This session is concerned with Poisson point process models. The lecturer’s R script is available here (right click and save).
rpoispp(100) generates realisations of the Poisson process with intensity λ = 100 in the unit square.
Repeat the command
plot(rpoispp(100)) several times to build your intuition about the appearance of a completely random pattern of points.
Try the same thing with intensity λ = 1.5.
Returning to the Japanese Pines data,
Fit the uniform Poisson point process model to the Japanese Pines data
Read off the fitted intensity. Check that this is the correct value of the maximum likelihood estimate of the intensity.
japanesepines dataset is believed to exhibit spatial inhomogeneity.
Plot a kernel smoothed intensity estimate.
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.
extract the fitted coefficients for these models using
Plot the fitted model intensity (using
perform the Likelihood Ratio Test for the null hypothesis of a loglinear intensity against the alternative of a log-quadratic intensity, using
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.
update command can be used to re-fit a point process model using a different model formula.
Type the following commands and interpret the results:
fit0 <- ppm(japanesepines ~ 1) fit1 <- update(fit0, . ~ x) fit1 fit2 <- update(fit1, . ~ . + y) fit2
step(fit2) and interpret the results.
bei dataset gives the locations of trees in a survey area with additional covariate information in a list
Fit a Poisson point process model to the data which assumes that the intensity is a loglinear function of terrain slope and elevation (hint: use
data = bei.extra in
Read off the fitted coefficients and write down the fitted intensity function.
Plot the fitted intensity as a colour image.
extract the estimated variance-covariance matrix of the coefficient estimates, using
Compute and plot the standard error of the intensity estimate (see
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|
Make image plots of the fitted intensities for the inhomogeneous models above.