Non-Parametric Field Estimation using Randomly Deployed, Noisy, Binary Sensors

The reconstruction of a deterministic data field from binary–quantized noisy observations of sensors which are randomly deployed over the field domain is studied. The study focuses on the extremes of lack of deterministic control in the sensor deployment, lack of knowledge of the noise distribution, and lack of sensing precision and reliability. Such adverse conditions are motivated by possible real–world scenarios where a large collection of low–cost, crudely manufactured sensors are mass–deployed in an environment where little can be assumed about the ambient noise. A simple estimator that reconstructs the entire data field from these unreliable, binary–quantized, noisy observations is proposed. Technical conditions for the almost sure and integrated mean squared error (MSE) convergence of the estimate to the data field, as the number of sensors tends to infinity, are derived and their implications are discussed. For finite–dimensional, bounded–variation, and Sobolev– differentiable function classes, specific integrated MSE decay rates are derived. For the first and third function classes these rates are found to be minimax order optimal with respect to infinite precision sensing and known noise distribution.