Lab 3: Intensity dependent on covariate

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

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
```
Plot the trees on top of an image of the elevation covariate.
Cut the study region into 4 areas according to the value of the terrain elevation, and make a texture plot of the result.
Convert the image from above to a tesselation, count the number of points in each region using quadratcount, and plot the quadrat counts.
Estimate the intensity in each of the four regions.

Assume that the intensity of trees is a function $\lambda(u) = \rho(e(u))$ 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)
```
Compute the predicted intensity based on this estimate of ρ.
Compute a non-parametric estimate by kernel smoothing and compare with the predicted intensity above.
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"))
```

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