  We study the problem of distributed adaptive estimation over networks where
nodes cooperate to estimate physical parameters that can vary over both space
and time domains. We use a set of basis functions to characterize the
space-varying nature of the parameters and propose a diffusion least
mean-squares (LMS) strategy to recover these parameters from successive time
measurements. We analyze the stability and convergence of the proposed
algorithm, and derive closed-form expressions to predict its learning behavior
and steady-state performance in terms of mean-square error. We find that in the
estimation of the space-varying parameters using distributed approaches, the
covariance matrix of the regression data at each node becomes rank-deficient.
Our analysis reveals that the proposed algorithm can overcome this difficulty
to a large extent by benefiting from the network stochastic matrices that are
used to combine exchanged information between nodes. We provide computer
experiments to illustrate and support the theoretical findings.
