  In this paper, we address the distributed filtering and prediction of
time-varying random fields represented by linear time-invariant (LTI) dynamical
systems. The field is observed by a sparsely connected network of
agents/sensors collaborating among themselves. We develop a Kalman filter type
consensus+innovations distributed linear estimator of the dynamic field termed
as Consensus+Innovations Kalman Filter. We analyze the convergence properties
of this distributed estimator. We prove that the mean-squared error of the
estimator asymptotically converges if the degree of instability of the field
dynamics is within a pre-specified threshold defined as tracking capacity of
the estimator. The tracking capacity is a function of the local observation
models and the agent communication network. We design the optimal consensus and
innovation gain matrices yielding distributed estimates with minimized
mean-squared error. Through numerical evaluations, we show that, the
distributed estimator with optimal gains converges faster and with
approximately 3dB better mean-squared error performance than previous
distributed estimators.
