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cases (see the highlighted trajectory); this demonstrates the need for truncated
functions to ensure that each interpolated location is always located within the
accessible spatial extent at any time. An interesting finding is that with all else
being equal, more interpolation time steps results in a smaller generated PPA. This
is because with the same overall travel distance, when the number of interpolated
time steps increases, the interval between two interpolation times will decrease;
therefore, given the Property 4 of the Wiener process, the standard deviations for
each interpolation will decrease and the generated PPA becomes smaller. Although
this can eliminate many trajectories that are not feasible given the prism constraints,
it may also exclude feasible trajectories. This suggests that the simulation of non-
truncated Brownian bridges is sensitive to the number of interpolation times, and
further investigation is required to find the relationship between the interpolation
times and the accuracy of simulated trajectories.
To restrict BBs within the space-time prism, we apply the concept of a truncated
distribution instead of artificially clipping the distribution by spatial extent of prism
at time
t
(Eq.
12.1
). A truncated distribution is a conditional distribution that results
from restricting the domain of some other probability distribution. For any instant
time
t
2
[
t
i
,
t
j
], the
x
and
y
coordinate of the mobile individual follows the truncated
normal distributions with density functions as:
N
0;
x
.t /
F.U
x
/
F.L
x
/
D
2
x
.t /
N
.0; 1/
Tr
.x.t //
erf
U
x
erf
L
x
(12.25)
x
.t /
p
2
x
.t /
p
2
0;
y
.t /
F
U
y
. x.t//
F
L
y
. x.t//
D
Tr
.y.t //
ˇ
ˇ
ˇ
x.t/
N
2
y
.t /
N
.0; 1/
erf
U
y
. x.t//
!
erf
L
y
. x.t//
!
y
.t /
p
2
y
.t /
p
2
(12.26)
where
erf
(
x
) is the error function based on normal distribution,
U
x
and
L
x
is
determined by the spatial extent of the prism at the time at time
t, Z
ij
(
t
), and
U
y
and
L
y
are determined by
Z
ij
(
t
) and a randomly simulated value for x.t/; x.t/.
Given these two truncated density functions, a BB within the space-time prism
can be simulated by interpolating locations at a set of time steps
f
t
k
g
. Give a time
series , the interpolated location at time
t
s
is calculated based on the interpolated
locations at times
t
u
and
t
t
. However, since the interpolation of (
x
(
t
s
),
y
(
t
s
)) uses
two random numbers that are independent from previous process (see Eqs.
12.25
and
12.26
), the stochastic process has independent increments between any two
locations, and the simulation process maintains the properties required by BBs.
The mathematical foundations demonstrated above applies and modifies BBs
with consideration of general restrictions that define a space-time prism. Compared
to Winter and Yin's (
2010b
) construction, it does not require assumptions about
the expected location, and avoids the bias by artificially clipping an unbounded
distribution by spatial extent of prism at time
t
. The truncated distribution preserves
the general properties of a BB as a stochastic process with independent increments,
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