1. Localization from Mere Connectivity Yi Shang (University of Missouri - Columbia); Wheeler Ruml (Palo Alto Research Center ); Ying Zhang; Markus Fromherz ACM Symposium on Mobile Ad Hoc Networking and Computing 2003 (MobiHoc'03)

2. Outline  Classic Multidimensional Scaling (MDS)  Localization using MDS  Experiment Results  Random placement  Grid placement

3. Multidimensional Scaling  What is MDS?  Given a set of proximity measure d i j for m objects, find a placement of the points( compute coordinates) in p (2 or 3 ) dimensions for the objects, such that the Euclidean distances between objects fit the given proximity measurements.

4. Multidimensional Scaling  Let the coordinates of n points in a p -dimensional Euclidean be X=(x 1, x 2,..,x n ), where x i = (x i1, x i2,..,x ip ).  proof

5. Multidimensional Scaling  B is symmetric matrix.  If B is positive semi-definite of rank p, then a configuration of points in p dimensional space can be found so that the given distances match the computed distance i.e., d ij =  ij  [Proofs are presented in de Leeuw and Heiser (1982) and Marda et al. (1979)]

6. Multidimensional Scaling- Examples P Q R P Q R 0 2 5 2 0 4 5 4 0 Given Proximity Measurements

7. Multidimensional Scaling- Examples O PQ 2 5 4 Compute coordinates for the objects

8. Algorithm - Only Connectivity information is available 1. Compute all-pairs shortest paths (hop count) to roughly estimate the distance between all pairs of nodes. The shortest path distances are used to construct the distance matrix for MDS. 2. Apply classical MDS to the distance matrix, retaining the largest 2 (or 3) largest eigenvalues and eigenvectors to construct a 2-D (or 3-D) relative map. 3. Given sufficient anchor nodes (3 or more for 2-D, 4 or more for 3-D), transform the relative map to an absolute map based on the absolute positions of anchors.

9. Algorithm - The distances with limited accuracy between neighbor nodes are known 1. Compute all-pairs shortest paths (estimated distances) to roughly estimate the distance between all pairs of nodes. The shortest path distances are used to construct the distance matrix for MDS. 2. Apply classical MDS to the distance matrix, retaining the largest 2 (or 3) largest eigenvalues and eigenvectors to construct a 2-D (or 3-D) relative map. 3. Given sufficient anchor nodes (3 or more for 2-D, 4 or more for 3-D), transform the relative map to an absolute map based on the absolute positions of anchors.

10. Experimental Results  Scenario 1:  200 nodes randomly placed in a 10 r  10 r square area, where R is radio range.

11. Experimental Results (Random Placement)  Random uniform placement using  connectivity only (left) or the distance measures between neighboring nodes with 5% errors (right).  The same four random anchors are used and the position estimation errors are 0.67r and 0.25r, respectively. Connectivity onlyDistance measure Anchor node

12. Experimental Results (Random C-Shaped Placement)  Scenario 2  160 nodes are randomly placed in an area of C shape within a 10 r  10 r square

13. Experimental Results (Random C-Shaped Placement)  connectivity only (left) or the distance measures between neighboring nodes with 5% errors (right).  The same four random anchors are used and the position estimation errors are 2.4r and 2.3r, respectively. Connectivity onlyDistance measure Anchor node

14. Experimental Results (Grid Placement)  Scenario 3:  grid placement – 100 nodes are placed on a grid with10% r placement errors. placement error

15. Experimental Results (Grid Placement)  connectivity only (left) or the distance measures between neighboring nodes with 5% errors (right).  The same four random anchors are used and the position estimation errors are 0.42r and 0.17r, respectively. Connectivity only Distance measure Anchor node

16. Experimental Results (Grid C-Shaped Placement)  Scenario 4  79 nodes are placed on a C shape grid with 10%r placement errors.

17. Experimental Results (Grid C-Shaped Placement)  connectivity only (left) or the distance measures between neighboring nodes with 5% errors (right).  The same four random anchors are used and the position estimation errors are 2.1 for both cases. Anchor node

18. Average Position Error V.S Connectivity Using proximity information only

19. Average Position Error V.S Connectivity Using distances between neighbors (5% range error)

20. Conclusion  This paper proposed a new method called, MDS-MAP  MDS-MAP builds a relative map of the nodes without anchor nodes. With three or more anchor nodes, the relative map can be transformed into absolute coordinates.

21. Related Papers  Yi Shang, Wheeler Ruml, INFOCOM 2004  Improved MDS-based Localization  Xiang Ji (Penn State University ), Hongyuan Zha, INFOCOM 2004  Sensor Positioning in Wireless Ad-hoc Sensor Networks with Multidimensional Scaling

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