Anchor uncertainty and space-time prisms on road networks

Space-time prisms capture all possible locations of a moving person or object between two known locations and times given the maximum travel velocities in the environment. These known locations or 'anchor points' can represent observed locations or mandatory locations because of scheduling constraints. The classic space-time prism as well as more recent analytical and computational versions in planar space and networks assume that these anchor points are perfectly known or fixed. In reality, observations of anchor points can have error, or the scheduling constraints may have some degree of pliability. This article generalizes the concept of anchor points to anchor regions: these are bounded, possibly disconnected, subsets of space-time containing all possible locations for the anchor points, with each location labelled with an anchor probability. We develop two algorithms for calculating network-based space-time prisms based on these probabilistic anchor regions. The first algorithm calculates the envelope of all space-time prisms having an anchor point within a particular anchor region. The second algorithm calculates, for any space-time point, the probability that a space-time prism with given anchor regions contains that particular point. Both algorithms are implemented in Mathematica to visualize travel possibilities in case the anchor points of a space-time prism are uncertain. We also discuss the complexity of the procedures, their use in analysing uncertainty or flexibility in network-based prisms and future research directions.