We propose a probabilistic model of pedestrian behavior for estimating the population at each road given the observed populations at a limited number of roads and a set of routes. Our proposed model has latent variables called route populations, which represent the pedestrian population who starts each route at each time period, and assumes that pedestrians stochastically walk through roads along the route. The pedestrian depends on the road’s congestion. The proposed model incorporates this dependence in a probabilistic framework, where the time length to pass a road is assumed to follow a Gaussian distribution that depends on road congestion. With the reproducing property of Gaussian distribution, we analytically derive the transition probability between roads on each route. The parameters of the proposed method, including the dependence between congestion and speed, congestion for each road, and the latent variables on the route population, are estimated from the given data by minimizing the error between the observed and estimated road populations based on gradient-based optimization methods. In experiments with simulated and real-world data sets, we demonstrate that the proposed model can estimate road populations more accurately than existing methods. We also confirm that it effectively estimates route populations, and the estimated route populations are useful for reproducing real-world population dynamics when used as inputs of a crowd simulator based on a multi-agent system.

We propose a probabilistic model for estimating population flow, which is defined as populations of the transition between areas over time, given aggregated spatio-temporal population data. Since there is no information about individual trajectories in the aggregated data, it is not straightforward to estimate population flow. With the proposed method, we utilize a collective graphical model with which we can learn individual transition models from the aggregated data by analytically marginalizing the individual locations. Learning a spatio-temporal collective graphical model only from the aggregated data is an ill-posed problem since the number of parameters to be estimated exceeds the number of observations. The proposed method reduces the effective number of parameters by modeling the transition probabilities with a neural network that takes the locations of the origin and the destination areas and the time of day as inputs. By this modeling, we can automatically learn nonlinear spatio-temporal relationships flexibly among transitions, locations, and times. With four real-world population data sets in Japan and China, we demonstrate that the proposed method can estimate the transition population more accurately than existing methods.