Probabilistic Extension of Allen's Relations Using the Hourglass Model

This paper presents a probabilistic extension of Allen's relations that enables reasoning with intervals whose boundaries are uncertain. Building on an earlier work on an hourglass model that depicts the geometry of interval relations, we first provide further evidence that support its qualitative properties. We then specify two orthogonal axes, which quantitatively describe respectively the relative position and relative size of intervals. The quantitative hourglass model is then used to define a consistent probabilistic set of all thirteen Allen's relations. Probabilistic relations between intervals and points are also accounted for in the developed framework.