Abstract
Space–time prisms are used to model the uncertainty of space–time locations of moving objects between (for instance, GPS-measured) sample points. However, not all space–time points in a prism are equally likely and we propose a simple, formal model for the so-called “visit probability” of space–time points within prisms. The proposed mathematical framework is based on a binomial random walk within one- and two-dimensional space–time prisms. Without making any assumptions on the random walks (we do not impose any distribution nor introduce any bias towards the second anchor point), we arrive at the conclusion that binomial random walk-based visit probability in space–time prisms corresponds to a hypergeometric distribution.
Highlights
Due to the discrete nature of moving object data, typically measured by GPS devices at distinct moments in time, there exists an inherent uncertainty between measured locations
At each point, has the choice between two directions: going left on the grid or going right on the grid. Unlike these authors, we make no assumptions on the random walks, we impose no distributions, we have no truncations and we do not introduce a bias towards the second anchor point
By defining the random walk on a grid and making no further assumptions, we arrive at a more natural conclusion: random walk-based visit probability in space–time prisms corresponds to a hypergeometric distribution
Summary
Due to the discrete nature of moving object data, typically measured by GPS devices at distinct moments in time, there exists an inherent uncertainty between measured locations. Winter and Yin [24] model visit probability in a prism using the theory of random walks and they arrive at the result that the internal distribution is of a bivariate multinomial nature at any given time. We propose a mathematical framework for random walk-based visit probability within one- and two-dimensional space–time prisms Like these authors, we consider movement in a prism that is constrained to what Burns calls fine-grid networks (see pages 33–34 of [30]). By defining the random walk on a grid and making no further assumptions, we arrive at a more natural conclusion: random walk-based visit probability in space–time prisms corresponds to a hypergeometric distribution.
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