This session covers tools for investigating intensity depending on a covariate.
The lecturer’s R script is available here (right click and save).

### Exercise 1

The bei dataset gives the locations of trees in a survey area with additional covariate information in a list bei.extra.

1. Assign the elevation covariate to a variable elev by typing

elev <- bei.extra\$elev
2. Plot the trees on top of an image of the elevation covariate.

3. Cut the study region into 4 areas according to the value of the terrain elevation, and make a texture plot of the result.

4. Convert the image from above to a tesselation, count the number of points in each region using quadratcount, and plot the quadrat counts.

5. Estimate the intensity in each of the four regions.

### Exercise 2

Assume that the intensity of trees is a function $$\lambda(u) = \rho(e(u))$$ where $$e(u)$$ is the terrain elevation at location u.

1. Compute a nonparametric estimate of the function ρ and plot it by

rh <- rhohat(bei, elev)
plot(rh)
2. Compute the predicted intensity based on this estimate of ρ.

3. Compute a non-parametric estimate by kernel smoothing and compare with the predicted intensity above.

4. 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"))

### Exercise 3

Continuing with the dataset bei conduct both Berman’s Z1 and Z2 tests for dependence on elev, and plot the results.