library(spatstat)

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.

    Let’s plot it three times:

    replicate(3, plot(rpoispp(lambda = 100), main = ""))

    As can be seen, the points (unsurprisingly) are much more random that want one might think. “Randomly” drawing points on a piece of paper one would usually draw a point pattern that is more regular (i.e. the points are repulsive).

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

    For brevity we only do it once here:

    plot(rpoispp(lambda = 1.5), main = "")

    Here we expect 1.5 points in the plot each time.

Exercise 2

Returning to the Japanese Pines data,

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

    ppm(japanesepines~1)

    We fit the Poisson process model with the given command and print the output:

    m.jp <- ppm(japanesepines ~ 1)
    print(m.jp)
    ## Stationary Poisson process
    ## Intensity: 65
    ##             Estimate      S.E.  CI95.lo  CI95.hi Ztest     Zval
    ## log(lambda) 4.174387 0.1240347 3.931284 4.417491   *** 33.65499
  2. Read off the fitted intensity. Check that this is the correct value of the maximum likelihood estimate of the intensity.

    We extract the coeficient with the coef function, and compare to the straightforward estimate obtained by `intensity``:

    unname(exp(coef(m.jp)))
    ## [1] 65
    intensity(japanesepines)
    ## [1] 65

    As seen, they agree exactly.

Exercise 3

The japanesepines dataset is believed to exhibit spatial inhomogeneity.

  1. Plot a kernel smoothed intensity estimate.

    Plot the kernel smoothed intensity estimate selecting the bandwidth with bw.scott:

    jp.dens <- density(japanesepines, sigma = bw.scott)
    plot(jp.dens)
    plot(japanesepines, col = "white", cex = .4, pch = 16, add = TRUE)

  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.

    We fit the two models with ppm:

    jp.m <- ppm(japanesepines ~ x + y)
    jp.m2 <- ppm(japanesepines ~ polynom(x, y, 2) )
  3. extract the fitted coefficients for these models using coef.

    coef(jp.m)
    ## (Intercept)           x           y 
    ##   4.0670790  -0.2349641   0.4296171
    coef(jp.m2)
    ## (Intercept)           x           y      I(x^2)    I(x * y)      I(y^2) 
    ##   4.0645501   1.1436854  -1.5613621  -0.7490094  -1.2009245   2.5061569
  4. Plot the fitted model intensity (using plot(predict(fit)))

    par(mar=rep(0,4))
    plot(predict(jp.m), main = "")

    plot(predict(jp.m, se=TRUE)$se, main = "")

    plot(predict(jp.m2), main = "")

    plot(predict(jp.m2, se=TRUE)$se, main = "")

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

    anova(jp.m, jp.m2)
    ## Analysis of Deviance Table
    ## 
    ## Model 1: ~x + y   Poisson
    ## Model 2: ~x + y + I(x^2) + I(x * y) + I(y^2)      Poisson
    ##   Npar Df Deviance
    ## 1    3            
    ## 2    6  3   3.3851
  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.

    par(mar=rep(0.5,4))
    plot(simulate(jp.m2, nsim=10), main = "")
    ## Generating 10 simulated patterns ...1, 2, 3, 4, 5, 6, 7, 8, 9,  10.