Geography Reference
In-Depth Information
1.
W
0
D
0.
2. With probability 1, the function
t
!
W
t
is continuous in
t
.
3. The process
f
W
t
g
has stationary, independent increments.
4. The increment
W
t
C
s
W
s
has the normal distribution
N
.0; t /.
For the two dimension case, the probability at any time
t
2
[
t
i
,
t
j
] follows bivariate
normal distribution:
x.t/
y.t/
u
x
.t /
u
y
.t /
;
"
x
.t / 0
0
y
.t /
#!
N
(12.21)
where
C
t
j
t
x
j
.t
t
i
/x
i
u
x
.t /
D
(12.22)
t
j
t
i
C
t
j
t
y
j
.t
t
i
/y
i
u
y
.t /
D
(12.23)
t
j
t
i
.t
t
i
/
t
j
t
x
.t /
D
y
.t /
D
(12.24)
t
j
t
i
The Eqs.
12.21
,
12.22
,
12.23
,and
12.24
are consistent with Winter and Yin's
(
2010b
) assumption of the visit probability distribution within the prism: (i) the
expected location at any time
t
is along the axis connecting the two anchors of
the space-time prism, and (ii) the visit probabilities follow a bivariate normal
distribution. However, our method has a firmer mathematical foundation, and the
distribution is generated from first principles instead of introduced as an assumption.
Figure
12.7
shows 10,000 non-truncated BB trajectories, each with 50 or 100
interpolation time steps. Note trajectories fall outside of the generated PPA in both
Fig. 12.7
Simulation results of non-truncated Brownian Bridges
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