4284 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007 Distributed Estimation Using Reduced- Dimensionality Sensor Observations Ioannis D. Schizas, Student Member, IEEE, Georgios B. Giannakis, Fellow, IEEE, and Zhi-Quan Luo, Fellow, IEEE Abstract—We derive linear estimators of stationary random signals based on reduced-dimensionality observations collected at distributed sensors and communicated to a fusion center over wireless links. Dimensionality reduction compresses sensor data to meet low-power and bandwidth constraints, while linearity in compression and estimation are well motivated by the limited computing capabilities wireless sensor networks are envisioned to operate with, and by the desire to estimate random signals from observations with unknown probability density functions. In the absence of fading and fusion center noise (ideal links), we cast this intertwined compression-estimation problem in a canonical cor- relation analysis framework and derive closed-form mean-square error (MSE) optimal estimators along with coordinate descent suboptimal alternatives that guarantee convergence at least to a stationary point. Likewise, we develop estimators based on reduced-dimensionality sensor observations in the presence of fading and additive noise at the fusion center (nonideal links). Performance analysis and corroborating simulations demonstrate the merits of the novel distributed estimators relative to existing alternatives. Index Terms—Canonical correlation analysis (CCA), distributed compression, distributed estimation, nonlinear optimization, wire- less sensor networks (WSNs). I. INTRODUCTION W ITH the popularity of battery-powered wireless sensor networks (WSNs), distributed estimation relying on sensor data processed at a fusion center (FC) has attracted increasing interest recently. Constrained by limited power and bandwidth resources, existing approaches either take advantage of spatial correlations across sensor data to reduce transmission requirements [2], [5], [11], [15], [16], or, rely on severely quantized (possibly down to one bit) digital WSN data to form distributed estimators of deterministic parameters, see, e.g., Manuscript received November 6, 2005; revised August 10, 2006. The as- sociate editor coordinating the review of this manuscript and approving it for publication was Prof. Javier Garcia-Frias. Prepared through collaborative par- ticipation in the Communications and Networks Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The work of Z.-Q. Luo is supported by the U.S. DoD Army, grant number W911NF-05-1-0567. The U.S. Government is authorized to reproduce and distribute reprints for Gov- ernment purposes notwithstanding any copyright notation thereon. Parts of the paper were presented at the Thirty-Ninth Asilomar Conference, Pacific Grove, CA, Oct. 30–November 2, 2005, and at the 2006 International Conference on Acoustics, Speech and Signal Processing (ICASSP), Toulouse, France, May 14–19, 2006. The authors are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: schizas@ece. umn.edu; georgios@ece.umn.edu; luozq@ece.umn.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2007.895987 [8], [10], [12], and references therein. Distributed estimation of random signals has also been considered by [5], [9], and [14]–[16], but results are restricted by one or more of the following assumptions: i) Gaussian signals and/or sensor data; ii) linear sensor observation models; and iii) ideal links; i.e., absence of fading in the sensor-FC channels and/or additive noise at the FC. Overcoming limitations i) and ii), our goal in this paper is to form estimates at the FC of a random stationary vector based on analog-amplitude multisensor observations. To enable es- timation under the stringent power and computing limitations of WSNs and develop methods that do not require knowledge of the sensor data probability density function (pdf), which, in a number of cases, may not be available, we seek linear di- mensionality reducing operators (data compressing matrices) per sensor along with linear operators at the FC in order to minimize the mean-square error (MSE) in estimation. We treat first the ideal channel case, where we formulate this intertwined compression-estimation task as a canonical correlation analysis (CCA) problem. CCA is a well-documented tool for data and model reduction problems encountered in various applications such as statistical data analysis, control, signal processing, to name a few [4, Ch. 10]. But our contribution here is to demon- strate that CCA provides a natural framework for estimating random signals based on reduced-dimensionality WSN obser- vations. The resultant estimators apply to possibly nonlinear and non-Gaussian setups and can be generalized to incorporate channel fading as well as FC noise effects, which necessitate tackling distributed CCA problems under a prescribed power budget per sensor. Specifically, we establish that with either decoupled or coupled multisensor observations communicated to the FC through ideal links, the problem formulation (Section II) lends itself naturally to CCA. In the decoupled case, we prove that the optimal solution amounts to compressing, via principal components analysis (PCA), the linear minimum mean-square error (LMMSE) signal estimate formed at each sensor (Section III). We further compare the MSE of this estimate-first compress-afterwards approach with suboptimal compress-first estimate-afterwards alternatives, including the scheme in [16]. With coupled (i.e., correlated) sensor data, optimal distributed estimation has been shown to be NP-hard when reduced-dimensionality sensor data are concatenated at the FC [9]. Interestingly, we establish that when the same data are superimposed at the FC, the CCA-based approach can provide closed-form solutions with low-order data reduction (Section IV-A). But since lower MSE estimates result when concatenating (rather than superimposing) compressed WSN 1053-587X/$25.00 © 2007 IEEE