Kmm function

Mark-weighted K-function

This is a functional data summary for marked point patterns that measures the joint pattern of points and marks at different scales determined by $r$. latin1

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

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

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 $r$.

Returns

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.

References

De la Cruz, M. 2008.

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

Author(s)

Marcelino de la Cruz Rot

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.

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)