The elements of probabilistic time geography

Time geography uses space-time volumes to represent the possible locations of a mobile agent over time in a x-y-t space. A volume is a qualitative representation of the fact that the agent is at a particular time t (i) inside of the volume's base at t (i) . Space-time volumes enable qualitative analysis such as potential encounters between agents. In this paper the qualitative statements of time geography will be quantified. For this purpose an agent's possible locations are modeled from a stochastic perspective. It is shown that probability is not equally distributed in a space-time volume, i.e., a quantitative analysis cannot be based simply on proportions of intersections. The actual probability distribution depends on the degree of a priori knowledge about the agent's behavior. This paper starts with the standard assumption of time geography (no further knowledge), and develops the appropriate probability distribution by three equivalent approaches. With such a model any analysis of the location of an agent, or relations between the locations of two agents, can be improved in expressiveness as well as accuracy.