If you have not already done so, you’ll need to start R and load the spatstat package by typing

library(spatstat)

Exercise 1

We will study a dataset that records the locations of Ponderosa Pine trees (Pinus ponderosa) in a study region in the Klamath National Forest in northern California. The data are included with spatstat as the dataset ponderosa.

  1. assign the data to a shorter name, like X or P;

  2. plot the data;

  3. find out how many trees are recorded;

  4. find the dimensions of the study region;

  5. obtain an estimate of the average intensity of trees (number of trees per unit area).

Exercise 2

The Ponderosa data, continued:

  1. When you type plot(ponderosa), the command that is actually executed is plot.ppp, the plot method for point patterns. Read the help file for the function plot.ppp, and find out which argument to the function can be used to control the main title for the plot;

  2. plot the Ponderosa data with the title Ponderosa Pine Trees above it;

  3. from your reading of the help file, predict what will happen if we type

    plot(ponderosa, chars="X", cols="green")

    then check that your guess was correct;

  4. try different values of the argument chars, for example, one of the integers 0 to 25, or a letter of the alphabet. (Note the difference between chars=3 and chars="+", and the difference between chars=4 and chars="X").

Exercise 3

The dataset japanesepines contains the locations of Japanese Black Pine trees in a study region.

  1. Plot the japanesepines data.

  2. What is the average intensity (the average number of points per unit area?

  3. Using density.ppp, compute a kernel estimate of the spatially-varying intensity function for the Japanese pines data, using a Gaussian kernel with standard deviation \(\sigma=0.1\) units, and store the estimated intensity in an object D say.

  4. Plot a colour image of the kernel estimate D.

  5. Most plotting commands will accept the argument add=TRUE and interpret it to mean that the plot should be drawn over the existing display, without clearing the screen beforehand. Use this to plot a colour image of the kernel estimate D with the original Japanese Pines data superimposed.

  6. Plot the kernel estimate without the ‘colour ribbon’.

  7. Try the following command

    persp(D, theta=70, phi=25, shade=0.4)

    and find the documentation for the arguments theta, phi and shade.

  8. Find the maximum and minimum values of the intensity estimate over the study region. (Hint: Use summary or range)

  9. The kernel estimate of intensity is defined so that its integral over the entire study region is equal to the number of points in the data pattern, ignoring edge effects. Check whether this is approximately true in this example. (Hint: use integral)

Exercise 4

The dataset hamster is a multitype pattern representing the locations of cells of two types, dividing and pyknotic.

  1. plot the pattern;

  2. plot the pattern again, changing the colours and symbols used to represent the two types of cells;

  3. plot the patterns of pyknotic and dividing cells separately using plot(split(hamster)).

  4. use relrisk to perform cross-validated bandwidth selection and computation of the relative intensity of pyknotic cells.

Exercise 5

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. 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 \(\rho\) and plot it by

    rh <- rhohat(bei, elev)
plot(rh)

  4. Compute the predicted intensity based on this estimate of \(\rho\).

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

  6. 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"))