1

Distortion-Rate for Non-Distributed and Distributed Estimation with WSNs

Presenter: Ioannis D. Schizas

May 5, 2005May 5, 2005EE8510 ProjectEE8510 Project PresentationPresentation

Acknowledgements: Profs. G. B. Giannakis and N. Jindal

2

Motivation and Prior WorkMotivation and Prior Work

Energy/Bandwidth constraints in WSN call for efficient compression-encoding

Bounds on minimum achievable distortion under prescribed rate important for:

Compressing and reconstructing sensor observations

• Best known inner and outer bounds in [Berger-Tung’78]

• Iterative determination of achievable D-R region [Gastpar et. al’04]

Estimating signals (parameters) under rate constraints

• The CEO problem

[Viswanathan et. al’97, Oohama’98, Chen et. al’04, Pandya et. al’04]

• Rate-constrained distributed estimation [Ishwar et. al’05]

3

Problem StatementProblem Statement

Linear Model:

s, n uncorrelated and Gaussian

and.

.

is known and full column rank.

Goal: Determine D-R function or more strict achievable D-R regions than obvious upper bounds when estimating s under rate constraints.

4

Point-to-Point Link (Single-Sensor)Point-to-Point Link (Single-Sensor)

Two non-distributed encoding options

Estimation errors

i. Compress-Estimate (C-E)

ii. Estimate-Compress (E-C)

= f (terms due to compression), =1,2

5

E-C outperforms C-EE-C outperforms C-E

Special Cases:

• Scalar case:

• Vector case (p=1): If

If

, then

• Matrix case:

Theorem 1:

, then

similar ‘threshold rates’ for which

6

Optimality of Estimate-CompressOptimality of Estimate-Compress

Extends the result in [Sakrison’68, Wolf-Ziv’70] in linear models & N>p.

Theorem 2:

7

0 10 20 30 400

0.5

1

1.5

2

2.5

3

Rate

Dis

tort

ion

SNR=2

0 10 20 30 400

0.5

1

1.5

2

2.5

3

Rate

Dis

tort

ion

SNR=4

C-E schemeE-C scheme

De0

Numerical ResultsNumerical Results

and

EC converges faster than CE to the D-R lower bound

8

Distributed SetupDistributed Setup

Desirable D-R

Treat as side info. with and

MMSE and

Let

Optimal output of encoder 1:

and

9

Distributed E-CDistributed E-C

Extends [Gastpar,et.al’04] to the estimation setup

Steps of iterative algorithm:

Initialize assuming each sensor works independently

Create M random rate increments r(i) s.t.

During iteration j:

• Retain pair of matrices with smallest distortion

Convergence to a local minimum is guaranteed

,

•Assign r(i) to the corresponding encoder

•Determine

10

Numerical ExperimentNumerical Experiment

SNR=2,

and

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5

Rate

Dis

tort

ion

D-R function for the estimation of the parameter s, SNR=2

Marginal

DistributedJoint

Distributed E-C yields tighter upper bound for D-R than the marginal E-C

11

ConclusionsConclusions

Comparison of two encoders for estimation from a D-R perspective

D-R function for the single-sensor non-distributed setup

Optimality of the estimate-first & compress-afterwards option

Numerical determination of an achievable D-R region, or, at best the D-R function for distributed estimation with WSNs

Thank You!