Description
1. Alt and Vach (1991) describe an archaeological investigation of an early medieval burial
ground in Germany. One question of interest was whether grave sites tended to be
placed according to family units. The archaeologists considered an inherited feature in
the teeth of the excavated skeletons; each grave has a location and an indicator variable
for whether the individual had this feature or not. The data for the point process is
in the file dental.reduced.dat. The second column is the indicator variable and the
last two columns are x and y location. The first column is an index you may ignore.
(a) Load the data and create two ppp objects from it, one for affected and one for
unaffected individuals. A key question is: what is the window? It is not available
for this data. For now, take the window to be the same for each ppp object: use a
rectangular region based on the range of x and y for both datasets. (Have a look
at the help file for owin.)
(b) For each dataset separately, create Monte Carlo simulation envelopes for the F and
G functions and plot them. Clearly label your plots and turn in a few sentences
describing any choices you made in creating the envelopes. Is their evidence
against CSR in this dataset? If so, what type of violation is suggested?
(c) Now let’s consider a more likely window: that these graves represent a complete
excavation of the area in which they appear, and that area is irregularly shaped.
Since we don’t know what it is, we can choose a rough polygon outline to surround
the points. You can create such an outline by plotting the locations and using
the locator function; again, see help(owin) for the details of how to specify a
polygon boundary. Create two new ppp objects with this new window.
(d) Repeat step (b) for the new datasets. What changes? Can you explain the reason
for this, based on the form of the test statistics?
2. Simulate four datasets on the unit square, from a homogeneous Poisson process with
a rate λ of your choosing. For each one, fit a kernel estimate of the intensity function
and plot it with the points overlaid.
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3. Read through sections 15.2 and 15.3 of the notes by Adrian Baddeley about fitting
point process models in R with the spatstat package (you can find it on YSCEC). In
15.3, follow along with the R code example for the bei dataset. In particular, find the
MLEs for inhomogeneous models with intensity functions:
Model 1 : λ(x) = exp{β0 + β1Z(x)}
Model 2 : λ(x) = βZ(x)
Plot a kernel density estimate of λ(x), ignoring covariates, as well as the fitted intensities under the two models. Put them all on the same color scale, and include a
sentence or two comparing them. Note: the data for this problem are available from
the spatstat package.
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