A comparison of non-linear regression and MDS-MAP
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The document compares the localization algorithms MDS-MAP and non-linear regression (NLR) using wireless sensor network experimental data. It describes the background of the localization problem, provides details of the MDS-MAP, MDS-MAP(P) and NLR algorithms, and explains the experimental setup using static sensor nodes with ranging and angle of arrival measurements. The results section shows that NLR achieved the lowest localization errors compared to the other algorithms in single-hop network scenarios.
![A Comparison of
MDS-MAP[1] Non-linear Regression
and MDS-MAP
C. Ellis, M. Hazas
Background
Algorithms
MDS-MAP
Uses the techniques of multidimensional scaling. MDS-MAP(P)
Non-linear Regression
Reduces a n × n dissimilarity matrix into a 2n coordinate Experimental Setup
system (x : y ) Results
Builds a dissimilarity matrix containing every node-by-node Discussion
measurement. Future Work
Conclusion
Fills in missing data using a shortest-path distance metric.
Centralised algorithm.
[1] Y. Shang, W. Ruml, Y. Zhang, and M.P.J. Fromherz.
Localization from mere connectivity. In Proceedings of the 4th
ACM international symposium on Mobile ad hoc networking &
computing, pages 201 - 212. ACM New York, NY, USA, 2003.
7/39](https://image.slidesharecdn.com/pres-100927040315-phpapp02/85/A-comparison-of-non-linear-regression-and-MDS-MAP-7-320.jpg)
![A Comparison of
MDS-MAP(P)[2] Non-linear Regression
and MDS-MAP
C. Ellis, M. Hazas
Background
Algorithms
MDS-MAP
MDS-MAP(P)
A distributed version of MDS-MAP. Non-linear Regression
Experimental Setup
Uses local neighbourhood dissimilarities rather than whole
Results
network.
Discussion
MDS-MAP ran on each patch of network, then the patches Future Work
are stitched together. Conclusion
Scales linearly with network size, cubicly with neighbourhood
density.
[2]Y. Shang and W. Ruml. Improved MDS-based localization. In
IEEE INFOCOM, volume 4, pages 2640 - 2651. INSTITUTE OF
ELECTRICAL ENGINEERS INC (IEEE), 2004
8/39](https://image.slidesharecdn.com/pres-100927040315-phpapp02/85/A-comparison-of-non-linear-regression-and-MDS-MAP-8-320.jpg)
![A Comparison of
MDS-MAP(P) Non-linear Regression
and MDS-MAP
C. Ellis, M. Hazas
Background
Steps of the algorithm: Algorithms
MDS-MAP
1. Measure pseudo ranges (or other metric) for local MDS-MAP(P)
neighbourhood. Non-linear Regression
Experimental Setup
1.1 Perform MDS-MAP on local dissimilarity matrix.
Results
1.2 Refine with least squares non-linear regression (LSQNONLIN
Discussion
in MATLAB, mrqmin() in Numerical Recipes[3]).
Future Work
2. Stitch together with other local maps Conclusion
3. (Optional) Refine global map with least squares non-linear
regression.
4. Translate to absolute coordinate system using anchors, if
available.
[3] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P.
Flannery. Numerical recipes in C: the art of scientific computing.
Cambridge Univ Pr, 1992.
9/39](https://image.slidesharecdn.com/pres-100927040315-phpapp02/85/A-comparison-of-non-linear-regression-and-MDS-MAP-9-320.jpg)
![A Comparison of
MDS-MAP(P) - Refinement Non-linear Regression
and MDS-MAP
C. Ellis, M. Hazas
Background
Steps of the algorithm:
Algorithms
1. Measure pseudo ranges (or other metric) for local MDS-MAP
MDS-MAP(P)
neighbourhood. Non-linear Regression
1.1 Perform MDS-MAP on local dissimilarity matrix. Experimental Setup
1.2 Refine with least squares non-linear regression Results
(LSQNONLIN in MATLAB, mrqmin() in Numerical Discussion
Recipes[3]). Future Work
Conclusion
2. Stitch together with other local maps
3. (Optional) Refine global map with least squares non-linear
regression.
4. Translate to absolute coordinate system using anchors, if
available.
[3] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P.
Flannery. Numerical recipes in C: the art of scientific computing.
Cambridge Univ Pr, 1992.
14/39](https://image.slidesharecdn.com/pres-100927040315-phpapp02/85/A-comparison-of-non-linear-regression-and-MDS-MAP-14-320.jpg)
![A Comparison of
Experimental Setup Non-linear Regression
and MDS-MAP
C. Ellis, M. Hazas
Three Scenarios: Background
Single-hop network Algorithms
MDS-MAP
Using datasets from the above experiments. MDS-MAP(P)
Non-linear Regression
Multi-hop network Experimental Setup
Using simulated datasets created using the real data’s node Results
link error distributions. Discussion
Future Work
Single-hop network with consistently over ranging link.
Conclusion
Using a real dataset and artificially over ranging a certain
node-node measure.
To test robustness in the presence of systematic errors
common in WSN.
As highlighted by Whitehouse and Culler[4] in their
experiments investigating systematic error in WSN.
[4] K. Whitehouse and D. Culler. A robustness analysis of
multi-hop ranging-based localization approximations. In
Proceedings of the 5th international conference on Information
processing in sensor networks. ACM, 2006.
