for the estimated Kmm function, the simulated envelopes and the mean of the simulated envelopes.
...	Additional graphical parameters passed to plot.

Details

Penttinnen (2006) defines Kmm(r), the mark-weighted K-function of a stationary marked point process X, so that

lambda*Kmm(r) = Eo[sum(mo*mn)]/mu^2

where lambda is the intensity of the process, i.e. the expected number of points of X per unit area, Eo[ ] denotes expectation (given that there is a point at the origin); m0 and mn are the marks attached to every two points of the process separated by a distance <= r and mu is the mean mark. It measures the joint pattern of marks and points at the scales determmined by r. If all the marks are set to 1, then lambda*Kmm(r) equals the expected number of additional random points within a distance r of a typical random point of X, i.e. Kmm becomes the conventional Ripley's K-function for unmarked point processes. As the K-function measures clustering or regularity among the points regardless of the marks, one can separate clustering of marks with the normalized weighted K-function

Kmm.normalized(r) = Kmm(r)/K(r)

If the process is independently marked, Kmm(r) equals K(r) so the normalized mark-weighted K-function will equal 1 for all distances r.

If nsim != NULL, Kmm computes 'simulation envelopes' from the simulated point patterns. These are simulated from nsim random permutations of the marks over the points coordinates. This is a kind of pointwise test of Kmm(r) == 1 or normalized Kmm(r) == 1 for a given r.

Value

Kmm returns an object of class 'ecespa.kmm', basically a list with the following items:

dataname	Name of the analyzed point pattern.
r	Sequence of distances at which Kmm is estimated.
nsim	Number of simulations for computing the envelopes, or NULL if none.
kmm	Mark-weighted K-function.
kmm.n	Normalized mark-weighted K-function.
kmmsim	Matrix of simulated mark-weighted K-functions, or or NULL if none.
kmmsim.n	Matrix of simulated normalized mark-weighted K-functions, or or NULL if none.

Note

This implementation estimates Kmm(r) without any correction of border effects, so it must be used with caution. However, as K(r) is also estimed without correction it migth compensate the border effects on the normalized Kmm-function.

Author(s)

Marcelino de la Cruz Rot

References

De la Cruz, M. 2008. Métodos para analizar datos puntuales. En: Introducción al Análisis Espacial de Datos en Ecología y Ciencias Ambientales: Métodos y Aplicaciones (eds. Maestre, F. T., Escudero, A. y Bonet, A.), pp 76-127. Asociación Española de Ecología Terrestre, Universidad Rey Juan Carlos y Caja de Ahorros del Mediterráneo, Madrid.

Penttinen, A. 2006. Statistics for Marked Point Patterns. In The Yearbook of the Finnish Statistical Society, pp. 70-91.

See Also

markcorr

Examples

  ## Figure 3.10 of De la Cruz (2008):
  # change r to r=1:100
  
   r = seq(1,100, by=5)
  
  data(seedlings1)
  
  data(seedlings2)
  
  s1km <- Kmm(seedlings1, r=r)
  
  s2km <- Kmm(seedlings2, r=r)
  
  plot(s1km, ylime=c(0.6,1.2), lwd=2, maine="", xlabe="r(cm)")

  plot(s2km,  lwd=2, lty=2, add=TRUE )

  abline(h=1, lwd=2, lty=3)
  
  legend(x=60, y=1.2, legend=c("Hs_C1", "Hs_C2", "H0"),
	 lty=c(1, 2, 3), lwd=c(3, 2, 2), bty="n")
## Not run: 
## A pointwise test of normalized Kmm == 1 for seedlings1:

   s1km.test <- Kmm(seedlings1, r=1:100, nsim=99)

   plot(s1km.test,  xlabe="r(cm)")

  
## End(Not run)

ecespa documentation built on Jan. 6, 2023, 1:21 a.m.