We develop a kernel density estimation method for estimating the density of points on a network and implement the method in the GIS environment. This method could be applied to, for instance, finding 'hot spots' of traffic accidents, street crimes or leakages in gas and oil pipe lines. We first show that the application of the ordinary two-dimensional kernel method to density estimation on a network produces biased estimates. Second, we formulate a 'natural' extension of the univariate kernel method to density estimation on a network, and prove that its estimator is biased; in particular, it overestimates the densities around nodes. Third, we formulate an unbiased discontinuous kernel function on a network, and fourth, an unbiased continuous kernel function on a network. Fifth, we develop computational methods for these kernels and derive their computational complexity. We also develop a plug-in tool for operating these methods in the GIS environment. Sixth, an application of the proposed methods to the density estimation of bag-snatches on streets is illustrated. Lastly, we summarize the major results and describe some suggestions for the practical use of the proposed methods.
SUMMARYA quantitative method for evaluating sport teamwork is proposed. The sports considered here are team sports in which players can move freely in the field, and two teams compete against each other. For this kind of sports, each player's dominant region has an important role in evaluating the teamwork. Therefore, here we propose some approaches to quantitative evaluation based on the concept of a generalized Voronoi diagram that divides space into dominant regions. We also construct a more realistic model of player's motion model based on experiments, and apply it to the evaluation.
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