Beyond the Boundary: New Insights from Inside the Space-Time Prism - Space-Time Integration in Geography and GIScience - page 197

and solves the issue of information loss outside the spatial extent caused by clipping
process. Therefore, it provides more theoretically consistent and unbiased estimates
of visit probabilities within the prism.

12.3.2 Velocity Distribution

A space-time path going through a given location in the prism has an instantaneous
travel velocity that may be different from other paths through the same location.
Hence, there is a velocity profile associated with each location in the prism. For a
location ( x , y ) in the spatial extent Z ij ( t ). , the probability to travel at velocity v among
all feasible velocities can be represented as P t ( v j ( x , y )), where v 2 [ v min , v max ]. We
start our analysis by assuming that an individual selected any velocity from feasible
velocities equally possible. This implies the following uniform velocity distribution:
(Note: “1” here indicates one unit of choice)

8
<

1
. v max v min / ;

if v max > v > v min

P t v ˇ ˇ ˇ

.x; y/

D (12.27)

1; if v D v max D v min

:

0; else

In most cases, the individual can choose to stay still or move at maximum
velocity; therefore, v min D 0and v max D V m respectively. A special consideration is
locations along the boundary of the prism, where individual has to travel at the
maximum velocity V m when not conducting stationary activities.
A uniform velocity assumption is straightforward but unsophisticated. A non-
uniform velocity distribution takes into account time pressure imposed by the
constraint of reaching the destination anchor by the required time t j . With longer
distance and less time to travel, the individual is more likely to travel with higher
velocities. This suggests that the probability to select a velocity v is affected by three
factors: (i) the total distance to travel from origin to destination passing location
( x , y ), D ( x , y ) ; (ii) the total time available for moving T m D ( t j t i t s ); and, (iii) the
maximum achievable travel velocity V m :
P t v ˇ ˇ ˇ

.x; y/ D f D .x;y/ ;T m ;V m ; where .x; y/ 2 Z ij .t /

(12.28)

There are several properties that this function should satisfy:

1. Since we have v 2 [0, V m ] for the entire space-time prism, given any location
within the prism, the individual has to choose from this velocity range, that is:

V m

Z

P t v ˇ ˇ ˇ .x; y/ D 1; for any .x; y/ 2 Z ij .t / (12.29)

0