Geography Reference
In-Depth Information
started to receive attention (Winter and Yin
2010a
,
b
). In the following sections,
we introduce analytical frameworks for two properties of locations within the prism
interior, namely, visit probabilities and velocity distributions, respectively.
12.3
Properties of the Prism Interior
12.3.1
Visit Probabilities
At a moment in time
t
2
[
t
i
,
t
j
]
,
the spatial extent
Z
ij
(
t
) delimits all locations that
an individual can access in space at that moment (Eq.
12.1
). For any two locations
within this spatial extent, the number of possible paths going through them may
be different, implying an unequal visit probability distribution
P
((
x
,
y
)
j
t
). Since an
individual cannot reach a moment in time without passing an earlier moment, the
probability to pass any time
t
2
[
t
i
,
t
j
] equals to one; therefore, the total probability
to visit all locations in
Z
ij
(
t
) equals to one:
“
P
.x; y/
ˇ
ˇ
ˇ
t
dxdy
D
1
P
Z
ij
.t /
D
2
t
i
;t
j
8
t
(12.14)
.
x;y
/
2
Z
ij
.t /
Winter and Yin (
2010a
) develop a mathematical foundation for modeling the
distribution of visit probabilities based on random walk theory. They start with an
undirected random walk from the origin anchor; this is a sequence of random steps
that approximate a continuous stochastic diffusion process. Figure
12.5
ashowsthe
case for one-dimensional space: the number of paths through a location in the next
step can be calculated as the sum of possible paths from itself and its neighbors in
the current step.
Extending the calculation to two-dimensional discrete space, the object can
move one step in one of the two dimensions, and the number of possible paths
Fig. 12.5
Visit probability using random walk theory (Based on Winter and Yin
2010a
)
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