This session covers tools for investigating intensity depending on a covariate. The lecturer’s R script is available here (right click and save).
library(spatstat)
The bei
dataset gives the locations of trees in a survey area with additional covariate information in a list bei.extra
.
Assign the elevation covariate to a variable elev
by typing
elev <- bei.extra$elev
OK, lets do that:
elev <- bei.extra$elev
Plot the trees on top of an image of the elevation covariate.
plot(elev, main = "")
plot(bei, add = TRUE, cex = 0.3, pch = 16, cols = "white")
Cut the study region into 4 areas according to the value of the terrain elevation, and make a texture plot of the result.
Z <- cut(elev, 4, labels=c("Low", "Med-Low", "Med-High", "High"))
textureplot(Z, main = "")
Convert the image from above to a tesselation, count the number of points in each region using quadratcount
, and plot the quadrat counts.
Y <- tess(image = Z)
qc <- quadratcount(bei, tess = Y)
Estimate the intensity in each of the four regions.
intensity(qc)
## tile
## Low Med-Low Med-High High
## 0.002259007 0.006372523 0.008562862 0.005843516
Assume that the intensity of trees is a function λ(u)=ρ(e(u)) where e(u) is the terrain elevation at location u.
Compute a nonparametric estimate of the function ρ and plot it by
rh <- rhohat(bei, elev)
plot(rh)
Repeating the R code:
rh <- rhohat(bei, elev)
plot(rh)
Compute the predicted intensity based on this estimate of ρ.
prh <- predict(rh)
plot(prh, main = "")
plot(bei, add = TRUE, cols = "white", cex = .2, pch = 16)
Compute a non-parametric estimate by kernel smoothing and compare with the predicted intensity above.
The kernel density estimate of the points is computed and plotted with the following code:
dbei <- density(bei, sigma = bw.scott)
plot(dbei, main = "")
plot(bei, add = TRUE, cols = "white", cex = .2, pch = 16)
Which seems to be quite different form the predicted intentisty.
Bonus info: To plot the two intensity estimates next to each other you collect the estimates as a spatial object list (solist
) and plot the result (the estimates are called pred
and ker
below):
l <- solist(pred, ker)
plot(l, equal.ribbon = TRUE, main = "",
main.panel = c("rhohat prediction", "kernel smoothing"))
l <- solist(prh, dbei)
plot(l, equal.ribbon = TRUE, main = "",
main.panel = c("rhohat prediction", "kernel smoothing"))
Continuing with the dataset bei
conduct both Berman’s Z1 and Z2 tests for dependence on elev
, and plot the results.
The tests are done straightforwardly with berman.test
:
Z1 <- berman.test(bei, elev)
print(Z1)
##
## Berman Z1 test of CSR in two dimensions
##
## data: covariate 'elev' evaluated at points of 'bei'
## Z1 = -0.72924, p-value = 0.4659
## alternative hypothesis: two-sided
plot(Z1)
Z2 <- berman.test(bei, elev, which = "Z2")
print(Z2)
##
## Berman Z2 test of CSR in two dimensions
##
## data: covariate 'elev' evaluated at points of 'bei'
## and transformed to uniform distribution under CSR
## Z2 = 2.4515, p-value = 0.01423
## alternative hypothesis: two-sided
plot(Z2)