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Real-time spatiotemporal population data is attracting a great deal of attention for understanding crowd movements in cities.The data is the aggregation of personal location information and consists of just areas and the number of people in each area at certain time instants. Accordingly, it does not explicitly represent crowd movement. This paper proposes a probabilistic model based on collective graphical models that can estimate crowd movement from spatiotemporal population data. There are two technical challenges: (i) poor estimation accuracy as the traditional approach means the model would have too many degrees of freedom, (ii) excessive computation cost. Our key idea for overcoming these two difficulties is to model the transition probability between grid cells (cells hereafter) in a geospatial grid space by using three factors: departure probability of cells, gathering score of cells, and geographical distance between cells. These advances enable us to reduce the degrees of freedom of the model appropriately and derive an efficient estimation algorithm.  To evaluate the performance of our method, we conduct experiments using real-world spatiotemporal population data. The results confirm the effectiveness of our method, both in estimation accuracy and computation cost.

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Collective graphical model (CGM) is a probabilistic model that provides a framework for analyzing aggregated count data. Maximum a posteriori (MAP) inference of unobserved variables under given observations is one of the essential operations in CGM. Because the MAP inference problem is known to be NP-hard in general, the current mainstream approach is to solve an alternative problem obtained by approximating the objective function and applying continuous relaxation. However, this approach has two significant drawbacks. First, the quality of the solution deteriorates when the values in the count data are negligible due to the inaccuracy of Stirling’s approximation. Second, the application of continuous relaxation causes the violation of integrality constraints. This paper proposes novel algorithms for MAP inference in CGMs on path graphs to overcome these problems. Our method is based on the discrete difference of convex algorithm (DCA); DCA is a general framework to minimize the sum of a convex function and a concave function by repeatedly minimizing surrogate functions. Utilizing the particular structure of path graphs, we efficiently solve the surrogate function minimization by minimum convex cost flow algorithms. Furthermore, our approach also leads to a new method of solving another important task; MAP inference of the sample size in CGM on path graphs. Our method is naturally applicable to this task, allowing us to design very efficient algorithms. Experimental results on synthetic and real-world datasets show the effectiveness of the proposed algorithms.