Sequential and Decentralized Estimation of Linear Regression Parameters in Wireless Sensor Networks

Abstract: Sequential estimation of a vector of linear regression coefficients is considered under both centralized and decentralized setups. In sequential estimation, the number of observations used for estimation is determined by the observed samples, hence is random, as opposed to fixed-sample-size estimation. Specifically, after receiving a new sample, if a target accuracy level is reached, we stop and estimate using the samples collected so far; otherwise we continue to receive another sample. It is known that finding an optimum sequential estimator, which minimizes the average sample number for a given target accuracy level, is an intractable problem with a general stopping rule that depends on the complete observation history. By properly restricting the search space to stopping rules that depend on a specific subset of the complete observation history, we derive the optimum sequential estimator in the centralized case via optimal stopping theory. However, finding the optimum stopping rule in this case requires numerical computations that {\em quadratically} scales with the number of parameters to be estimated. For the decentralized setup with stringent energy constraints, under an alternative problem formulation that is conditional on the observed regressors, we first derive a simple optimum scheme whose computational complexity is {\em constant} with respect to the number of parameters. Then, following this simple optimum scheme we propose a decentralized sequential estimator whose computational complexity and energy consumption scales {\em linearly} with the number of parameters. Specifically, in the proposed decentralized scheme a close-to-optimum average stopping time performance is achieved by infrequently transmitting a single pulse with very short duration.