Template for MCMA Poster Slides

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Estimation Bounds for Localization
October 7th, 2004 Cheng Chang EECS Dept ,UC
Berkeley cchang_at_eecs.berkeley.edu Joint work
with Prof. Anant Sahai (part of BWRC-UWB project
funded by the NSF)
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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

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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

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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

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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

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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 ?(xi ,yi)T i ? S is a parameter vector,
  ?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

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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
  Patterson02)
• Cramer-Rao Lower bound (CRB)
• For any unbiased estimator

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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

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FIM for localization (a geometric interpretation)

• FIM of the Localization Problem (anchored and
  anchor-free)
• ni,j are modeled as iid Gaussian N(0,s2).
• Let ?(x1,y1,xm,ym) be the parameter vector, 2m
  parameters.

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Properties of CRB for Anchored Localization

• The standard Cramer-Rao bound analysis works.
  (FIM nonsingular in general)
• V(xi)J(?) -1 2i-1,2i-1 and V(yi) J(?) -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(?) J(?) , if (xi,yi)(xi ,yi)(Tx ,Ty)
• V(xi)V(yi)V(xi)V(yi), if (xi,yi)(xi
  ,yi)R, where RRTI2
• J(?) J(a?) , if a ? 0

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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

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Local lower bounds how good can you do?

• A lower bound on the Cramer-Rao bound , write ?l
  (xl ,yl)

• Jl is a 22 sub-matrix of J(?) . Then for any
  unbiased estimator , E(( -?l) T( -?l))
  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 (Wadj(l)), then

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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.

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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.

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Lower bound how good can you do?

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

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Upper Bound whats 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.

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Upper Bound whats the best you can do with
local information.
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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

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Equivalent class in the anchor-free localization

• If a(xi ,yi)T i ? S , ßR(xi ,yi)T(a,b) i
  ? S is equivalent to a, where RRT I2 .
• Same inter-node disances
• A parameter vector ?a is an equivalent class
  ?a ß ßR(xi ,yi)T(a,b) i ?
  S

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Estimation Bound on Anchor-free Localization

• The Fisher Information Matrix J(?) is singular
  (Moses and Patterson02)
• m nodes with unknown position
• J(?) has rank 2m-3 in general
• J(?) has 2m-3 positive eigenvalues ?i, i1,2m-3,
  and they are invariant under rotation,
  translation and zooming on the whole sensor
  network.
• The error between ? and is defined as
• Total estimation bounds

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Estimation Bound on Anchor-free Localization

• The number of the nodes doesnt matter
• The shape of the sensor network affects the total
  estimation bound.
• Nodes are uniformly distributed in a rectangular
  region (RL1/L2)
• All inter-node distances are measured

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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

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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

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Cramer-Rao Bounds on Localization in Different
Propagation Models

• So far, have assumed that the noise variance is
  constant s2.
• 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(c?) J(?)/ca, so the Cramer-Rao bound on the
  estimation of a single node ca V(xi), ca V(yi).
• Received power per node

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Cramer-Rao Bounds on Localization in Different
Propagation Models

• PR converges for agt2 , diverges for a2.
• Consistent with the CRB (anchor-free).

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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

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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.
• Its possible to compute bounds locally.

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Some open questions

• Noise model
• Correlated ranging noises (interference)
• Non-Gaussian ranging noises
• Achievability
• Bottleneck of localization
• Sensitivity to a particular measurement
• Energy allocation