  This work studies distributed (probability) density estimation of large-scale
systems. Such problems are motivated by many density-based distributed control
tasks in which the real-time density of the swarm is used as feedback
information, such as sensor deployment and city traffic scheduling. This work
is built upon our previous work [1] which presented a (centralized) density
filter to estimate the dynamic density of large-scale systems through a novel
integration of mean-field models, kernel density estimation (KDE), and
infinite-dimensional Kalman filters. In this work, we further study how to
decentralize the density filter such that each agent can estimate the global
density only based on its local observation and communication with neighbors.
This is achieved by noting that the global observation constructed by KDE is an
average of the local kernels. Hence, dynamic average consensus algorithms are
used for each agent to track the global observation in a distributed way. We
present a distributed density filter which requires very little information
exchange, and study its stability and optimality using the notion of
input-to-state stability. Simulation results suggest that the distributed
filter is able to converge to the centralized filter and remain close to it.
