This session is concerned with Poisson point process models.
The lecturer’s R script is available here (right click and save).

Exercise 1

The command rpoispp(100) generates realisations of the Poisson process with intensity \(\lambda = 100\) in the unit square.

  1. Repeat the command plot(rpoispp(100)) several times to build your intuition about the appearance of a completely random pattern of points.

  2. Try the same thing with intensity \(\lambda = 1.5\).

Exercise 2

Returning to the Japanese Pines data,

  1. Fit the uniform Poisson point process model to the Japanese Pines data

  2. Read off the fitted intensity. Check that this is the correct value of the maximum likelihood estimate of the intensity.

Exercise 3

The japanesepines dataset is believed to exhibit spatial inhomogeneity.

  1. Plot a kernel smoothed intensity estimate.

  2. Fit the Poisson point process models with loglinear intensity (trend formula ~x+y) and log-quadratic intensity (trend formula ~polynom(x,y,2)) to the Japanese Pines data.

  3. extract the fitted coefficients for these models using coef.

  4. Plot the fitted model intensity (using plot(predict(fit)))

  5. perform the Likelihood Ratio Test for the null hypothesis of a loglinear intensity against the alternative of a log-quadratic intensity, using anova.

  6. Generate 10 simulated realisations of the fitted log-quadratic model, and plot them, using plot(simulate(fit, nsim=10)) where fit is the fitted model.

Exercise 4

The update command can be used to re-fit a point process model using a different model formula.

  1. Type the following commands and interpret the results:

    fit0 <- ppm(japanesepines ~ 1)
    fit1 <- update(fit0, . ~ x)
    fit2 <- update(fit1, . ~ . + y)
  2. Now type step(fit2) and interpret the results.

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 both terrain elevation and slope (gradient) sensible names

    elev <- bei.extra$elev
    grad <- bei.extra$grad
  2. Fit a Poisson point process model to the data which assumes that the intensity is a loglinear function of terrain slope and elevation

  3. Read off the fitted coefficients and write down the fitted intensity function.

  4. Plot the fitted intensity as a colour image.

  5. extract the estimated variance-covariance matrix of the coefficient estimates, using vcov.

  6. Compute and plot the standard error of the intensity estimate (see help(predict.ppm)).

Exercise 6

Fit Poisson point process models to the Japanese Pines data, with the following trend formulas. Read off an expression for the fitted intensity function in each case.

Trend formula Fitted intensity function
~factor(x < 0.4)

Exercise 7

Make image plots of the fitted intensities for the inhomogeneous models above.