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Optimum Distributed Estimation of a Spatially Correlated Random Field

The distributed estimation of a spatially correlated random field with decentralized sensor networks is studied in this paper. Nodes in the network take spatial samples of the random field, then each node estimates the values of arbitrary points on the random field by iteratively exchanging information with each other. The objective is to minimize the estimation mean squared error (MSE) while ensuring all nodes reach a distributed consensus on the estimation results. We propose a distributed iterative linear minimum mean squared error (LMMSE) algorithm that contains a state consensus stage and a local estimation stage in each iteration. The proposed algorithm requires the knowledge of the second-order statistics of the random field, and they are estimated by using a distributed learning algorithm with the help of distributed consensus. The key parameters of the algorithm, including an edge weight matrix and a sample weight matrix, are designed to minimize an MSE upper bound at all nodes when the number of iterations is large. It is shown that the optimum performance can be achieved by distributively mapping the high dimension measurement samples from all nodes into a low dimension subspace related to the covariance matrices of data and noise samples. The low-dimension mapping is achieved in a distributed manner through iterative information propagation. The low dimension mapping can significantly reduce the amount of data exchanged in the network, thus improve the convergence speed of the iterative algorithm.