Kmm {ecespa} | R Documentation |
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.
Kmm(mippp, r)
mippp |
A marked point pattern. An object with the ppp format of spatstat. |
r |
Sequence of distances at which Kmm is estimated. |
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.
r |
Sequence of distances at which Kmm is estimated. |
Kmm |
Mark-weighted K-function. |
Kmm.n |
Normalized mark-weighted K-function. |
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.
Marcelino de la Cruz Rot marcelino.delacruz@upm.es
Penttinen, A. 2006. Statistics for Marked Point Patterns. In The Yearbook of the Finnish Statistical Society, pp. 70-91.
## Not run: data(seedlings1) data(seedlings2) s1km <- Kmm(seedlings1, r=1:100) s2km <- Kmm(seedlings2, r=1:100) plot(s1km$r, s1km$Kmm.n, type="l", lty=1, lwd=3, ylim=c(0.6, 1.2), xlab="r (cm)", ylab= expression (K[mm](r)), main="Mark-weighted K-function of Hs seedling cohorts") lines(s2km$r, s2km$Kmm.n, lty=2,lwd=3) abline(h=1, lwd=2, lty=3) legend(x=60, y=1.2, legend=c("HsC1", "HsC2", "Ho:"), lty=c(1, 2, 3), lwd=c(3, 2, 2), bty="n") ## End(Not run)