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)
  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.