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1 Closed-Form MSE Performance of the Distributed LMS Algorithm Gonzalo Mateos, Ioannis Schizas and Georgios B. Giannakis ECE Department, University of Minnesota Acknowledgment: ARL/CTA grant no. DAAD19-01-2-0011 USDoD ARO grant no. W911NF-05-1-0283

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2 Motivation Estimation using ad hoc WSNs raises exciting challenges Communication constraints Limited power budget Lack of hierarchy / decentralized processing Consensus Unique features Environment is constantly changing (e.g., WSN topology) Lack of statistical information at sensor-level Bottom line: algorithms are required to be Resource efficient Simple and flexible Adaptive and robust to changes Single-hop communications

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3 Prior Works Single-shot distributed estimation algorithms Consensus averaging [Xiao-Boyd ’ 05, Tsitsiklis-Bertsekas ’ 86, ’ 97] Incremental strategies [Rabbat-Nowak etal ’ 05] Deterministic and random parameter estimation [Schizas etal ’ 06] Consensus-based Kalman tracking using ad hoc WSNs MSE optimal filtering and smoothing [Schizas etal ’ 07] Suboptimal approaches [Olfati-Saber ’ 05], [Spanos etal ’ 05] Distributed adaptive estimation and filtering LMS and RLS learning rules [Lopes-Sayed ’ 06 ’ 07]

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4 Problem Statement Ad hoc WSN with sensors Single-hop communications only. Sensor ‘ s neighborhood Connectivity information captured in Zero-mean additive (e.g., Rx) noise Goal: estimate a signal vector Each sensor, at time instant Acquires a regressor and scalar observation Both zero-mean and spatially uncorrelated Least-mean squares (LMS) estimation problem of interest

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5 Power Spectrum Estimation Find spectral peaks of a narrowband (e.g., seismic) source AR model: Source-sensor multi-path channels modeled as FIR filters Unknown orders and tap coefficients Observation at sensor is Define: Challenges Data model not completely known Channel fades at the frequencies occupied by

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6 A Useful Reformulation Introduce the bridge sensor subset For all sensors, such that For, a path connecting them devoid of edges linking two sensors Consider the convex, constrained optimization Proposition [Schizas etal ’ 06]: For satisfying 1)-2) and the WSN is connected, then

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7 Algorithm Construction Associated augmented Lagrangian Two key steps in deriving D-LMS Resort to the alternating-direction method of multipliers Gain desired degree of parallelization Apply stochastic approximation ideas Cope with unavailability of statistical information

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8 D-LMS Recursions and Operation In the presence of communication noise, for and Simple, distributed, only single-hop exchanges needed Step 1: Step 2: Step 3: Sensor Rx from Tx to Bridge sensor Tx to Rx from Steps 1,2:Step 3:

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9 Error-form D-LMS Study the dynamics of Local estimation errors: Local sum of multipliers: (a1) Sensor observations obey where the zero-mean white noise has variance Introduce and Lemma : Under (a1), for then where and consists of the blocks and with

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10 Performance Metrics Local (per-sensor) and global (network-wide) metrics of interest (a2) is white Gaussian with covariance matrix (a3) and are independent Define Customary figures of merit EMSEMSD Local Global

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11 Tracking Performance (a4) Random-walk model: where is zero- mean white with covariance ; independent of and Let where Convenient c.v.: Proposition: Under (a2)-(a4), the covariance matrix of obeys with. Equivalently, after vectorization where

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12 Stability and S.S. Performance MSE stability follows Intractable to obtain explicit bounds on From stability, has bounded entries The fixed point of is Enables evaluation of all figures of merit in s.s. Proposition: Under (a1)-(a4), the D-LMS algorithm is MSE stable for sufficiently small Proposition: Under (a1)-(a4), the D-LMS algorithm achieves consensus in the mean, i.e., provided the step-size is chosen such that with

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13 Step-size Optimization If optimum minimizing EMSE Not surprising Excessive adaptation MSE inflation Vanishing tracking ability lost Recall Hard to obtain closed-form, but easy numerically (1-D).

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14, D-LMS: Simulated Tests node WSN, Rx AWGN w/, Random-walk model: Time-invariant parameter: Regressors: w/ ; i.i.d.; w/ Observations: linear data model, WGN w/

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15 Concluding Summary Developed a distributed LMS algorithm for general ad hoc WSNs Detailed MSE performance analysis for D-LMS Stationary setup, time-invariant parameter Tracking a random-walk Analysis under the simplifying white Gaussian setting Closed-form, exact recursion for the global error covariance matrix Local and network-wide figures of merit for and in s.s. Tracking analysis revealed minimizing the s.s. EMSE Simulations validate the theoretical findings Results extend to temporally-correlated (non-) Gaussian sensor data

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