Determining high-risk zones by using spatial point process methodology

Methods for constructing high-risk zones, which can be used in
situations where a spatial point pattern has been observed
incompletely, are introduced and evaluated with regard to
unexploded bombs in federal properties in Germany. Unexploded bombs
from the Second World War represent a serious problem in Germany.
It is desirable to search high-risk zones for unexploded bombs, but
this causes high costs, so the search is usually restricted to
carefully selected areas. If suitable aerial pictures of the area
in question exist, statistical methods can be used to determine
such zones by considering patterns of exploded bombs as
realisations of spatial point processes. The patterns analysed in
this thesis were provided by Oberfinanzdirektion Niedersachsen,
which supports the removal of unexploded ordnance in federal
properties in Germany. They were derived from aerial pictures taken
by the Allies during and after World War II. The main task consists
of finding as small regions as possible containing as many
unexploded bombs as possible. In this thesis, an approach based on
the intensity function of the process is introduced: The high-risk
zones consist of those parts of the observation window where the
estimated intensity is largest, i.e. the estimated intensity
function exceeds a cut-off value c. The cut-off value can be
derived from the risk associated with the high-risk zone. This risk
is defined as the probability that there are unexploded bombs
outside the zone. A competing approach for determining high-risk
zones consists in using the union of discs around all exploded
bombs as high-risk zone. The radius is chosen as a high quantile of
the nearest-neighbour distance of the point pattern. In an
evaluation procedure, both methods yield comparably good results,
but the theoretical properties of the intensity-based high-risk
zones are considerably better. A further goal is to perform a risk
assessment of the investigated area by estimating the probability
that there are unexploded bombs outside the high-risk zone. This is
especially important as the estimation of the intensity function is
a crucial issue for the intensity-based method, so the risk cannot
be determined exactly in advance. A procedure to calculate the risk
is introduced. By using a bootstrap correction, it is possible to
decide on acceptable risks and find the optimal, i.e. smallest,
high-risk zone for a fixed probability that not all unexploded
bombs are located inside the high-risk zone. The consequences of
clustering are investigated in a sensitivity analysis by exploiting
the procedure for calculating the risk. Furthermore, different
types of models which account for clustering are fitted to the
data, classical cluster models as well as a mixture of bivariate
normal distributions.