Modelling Visit Probabilities within Space-Time Prisms using Directed Random Walk and Truncated Brownian Bridges

The space-time prism demarcates locations that a mobile object can occupy given origin and destination anchors, the earliest origin departure and the latest destination arrival time, and the maximum travel velocity n Lenntorp 1976). The prism boundary has been used widely to delimit individual ’ accessibility (Miller 1991, Kwan 1998, Geurs and Wee 2004). However, little attention has been paid to the prism interior. Modeling interior properties such as visit probabilities can help refine the prism as a space-time accessibility measure by better representing potential mobility behavior. In pioneering papers, Winter and Yin (2010a, 2010b) demonstrate that the probabilities to visit locations within the prism interior are not equally distributed. They develop a mathematical foundation for modelling probabilistic space-time prisms in both discrete and continuous spacetime. Although their results are intuitive, their methodology has several weaknesses. First, they use an unsophisticated directed random walk process that assumes biased movements without a corresponding micro-scale foundation. Then, they use an ad-hoc procedure for truncating the distribution at the prism boundary. Finally, since they rely on the random walk theory, they restrict their attention to discrete rather than continuous time. This paper advances theory and methods for modelling visit probabilities within classic space-time prisms (planar space prisms with constant maximum velocity). We first develop an improved directed random walk method for the discrete case. Then, we develop the method for continuous space-time using Brownian Bridges methods. We also develop a truncated Brownian Bridges technique that allows the prism boundary to emerge naturally instead of being imposed artificially. We provide some preliminary results to illustrate the methods.