Estimation Bounds for Localization

1

IEEE SECON 2004

Estimation Bounds for Localization

October 7th, 2004

Cheng Chang

EECS Dept ,UC Berkeley [email protected]

Joint work with Prof. Anant Sahai

(part of BWRC-UWB project funded by the NSF)

2

IEEE SECON 2004

Outline

Introduction– Range-based Localization as an Estimation Problem

– Cramer-Rao Bounds (CRB)

Estimation Bounds on Localization– Properties of CRB on range-based localization

– Anchored Localization (3 or more nodes with known positions) Lower Bounds Upper Bounds

– Anchor-free Localization

– Different Propagation Models

Conclusions

3

IEEE SECON 2004

Localization Overview

What is localization?– Determine positions of the nodes (relative or absolute)

Why is localization important?– Routing, sensing etc

Information available– Connectivity

– Euclidean distances and angles

– Euclidean distances (ranging) only

Why are bounds interesting– Local computable

4

IEEE SECON 2004

Range-based localization

Range-based Localization– Positions of nodes in set F (anchors, beacons) are known, positions of

nodes in set S are unknown

– Inter-node distances are known among some neighbors

5

IEEE SECON 2004

Localization is an estimation problem

Knowledge of anchor positions

Range observations:– adj(i) = set of all neighbor nodes of node I

– di,j : distance measurement between node i and j

– di,j = di,j true+ ni,j (ni,j is modeled as iid Gaussian throughout most of the talk)

Parameters to be estimated :

6

IEEE SECON 2004

Anchored vs Anchor-free

Anchored localization ( absolute coordinates)– 3 or more anchors are needed

– The positions of all the nodes can be determined.

Anchor-free localization (relative coordinates)– No anchors needed

– Only inter-node distance measurements are available.

– If θ={(xi ,yi)T| i є S} is a parameter vector, θ*={R(xi ,yi)T+(a,b) | i ∊ S} is an equivalent parameter vector, where RRT =I2 .

Performance evaluation– Anchored: Squared error for individual nodes

– Anchor-free: Total squared error

7

IEEE SECON 2004

Fisher Information and the Cramer-Rao Bound

Fisher Information Matrix (FIM)– Fisher Information Matrix (FIM) J provides a tool to compute the best

possible performance of all unbiased estimators

– Anchored: FIM is usually non-singular.

– Anchor-free: FIM is always singular (Moses and Patterson’02)

Cramer-Rao Lower bound (CRB)– For any unbiased estimator :

, ( ) cov log ( | ), log ( | )i ji j

J p x p x

1)())ˆ)(ˆ(( JE T

8

IEEE SECON 2004

Outline







Conclusions and Future Work

9

IEEE SECON 2004

FIM for localization (a geometric interpretation)

FIM of the Localization Problem (anchored and anchor-free)– ni,j are modeled as iid Gaussian ~N(0,σ2).

– Let θ=(x1,y1,…xm,ym) be the parameter vector, 2m parameters.

10

IEEE SECON 2004

The standard Cramer-Rao bound analysis works. (FIM nonsingular in general)

– V(xi)=J(θ) -1 2i-1,2i-1 and V(yi) =J(θ) -1 2i,2i are the Cramer-Rao bound on the coordinate-estimation of the i th node.

– CRB is not local because of the inversion.

Translation, rotation and zooming, do not change the bounds.

– J(θ)= J(θ*) , if (x*i,y*i)=(xi ,yi)+(Tx ,Ty)

– V(xi)+V(yi)=V(x*i)+V(y*i), if (x*i,y*i)=(xi ,yi)R, where RRT=I2

– J(θ)= J(aθ) , if a ≠ 0

Properties of CRB for Anchored Localization

11

IEEE SECON 2004

Outline







Conclusions

12

IEEE SECON 2004

Local lower bounds: how good can you do?

