Geography Reference
In-Depth Information
through a location (
x
k
,
y
k
) at time
t
k
can be calculated by adding up the number of
paths through its direct predecessor and four neighbors at time
t
k
1
(Fig.
12.5
b)
Therefore, the space-time cone between
t
0
and
t
n
consists of
n
C
1 layers can be
computed as:
P
t
..x; y/
j
t
k
/
0
if
j
x
x
0
j C j
y
y
0
j
>k
D
j
t
k
1
/
˝
hif
j
x
x
0
j
C
j
y
y
0
j
k
;k
D
1; 2; 3; ::::; n
(12.15)
p ..x; y/
where
h
is the normalized kernel for undirected/unbiased random walk:
2
3
0:0
0:2
0:0
4
5
h
D
0:2
0:2
0:2
(12.16)
0:0
0:2
0:0
Based on this construction, the probability to visit accessible locations at each
layer follows a bivariate multinomial distribution in discrete space and time.
Based on this framework, Winter and Yin (
2010b
) consider the case of directed
movement from the first anchor to the second anchor in the prism. But, rather
than modifying the fundamental random walk process to account for directionality,
they assume that the uneven visit probabilities implied by the undirected case
translates to the directed space-time axis connecting the two anchors. To describe
visit probabilities in continuous space, they employ the continuous analogue
of the multinomial distribution, that is, the multivariate normal distribution. In
two-dimensional space, this is a bivariate normal distribution. However, these
distributions are unbounded: this is incongruent with a bounded space-time prism.
To resolve this, Winter and Yin (
2010b
) use the spatial extent of the prism at a given
moment in time to modify the standard deviation at that moment and to clip the
Bivariate normal distribution.
The conclusions reached by Winter and Yin (
2010b
) regarding visit probabilities
within planar space prisms are intuitive, but their argument lacks a solid mathemat-
ical foundation. The transition from a discrete space random walk to continuous
space is accomplished by analogy only: there is no explanation of why the expected
location within the prism with respect to time (the mean of bivariate normal
distribution) should be along the space-time axis linking the two anchors of the
prism; this is simply assumed. Also, clipping the bivariate normal distribution using
the spatial extent
Z
ij
(
t
) is an artificial construction that is not based on fundamental
principles. Bias may be introduced by cutting off part of the original distribution
outside the extent, especially when the prism spatial extent is not symmetric with
respect to the axis connecting the anchor points.
In the next subsections of this paper, we develop a more solid foundation
for calculating visit probabilities within space-time prism in two steps. First, we
maintain discrete space and time, but modify the undirected random walk to
Search WWH ::
Custom Search