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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, MNJuly 29, 2009

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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

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FC-based WSN Ad hoc WSN

Two Prevailing Topologies

Why ad hoc WSNs? Less power consumption as WSN scales (geographically) Improved robustness to sensor failures

+

= Increased life expectancy of the WSN

Ad hoc WSN

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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

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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

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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)

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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

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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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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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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

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Algorithmic Construction

Consider the convex, constrained optimization

Equivalent for connected WSN:

Two key steps in deriving D-LMS

1) Resort to the AD-MoM [Glowinski ‘75]Gain desired degree of parallelization

2) Apply stochastic approximation ideasCope with unavailability of statistical

information

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D-LMS Recursions and Operation In the presence of communication noise, for and

Reduced communications possible with `bridge’ sensors

Step 1:

Step 2:

Rx from Tx to Rx from Tx to

Step 1: forming Step 2: forming

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Consensus Controller Interpretation Consensus error at sensor :

Local LMS Algorithm

Sensor j

PI RegulatorTo

Consensus Loop

Superposition of two learning mechanisms Purely local LMS-type of adaptation PI consensus loop: tracks the consensus reference

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D-LMS in Action node WSN, Regressors: i.i.d.Observations:

D-LMS:

True time-varying weight:

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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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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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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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Stability and Steady-State 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

with

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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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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

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, 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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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

Distributed RLS Estimation

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D-RLS Algorithm In the presence of communication noise, for and

Recursively compute

When , updated recursively in operations

Step 1:

Step 2:

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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

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D-RLS:

Diffusion RLS: Metropolis weights

D-RLS in Action node WSN, Regressors: i.i.d.Observations:

0 50 100 150 200 250 300

10-4

10-3

10-2

10-1

100

Time index t

Lear

ning

cur

ve

Centralized RLS

D-RLS in ideal links

D-RLS w/ com noise (15 dB)

Diffusion RLS in ideal links

Diffusion RLS w/ com noise (15 dB)

0 50 100 150 200 250 30010

-8

10-6

10-4

10-2

100

102

Time index t

Nor

mal

ized

est

imat

ion

erro

r

Centralized RLS D-RLS, ideal links

Diffusion RLS, ideal links

D-RLS w/ com noise (15 dB)

Diffusion RLS w/ com noise (15 dB)

Global MSE(t) evolution: Global MSD(t) evolution:

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Spectrum Estimation Task node WSNSource: is AR(4)

Channels: . Sensors 3, 7, 15 and 27 have a zero at

D-LMS estimates (sensor 15) Global MSE(t) evolution:

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D-RLS with Ideal Links Recall:

If and

Local estimate updates simplify to

Introduce

Savings: multipliers not exchanged

Step 1:

Step 2:

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Alternating Minimization Algorithm Consider the convex separable problem

Lagrangian function: Augmented Lagrangian:

AD-MoM [Glowinski ‘75]:

AMA [Tseng ’91]: [S1]

[S2]

[S3]

[S2]

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Because

Goal: reduce complexity in updating

Setting , then D-RLS = L-RLS Apply AMA (EWLSE cost strictly convex):

Savings: for all , complexity is

unless

AMA-based D-RLS

Step 1:

Step 2:

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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

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Overview of Results

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

Proposition: Under (a1)-(a3) and for , the D-RLS algorithm achieves consensus in the mean, i.e., provided with

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D-LMS: ,

Simulated Tests node WSN, Rx AWGN w/ , ,

Regressors: w/

; i.i.d.; w/

Observations: linear data model, WGN w/

D-RLS: , ,

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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

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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.

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Deriving D-LMS Write constraints as Augmented Lagrangian

AD-MoM

[S1]

[S3]

[S2]

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Deriving D-LMS (cont.) [S1]-[S3] boil down to: ( redundant)

First order optimality condition

Obtain recursion via Robbins-Monro iteration