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In-Depth Information
The second operational problem that was faced regards transition probabilities [4].
In fact, the set of rules obtained from the real transitions' analysis is not deterministic;
this means that the same condition in the
if...and
part of the rule brings to different
final class, and this characteristic must be inherited by one-step transition rules.
Unfortunately, probabilities gained from the comparison of successive maps refer to
the real transition itself, and cannot be used to weight one-step transition rules.
Modeling a transition as a
Markov Chain
, and assuming that transition probabilities
are the same for every step belonging to the same real transition, a good estimation is
p
(
1
)
ˆ
ij
=
p
ˆ
=
OSS
n
ii
ii
ij
OSS
n
∑
=
n
−
k
k
ˆ
ˆ
ii
jj
k
0
where p
ii
is the probability to remain in the same class i, p
ij
is the probability to move
from class i to class j, and n is the number of steps in the observed transition.
The third and last issue to solve refers to the role of neighborhoods in transitions.
In fact, till now we have said nothing about this role, because we have analyzed
transitions in terms of
how many
cells change/do not change their class without
considering
which
cells change; modeling through a CA requires the examination of
the influence of the neighborhood as well. So, due to the lack of data (see above), as a
first approximation we assume that the neighborhood does not change during an
"observed" transition; for instance, considering the first transition of Grenoble from
1948 to 1960, the only neighborhood we are able to calculate is the one in 1948, but
we assume that it keeps the same in 1951, 1954, 1957. This hypothesis involves that
the number of changes of land use does not affect the prevalent class of the
neighborhood itself, and allows us to use the rules obtained from real transitions'
analysis, that explain
how and why
the change took place,
weighted
by the
probabilities referred to one-step transition.
4.3
Results
Here the major results, that emerged from the analysis, follow:
1.
transition rules are stochastic, as mentioned at the end of the previous paragraph;
2.
rules can be grouped in five different types, depending on the relations among the
classes involved:
•
a,a,a: the cell has a neighborhood homogeneous to her class and remains in that
class;
•
a,b,a: the cell does not change class even if its neighborhood belongs to a different
class;
•
a,b,b: the cell is absorbed by its neighborhood;
•
a,a,b: the cell moves to a different class, even if the neighborhood is homogeneous
to its starting class;
