An isotropic universal decentralized estimation scheme for a bandwidth constrained ad hoc sensor network

Consider a decentralized estimation problem whereby an ad hoc network of K distributed sensors wish to cooperate to estimate an unknown parameter over a bounded interval [-U, U]. Each sensor collects one noise-corrupted sample, performs a local data quantization according to a fixed (but possibly probabilistic) rule, and transmits the resulting discrete message to its neighbors. These discrete messages are then percolated in the network and used by each sensor to form its own minimum mean squared error (MMSE) estimate of the unknown parameter according to a fixed fusion rule. In this paper, we propose a simple probabilistic local quantization rule: each sensor quantizes its observation to the first most significant bit (MSB) with probability 1/2, the second MSB with probability 1/4, and so on. Assuming the noises are uncorrelated and identically distributed across sensors and are bounded to [-U, U], we show that this local quantization strategy together with a fusion rule can guarantee a MSE of 4U² / K, and that the average length of local messages is bounded (no more than 2.5 bits). Compared with the worst case Cramer-Rao lower bound of U² / K (even for the centralized counterpart), this is within a factor of at most 4 to the minimum achievable MSE. Moreover, the proposed scheme is isotropic and universal in the sense that the local quantization rules and the final fusion rules are independent of sensor index, noise distribution, network size, or topology. In fact, the proposed scheme allows sensors in the network to operate identically and autonomously even when the network undergos changes in size or topology.