Localization from Mere Connectivity

1. Localization from Mere Connectivity Yi Shang, Ying Zhang, and Wheeler Ruml Presented By Tu Tran

2. Outline • Introduction • Contribution • Multidimensional Scaling (MDS) • MDS-MAP Algorithm • Experimental Results • Conclusions

3. Introduction • Node localization – a topic of active research in WSN • Tracking or even-detection applications need to know where events have occurred. • Problems: • Manually configuring location: not a good choice. • Global Positioning System (GPS): not a good option – cost and deployment limitations

4. Contribution • Presenting the algorithm (MDS-MAP) for computing the positions of nodes given connectivity information (which nodes are within communication range of which others) • MDS-MAP can incorporate distance information between neighboring nodes. • More robust to measurement error and produce relative coordinates (anchor nodes not available) than previous proposals.

5. Multidimensional Scaling (MDS) • A set of data analysis techniques that display the structure of distance-like data as a geometrical picture. • To approximate the distances between pairs of the objects. • Each object or event is represented by a point in a multidimensional space. • Classical MDS (CMDS) is that there is only one similarity matrix.

6. MDS Cont’d • CMDS employs Euclidean distance to model dissimilarity. • The distance dij between points i and j is defined as • dij is related to the proximity by a transformation dij = f(pij ) • In CMDS, dij = a + bpij

7. MDS Cont’d • Adapt this concept, they define I(P) = D + E • I(P): a linear transformation of the proximities • E: matrix of errors • D: function of the coordinates X • Calculate X: the sum of squares of E is minimized. • In CMDS, P is shifted to the center and X can be computed from the double centered P by performing singular value decomposition.

8. MDS Cont’d • Performing singular value decomposition gives • Node localization problem is classified into two different cases: • Only proximity (connectivity) information is available (knowing nodes nearby). • Knowing the distances.

9. MDS Cont’d • Both cases: • The network is represented as an undirected graph with vertices V and edges E. • V correspond to the nodes • E associated with connectivity information • Assume that there is a path between every pair of nodes.

10. MDS-MAP Algorithm • 1. Compute shortest path between all pairs of nodes. The shortest path distances are used to construct the distance matrix for MDS • 2. Apply CMDS to the distance matrix and retain the first 2 or 3 largest eigenvalues & eigenvectors to construct 2-D or 3-D relative map. • 3. Given sufficient anchor nodes ( 3 or more 2-D, 4 or more 4-D) transform the relative map to an absolute map based on the absolute positions of anchors.

11. Experimental Results

12. Experimental Results Cont’d

13. Experimental Results Cont’d

14. Experimental Results Cont’d • Position estimates by MDS-MAP have an average error under 100%R in scenarios with just 4 anchor nodes and an average connectivity level of 8.9 or greater. • When the connectivity level is 12.2 or greater, the errors with just 3 anchors is close to or better than 50%. • Compare with MDS-MAP, the convex optimization approach has twice an average estimation error.

15. Conclusions • Pros: • Introduce a new localization method (MDS-MAP) that works well with mere connectivity information. • MDS-MAP can be extended with other MDS techniques (ordinal MDS). • MDS-MAP can be used get a good initial estimates of node positions. • Better performance when few anchor nodes available comparing with previous methods.

16. Conclusions Cont’d • Cons: • Global connectivity information needs to be obtained first. • Centralized approach, not distributed. • If number of anchor nodes is large, MDS-MAP does not perform well compared with other methods (constraint-based approach).

17. Questions Questions or Comments? Thank You!