1 / 36

Distributed Adaptive Estimation and Tracking using Ad Hoc WSNs

Distributed Adaptive Estimation and Tracking using Ad Hoc WSNs. Gonzalo Mateos ECE Department, University of Minnesota Acknowledgment: ARL/CTA grant no. DAAD19-01-2-0011 USDoD ARO grant no. W911NF-05-1-0283. Minneapolis, MN July 29, 2009. Wireless Sensor Networks (WSNs).

gunnar
Download Presentation

Distributed Adaptive Estimation and Tracking using Ad Hoc WSNs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Distributed Adaptive Estimation and Tracking using Ad Hoc WSNs Gonzalo Mateos ECE Department, University of Minnesota Acknowledgment: ARL/CTA grant no. DAAD19-01-2-0011 USDoD AROgrant no. W911NF-05-1-0283 Minneapolis, MNJuly 29, 2009

  2. Wireless Sensor Networks (WSNs) • Large number of wireless sensors • Randomly deployed • Inexpensive • Resource constrained • Unique feature: cooperative effort of sensors • Promising technology for crucial applications • Environmental monitoring • Fault diagnosis in process industry • Protection of critical infrastructure • Surveillance systems • Renewed interest in distributed computing

  3. FC-based WSN Ad hoc WSN Ad hoc WSN + = Increased life expectancy of the WSN Two Prevailing Topologies • Why ad hoc WSNs? • Less power consumption as WSN scales (geographically) • Improved robustness to sensor failures

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

  5. Subject of the Thesis • Distributed estimation/tracking algorithms using ad hoc WSNs • In-network processing of sensor observations • Stability/convergence analysis • Quantifiable MSE (tracking) performance • Distributed (D-) least mean-square (LMS) & recursive least-squares (RLS) • Affordable complexity • Do not require a data model to be applicable • Online data enriches the estimation process • Can track slowly time-varying processes • Explore the complexity vs. performance tradeoff

  6. This Work in Context • 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-Cattivelli-Sayed ’06-08] • Optimization tools in distributed estimation • Incremental strategies • Primal-dual approaches • Alternating-direction method of multipliers (AD-MoM)

  7. Outline • Part I: The D-LMS algorithm • Algorithm construction and operation • Stability results • Tracking performance analysis • Part II: The D-RLS algorithm • Reduced complexity variants • Stability and steady-state MSE performance analysis • Concluding remarks and future research directions

  8. 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

  9. 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

  10. Centralized Approaches • If , jointly stationary with Wiener solution • If , are available Steepest-descent converges avoiding matrix inversion • If (cross-) covariance info. not available or time-varying Low complexity suggests (C-) LMS adaptation Goal:develop a distributed (D-) LMS algorithm for ad hoc WSNs

  11. Algorithmic Construction • Consider the convex, constrained optimization • Equivalent for connected WSN: • Two key steps in deriving D-LMS • Resort to the AD-MoM [Glowinski ‘75] Gain desired degree of parallelization • Apply stochastic approximation ideas Cope with unavailability of statistical information

  12. Rx from Tx to Tx to D-LMS Recursions and Operation • In the presence of communication noise, for and • Reduced communications possible with `bridge’ sensors Step 1: Step 2: Step 1: forming Step 2: forming Rx from

  13. Sensor j Local LMS Algorithm Consensus Loop To PI Regulator Consensus Controller Interpretation • Consensus error at sensor : • Superposition of two learning mechanisms • Purely local LMS-type of adaptation • PI consensus loop: tracks the consensus reference

  14. D-LMS in Action True time-varying weight: node WSN, Regressors: i.i.d. Observations: D-LMS:

  15. 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

  16. MSD EMSE Local Global 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

  17. 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 .

  18. Proposition: Under (a1)-(a4), the D-LMS algorithm is MSE stable for sufficiently small Stability and Steady-State Performance Proposition: Under (a1)-(a4), the D-LMS algorithm achieves consensus in the mean, i.e., provided with • 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.

  19. 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).

  20. Available Extensions • Results hold when communication noise is present • Tracking an AR(1) signal vector • Time-correlated, stationary ergodic regressors • Estimation errors are weakly stochastic bounded [Solo’97] • Almost sure exponential stability in the absence of noise • MSE performance analysis via stochastic averaging

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

  22. Distributed RLS Estimation • Motivation: fast convergence, increased complexity affordable • Second-order approach: exponentially-weighted LS (EWLS) estimator • is the `forgetting’ factor. Tracking with • is a regularization matrix (small ) • Equivalent reformulation for connected ad hoc WSN • Solve via AD-MoM

  23. D-RLS Algorithm Step 1: • In the presence of communication noise, for and • Recursively compute • When , updated recursively in operations Step 2:

