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Estimation Bounds for Localization

This paper discusses estimation bounds in localization, ranging from Cramer-Rao Bounds to anchored and anchor-free localization methods, evaluating lower and upper bounds, and exploring different propagation models. It also covers Fisher Information and Cramer-Rao Bound analysis, providing insights into the performance evaluation of localization techniques.

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Estimation Bounds for Localization

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  1. Estimation Bounds for Localization October7th, 2004 Cheng Chang EECS Dept ,UC Berkeleycchang@eecs.berkeley.edu Joint work with Prof. AnantSahai (part of BWRC-UWB project funded by the NSF) IEEE SECON 2004

  2. 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 IEEE SECON 2004

  3. 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 IEEE SECON 2004

  4. 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 IEEE SECON 2004

  5. 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,jtrue+ ni,j (ni,j is modeled as iid Gaussian throughout most of the talk) • Parameters to be estimated : IEEE SECON 2004

  6. 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 IEEE SECON 2004

  7. 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 : IEEE SECON 2004

  8. 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 and Future Work IEEE SECON 2004

  9. 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. IEEE SECON 2004

  10. Properties of CRB for Anchored Localization • The standard Cramer-Rao bound analysis works. (FIM nonsingular in general) • V(xi)=J(θ) -12i-1,2i-1and V(yi) =J(θ) -12i,2iare 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 IEEE SECON 2004

  11. 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 IEEE SECON 2004

  12. 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 IEEE SECON 2004

  13. 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. IEEE SECON 2004

  14. 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. IEEE SECON 2004

  15. Lower bound: how good can you do? • 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) IEEE SECON 2004

  16. Upper Bound: what’s the best you can do with local information. • Anupper bound on the Cramer-Rao bound . • Using partial information can only make the estimation less accurate. IEEE SECON 2004

  17. Upper Bound: what’s the best you can do with local information. IEEE SECON 2004

  18. 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 IEEE SECON 2004

  19. 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 θαis an equivalent class θα={β |β={R(xi ,yi)T+(a,b) | i ∊ S} } IEEE SECON 2004

  20. 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 invariantunder rotation, translation and zooming on the whole sensor network. • The error between θand is defined as • Total estimation bounds IEEE SECON 2004

  21. Estimation Bound on Anchor-free Localization • The number of the nodes doesn’t matter • The shape of the sensor network affects thetotal estimation bound. • Nodes are uniformly distributed in a rectangular region (R=L1/L2) • All inter-node distances are measured IEEE SECON 2004

  22. 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 IEEE SECON 2004

  23. 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 IEEE SECON 2004

  24. Cramer-Rao Bounds on Localization in Different Propagation Models • 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: IEEE SECON 2004

  25. Cramer-Rao Bounds on Localization in Different Propagation Models • PR converges for a>2 , diverges for a≤2. • Consistent with the CRB (anchor-free). IEEE SECON 2004

  26. 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 IEEE SECON 2004

  27. 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. IEEE SECON 2004

  28. Some open questions • Noise model • Correlated ranging noises (interference) • Non-Gaussian ranging noises • Achievability • Bottleneck of localization • Sensitivity to a particular measurement • Energy allocation IEEE SECON 2004

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