Geography Reference
In-Depth Information
2. Given the same
T
m
and
V
m
,
as the total moving distance
D
(
x
,
y
)
increases, the
individual intends to move faster at location (
x
,
y
);
3. Given the same
D
(
x
,
y
)
and
V
m
,
as the individual has more time to move
(
T
m
increases), s/he is more likely to take his/her time and move with lower
velocities.
Given these properties, a consistent probability function is the following normal
distribution:
D
.x;y/
T
m
;
v
.t /
P
t
v
ˇ
ˇ
ˇ
.x; y/
2
Z
ij
.t /
N
(12.30)
where
v
(
t
) is determined by the velocity range [0,
V
m
] and is uniform across all
location within the prism except those along the prism boundary. This function
is reasonable given that the individual intends to move smoothly between two
locations.
Another approach to calculate the velocity distribution at a given location is to
utilize the distribution of visit probabilities in the prism (see Sect.
12.3.1
). Given
a moment in time
t
2
[
t
i
,
t
j
] and a later time (
t
C
t
)
2
[
t
i
,
t
j
], we can estimate all
locations at time
t
that can reach a given location (
x
,
y
) at time (
t
C
t
). In planar
space, the spatial extent of all these location can be calculated as:
x
0
;y
0
ˇ
ˇ
ˇ
q
.x
0
x/
2
t
V
m
C
.y
0
y/
2
Z
ij
.t /
D
Z
ij
.t /
\
(12.31)
If the time difference t is small enough, travelling from any location
(
x
0
,
y
0
)
2
Z
0
ij
(
t
) at time
t
to location (
x
,
y
) at time (
t
C
t
) generate an instantaneous
velocity:
q
.x
0
x/
2
v
t
.x; y/
ˇ
ˇ
ˇ
x
0
;y
0
C
.y
0
y/
2
t
D
lim
t
!
0
(12.32)
Along with the probability to visit location (
x
0
,
y
0
) at time
t
calculated by ana-
lytical framework discussed in Sect.
12.3.2
, we can derive the velocity distribution
for location (
x
,
y
) at time (
t
C
t
). Although this method is sensitive to the method
used to derive the visit probability, the estimated velocity distribution calculated has
a more solid theoretical foundation than the former two methods and is consistent
with the unequal visit probability case.
The visit probability and velocity distribution discussed in this section are the
two example properties of locations within the prisms derived from the object's
movement potential. At any instant time within the time budget, the spatial extent
of all accessible locations can be calculated, and each location is expected to have a
visit probability that follows binomial distribution for the discrete case and truncated
bivariate normal distribution for the continuous case. The velocity distribution
can be calculated based on the derived visit probability, and is expected to be
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