  24. Remarks • Communication exchanges and cost identical to D-LMS • Cost is , no matrices exchanged • Raw data not exchanged comm. noise resilience • Provides its own regularization can use • Multiplier updates identical to D-LMS • Increased cost in updating local estimates • Cost is for D-LMS • Cost is for D-RLS ( when ) • D-LMS/D-RLS do not require a Hamiltonian cycle

  25. D-RLS in Action D-RLS: Diffusion RLS: Metropolis weights node WSN, Regressors: i.i.d. Observations: Global MSD(t) evolution: Global MSE(t) evolution:

  26. Spectrum Estimation Task node WSN Source: is AR(4) Channels: . Sensors 3, 7, 15 and 27 have a zero at D-LMS estimates (sensor 15) Global MSE(t) evolution:

  27. D-RLS with Ideal Links • Recall: • If and • Local estimate updates simplify to • Introduce • Savings: multipliers not exchanged Step 1: Step 2:

  28. Alternating Minimization Algorithm • Consider the convex separable problem • Lagrangian function: • Augmented Lagrangian: AMA[Tseng ’91]: [S1] [S2] [S3] AD-MoM[Glowinski ‘75]: [S2]

  29. AMA-based D-RLS • Because • Goal: reduce complexity in updating • Setting , then D-RLS = L-RLS • Apply AMA (EWLSE cost strictly convex): • Savings:for all , complexity is unless Step 1: Step 2:

  30. MSE Analysis Preliminaries • Analysis challenging due to: • Finding the distribution of is typically intractable • Resort to simplifying assumptions (a1) Sensor observations obey where the zero-mean white noise has variance (a2) is white with covariance matrix (a3) , , and are independent and approximations for and • Approach:form `averaged’ error-form D-RLS system

  31. Overview of Results Proposition: Under (a1)-(a3) and for , the D-RLS algorithm achieves consensus in the mean, i.e., provided with • As for D-LMS, closed-form recursion for • Approximation only valid for large • Vectorized recursion sufficient condition for MSE stability • Solve for from a fixed-point equation • Enables evaluation of all figures of merit in s.s. • Results account for communication noise

  32. Observations: linear data model, WGN w/ Simulated Tests Regressors: w/ ; i.i.d.; w/ node WSN, Rx AWGN w/ , , D-LMS: , D-RLS: , ,

  33. Concluding Summary • Developed D-LMS/D-RLS algorithms for general ad hoc WSNs • Estimators expressed as separable minimization problems • Detailed stability and MSE performance analysis for D-LMS • Stationary setup, time-invariant parameter vector • Tracking a random-walk/stable AR(1) process • D-RLS: complexity vs. performance tradeoff • Reduced complexity variants • Local and network-wide figures of merit for in s.s. • Ongoing research • Tracking s.s. performance analysis for D-RLS • Distributed lasso for estimation of sparse signals

  34. Related Publications • Journal publications: • I. D. Schizas, G. Mateos and G. B. Giannakis, ``Distributed LMS for Consensus-Based In-Network Adaptive Processing,'' IEEE Transactions on Signal Processing, vol. 57, no. 6, pp. 2365-2381, June 2009. • G. Mateos, I. D. Schizas, and G. B. Giannakis, ``Distributed Recursive Least-Squares for Consensus-Based In-Network Adaptive Estimation,'' IEEE Transactions on Signal Processing, 2009 (to appear) • G. Mateos, I. D. Schizas, and G. B. Giannakis, ``Performance Analysis of the Consensus-Based Distributed LMS Algorithm,'' EURASIP Journal on Advances in Signal Processing, submitted May 2009. • Conference papers: • G. Mateos, I. D. Schizas and G. B. Giannakis, ``Distributed Least-Mean Square Algorithm Using Wireless Ad Hoc Networks,'' Proc. of 45th Allerton Conf., Univ. of Illinois at U-C, Monticello, IL, Sept. 26-28, 2007. • I. D. Schizas, G. Mateos and G. B. Giannakis, ``Distributed Recursive Least-Squares Using Wireless Ad Hoc Sensor Networks,'' Proc. of 41st Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 4-7, 2007. • I. D. Schizas, G. Mateos and G. B. Giannakis, ``Stability analysis of the consensus-based distributed LMS algorithm,'' Proc. of Intl. Conf. on Acoustics, Speech and Signal Processing, Las Vegas, NV, March 30-April 4, 2008. • G. Mateos, I. D. Schizas, and G. B. Giannakis, ``Closed-Form MSE Performance of the Distributed LMS Algorithm,'' Proc. of DSP Workshop, Marco Island, FL, January 4-7, 2009.

  35. Deriving D-LMS • Write constraints as • Augmented Lagrangian • AD-MoM [S1] [S2] [S3]

  36. Deriving D-LMS (cont.) • [S1]-[S3] boil down to: ( redundant) • First order optimality condition • Obtain recursion via Robbins-Monro iteration

More Related