A lower bound on the Cramer-Rao bound , write θl =(xl ,yl)

Jl is a 2×2 sub-matrix of J(θ) . Then for any unbiased estimator , E(( -θl) T( -θl))≥ Jl

-1

Jl only depends on (xl ,yl) and (xi ,yi) , i ∊ adj(l) so we can give a performance bound on the estimation of (xl ,yl) using only the geometries of sensor l 's neighbors.

Sensor l has W neighbors (W=|adj(l)|), then

ll l

13

IEEE SECON 2004

Lower bound: how good can you do?

Jl is the FIM of another estimation problem of (xl ,yl): knowing the positions of all neighbors and inter-node distance measurements between node l and its neighbors.

14

IEEE SECON 2004


Jl is the FIM of another estimation problem of (xl ,yl): knowing the positions of all neighbors and inter-node distance measurements between node l and its neighbors.

15

IEEE SECON 2004


Jl is the ‘one-hop’ sub-matrix of J(θ), Using multiple-hop sub-matrices , we can get tighter bounds.(figure out the computations of it)

16

IEEE SECON 2004

Upper Bound: what’s the best you can do with local information.

An upper bound on the Cramer-Rao bound .– Using partial information can only make the estimation less

accurate.

17

IEEE SECON 2004

Upper Bound: what’s the best you can do with local information.

18

IEEE SECON 2004

Outline







Conclusions

19

IEEE SECON 2004

Equivalent class in the anchor-free localization

If α={(xi ,yi)T| i є S} , β={R(xi ,yi)T+(a,b) | i ∊ S} is equivalent to α, where RRT =I2 .– Same inter-node disances

A parameter vector θα is an equivalent class θα ={β |β={R(xi ,yi)T+(a,b) | i ∊ S} }

20

IEEE SECON 2004

Estimation Bound on Anchor-free Localization

The Fisher Information Matrix J(θ) is singular (Moses and Patterson’02)

m nodes with unknown position:

– J(θ) has rank 2m-3 in general

– J(θ) has 2m-3 positive eigenvalues λi, i=1,…2m-3, and they are invariant under rotation, translation and zooming on the whole sensor network.

The error between θ and is defined as

Total estimation bounds

2

2',inf)ˆ,( D

)(1

))ˆ,((32

1

VDEm

i i

21

IEEE SECON 2004

Estimation Bound on Anchor-free Localization

The number of the nodes doesn’t matter

The shape of the sensor network affects the total estimation bound.

– Nodes are uniformly distributed in a rectangular region (R=L1/L2)

– All inter-node distances are measured

22

IEEE SECON 2004

To Anchor or not to Anchor To give absolute positions to the nodes is more challenging.

Bad geometry of anchors results in bad anchored-localization.

– 195.20 vs 4.26

23

IEEE SECON 2004

Outline







Conclusions

24

IEEE SECON 2004

So far, have assumed that the noise variance is constant σ2.

Physically, the power of the signal can decay as 1/da

Consequences:

– Rotation and Translation still does not change the Cramer-Rao bounds V(xi)+V(yi)

– J(cθ) =J(θ)/ca, so the Cramer-Rao bound on the estimation of a single node : ca V(xi), ca V(yi).

Received power per node:

Cramer-Rao Bounds on Localization in Different Propagation Models

25

IEEE SECON 2004

PR converges for a>2 , diverges for a≤2.

Consistent with the CRB (anchor-free).

Cramer-Rao Bounds on Localization in Different Propagation Models

26

IEEE SECON 2004

Outline







Conclusions

27

IEEE SECON 2004

Conclusions

Implications on sensor network design:– Bad local geometry leads to poor localization performance.

– Estimation bounds can be lower-bounded using only local geometry.

Implications on localization scheme design: – Distributed localization might do as well as centralized

localization.

– Using local information, the estimation bounds are close to CRB.

Localization performance per-node depends roughly on the received signal power at that node.

It’s possible to compute bounds locally.

28

IEEE SECON 2004

Some open questions

Noise model

– Correlated ranging noises (interference)

– Non-Gaussian ranging noises

Achievability

Bottleneck of localization

– Sensitivity to a particular measurement

– Energy allocation

Documents

Estimation Bounds for Localization