Asymptotic Optimality of Multicenter Voronoi Configurations for Random Field Estimation

This technical note deals with multi-agent networks performing estimation tasks. Consider a network of mobile agents with sensors that can take measurements of a spatial stochastic process. Using the kriging statistical technique, a field estimate may be calculated over the environment, with an associated error variance at each point. We study a single-snapshot scenario, in which the spatial process mean is known and each agent can only take one measurement. We consider two optimization problems with respect to the measurement locations, using as objective functions the maximum error variance and the extended prediction variance. As the correlation between distinct locations vanishes, we show that circumcenter and incenter Voronoi configurations become network configurations that optimize the maximum error variance and the extended prediction variance, respectively. We also present distributed coordination algorithms that steer the network towards these configurations.