Kmm: 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 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.

Sequence of distances at which Kmm is estimated.

nsim

Number of simulated point patterns to be generated when computing the envelopes.

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).

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.

Value

Kmm returns an object of class 'ecespa.kmm', basically a list with the following items:
datanameName of the analyzed point pattern.
rSequence of distances at which Kmm is estimated.
nsimNumber of simulations for computing the envelopes, or NULL if none.
kmmMark-weighted $K$-function.
kmm.nNormalized mark-weighted $K$-function.
kmmsimMatrix of simulated mark-weighted $K$-functions, or or NULL if none.
kmmsim.nMatrix of simulated normalized mark-weighted $K$-functions, or or NULL if none.

encoding

latin1

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="" joint="" pattern="" of="" marks="" points="" at="" scales="" determmined="" by="" $r$.="" if="" all="" are="" set="" to="" 1,="" then="" $lambda*kmm(r)$="" equals="" expected="" number="" additional="" random="" within="" a="" distance="" $r$="" typical="" point="" $x$,="" i.e.="" $kmm$="" becomes="" conventional="" ripley's="" $k$-function="" for="" unmarked="" processes.="" as="" clustering="" or="" regularity="" among="" regardless="" marks,="" one="" can="" separate="" with="" 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 $r$.

References

De la Cruz, M. 2008. M�todos para analizar datos puntuales. In Introducci�n al An�lisis Espacial de Datos en Ecolog�a y Ciencias Ambientales: M�todos y Aplicaciones (eds. Maestre, F.T., Escudero, A. and Bonet, A.). Asociaci�n Espa�ola de Ecolog�a Terrestre, Universidad Rey Juan Carlos y Caja de Ahorros del Mediterr�neo. Madrid.{De la Cruz, M. 2008. Metodos para analizar datos puntuales. In Introduccion al Analisis Espacial de Datos en Ecologia y Ciencias Ambientales: Metodos y Aplicaciones (eds. Maestre, F.T., Escudero, A. and Bonet, A.). Asociacion Espa�ola de Ecologia Terrestre, Universidad Rey Juan Carlos y Caja de Ahorros del Mediterraneo. Madrid.} Penttinen, A. 2006. Statistics for Marked Point Patterns. In The Yearbook of the Finnish Statistical Society, pp. 70-91.

Examples

Run this code

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

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

  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")
 
## A pointwise test of normalized Kmm == 1 for seedlings1:

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

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

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