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
 
Search WWH ::




Custom Search