Asymptotic optimality of multicenter Voronoi configurations for random field estimation

This paper deals with multi-agent networks performing optimal estimation tasks. Consider a network of mobile agents with sensors that can take measurements of a spatial process in an environment of interest. Using the measurements, one can construct a kriging interpolation of the spatial field over the whole environment, with an associated prediction error at each point. We study the continuity properties of the prediction error, and consider as global objective functions the maximum prediction error and the generalized prediction variance. We study the network configurations that give rise to optimal field interpolations. Specifically, we show how, as the correlation between any two different locations vanishes, circumcenter and incenter Voronoi configurations become network configurations that optimize the maximum prediction error and the generalized prediction variance, respectively. The technical approach draws on tools from geostatistics, computational geometry, linear algebra, and dynamical systems.