Ecole Nationale Supérieure Agronomique de Montpellier ... - CIAM

2.4 grid approximation
In practice, one very often does not observe the area process in W ,
but observe its realization on a regular grid, the realization at the
center of each cell being 1 or 0 whether the cell intersects the boolean
process or not. Let U = (us)s∈S be the intersection of the grid with
W , vs = {x ∈ W ; || x − us ||= mins
′(|| x − us ′ ||)}
This is for example the case when mapping bush coverage by using
aerial photography, as presented in the following. The observed
process is then X = (xs)s∈S, where xs = 1I {vs∩B=∅}
This can be also the case when looking at a plant disease transmitted
by insects in orchards. Suppose that the trees are old enough
so that their canopies form a continuous surface. Orchard trees being
of the same age, species and variety, the canopies of all trees are more
or less similar, and equal to v0, the area of the cell centered at 0, as
soon as they do not overlap. If insects arrive and move independently
in the canopy, let us denote Y = (Yi) the process of first insect arrival,
Zi = ∪j∈Ji uij the set of the successive relative positions of insect i
from Yi. The sets Yi + Zi are independent from each other. If insects
arrive in independent identically distributed small groups, let Yi be
the center of a group, Zi the relative positions of insects of group i
around Yi. Then, one gets also that the sets Yi + Zi are independent
from each other. Suppose that these insects feed on trees and transmit
an illness observed at the tree level when feeding. The observed
process of tree illness X is then the intersection of us with the boolean
process Y + Z + v0.
In all these cases, if the value of Xs1 , .., Xsn on n consecutive points
of the grid is null, then the intersection of the original boolean process
with the segment [Xs1 , Xsn] is empty, and two consecutive empty segments
on the grid are of independent length. The same tests as above
can be applied by replacing randomness of segment limits on [0, Li]
by randomness of segment limits on [0, Li]∩IN and conditioning of the
number of segments.
2.5 testing with an assumption as general as
possible
There are only few methods for testing the boolean model (see for
example Molchanov 1997) and the convexity of the grain is essential
for all the methods. The most popular approach comes from Laslett’s