19/39](https://image.slidesharecdn.com/pres-100927040315-phpapp02/85/A-comparison-of-non-linear-regression-and-MDS-MAP-19-320.jpg)
A comparison of non-linear regression and MDS-MAP
- 1. A Comparison of Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP A Comparison of Non-linear Regression and MDS-MAP(P) Non-linear Regression MDS-MAP Experimental Setup Results Discussion Future Work Carl Ellis Mike Hazas Conclusion School of Computing and Communications Lancaster University Lancaster, UK September 16, 2010 1/39
- 2. A Comparison of Overview Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Background Algorithms MDS-MAP MDS-MAP(P) Algorithms Non-linear Regression MDS-MAP Experimental Setup MDS-MAP(P) Results Non-linear Regression Discussion Future Work Experimental Setup Conclusion Results Discussion Future Work Conclusion 2/39
- 3. A Comparison of Overview Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Background Algorithms MDS-MAP MDS-MAP(P) Algorithms Non-linear Regression MDS-MAP Experimental Setup MDS-MAP(P) Results Non-linear Regression Discussion Future Work Experimental Setup Conclusion Results Discussion Future Work Conclusion 3/39
- 4. A Comparison of Background of the Problem Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms A common problem with wireless sensor networks (WSNs) is MDS-MAP MDS-MAP(P) localisation. Non-linear Regression GPS is too expensive and power hungry for small sensor Experimental Setup nodes, and is fairly inaccurate indoors. Results Localisation has been comprehensively covered in the WSN Discussion literature. Future Work Conclusion The problem becomes one of reducing many (up to n2 ) node-to-node measurements into a single, global coordinate system (Graph Reduce). As with all algorithms for wireless sensor networks, a number of design objectives exist: Low complexity Low communication overhead 4/39
- 5. A Comparison of Overview Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Background Algorithms MDS-MAP MDS-MAP(P) Algorithms Non-linear Regression MDS-MAP Experimental Setup MDS-MAP(P) Results Non-linear Regression Discussion Future Work Experimental Setup Conclusion Results Discussion Future Work Conclusion 5/39
- 6. A Comparison of Common WSN Localisation Algorithms Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP Bounding Box MDS-MAP(P) Non-linear Regression Builds constraints upon the possible location of a static node Experimental Setup based upon ranging data. Results Robust Quads Discussion Future Work Uses trilateration techniques to gain locations, uses robust Conclusion quads to ensure no flip ambiguities. MDS-MAP The WSN community have embraced MDS-MAP as the ”go-to” algorithm. This presentation hopes to highlight the disadvantages of the above algorithm compared to over methods. 6/39
- 7. A Comparison of MDS-MAP[1] Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP Uses the techniques of multidimensional scaling. MDS-MAP(P) Non-linear Regression Reduces a n × n dissimilarity matrix into a 2n coordinate Experimental Setup system (x : y ) Results Builds a dissimilarity matrix containing every node-by-node Discussion measurement. Future Work Conclusion Fills in missing data using a shortest-path distance metric. Centralised algorithm. [1] Y. Shang, W. Ruml, Y. Zhang, and M.P.J. Fromherz. Localization from mere connectivity. In Proceedings of the 4th ACM international symposium on Mobile ad hoc networking & computing, pages 201 - 212. ACM New York, NY, USA, 2003. 7/39
- 8. A Comparison of MDS-MAP(P)[2] Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) A distributed version of MDS-MAP. Non-linear Regression Experimental Setup Uses local neighbourhood dissimilarities rather than whole Results network. Discussion MDS-MAP ran on each patch of network, then the patches Future Work are stitched together. Conclusion Scales linearly with network size, cubicly with neighbourhood density. [2]Y. Shang and W. Ruml. Improved MDS-based localization. In IEEE INFOCOM, volume 4, pages 2640 - 2651. INSTITUTE OF ELECTRICAL ENGINEERS INC (IEEE), 2004 8/39
- 9. A Comparison of MDS-MAP(P) Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Steps of the algorithm: Algorithms MDS-MAP 1. Measure pseudo ranges (or other metric) for local MDS-MAP(P) neighbourhood. Non-linear Regression Experimental Setup 1.1 Perform MDS-MAP on local dissimilarity matrix. Results 1.2 Refine with least squares non-linear regression (LSQNONLIN Discussion in MATLAB, mrqmin() in Numerical Recipes[3]). Future Work 2. Stitch together with other local maps Conclusion 3. (Optional) Refine global map with least squares non-linear regression. 4. Translate to absolute coordinate system using anchors, if available. [3] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical recipes in C: the art of scientific computing. Cambridge Univ Pr, 1992. 9/39
- 10. A Comparison of Non-linear Regression Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Non-linear regression is a textbook algorithm for finding a Results solution from non-linear and noisy data. Discussion Future Work Observational data is modelled by a given function tailored to Conclusion an application. A solution is found iteratively by using successive approximations. 10/39
- 11. A Comparison of Non-linear Regression Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Results Non-linear regression requires: Discussion A set of observations (e.g. node-to-node distances) Future Work Initial conditions Conclusion Modelling functions which relate observations to parameters to be estimated (e.g. location coordinates, orientations) 11/39
- 12. A Comparison of Non-linear Regression Non-linear Regression and MDS-MAP In our implementation of NLR, the modelling functions used are: C. Ellis, M. Hazas Background min ( (xj − xi )2 + (yj − yi )2 − ri,j )2 (1) Algorithms i<j<n MDS-MAP MDS-MAP(P) yj − yi Non-linear Regression min (atan( ) − θi − Φij )2 (2) Experimental Setup xj − xi i<j<n Results Discussion transmitter Future Work Conclusion receiver 12/39
- 13. A Comparison of Non-linear Regression Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup In our experiments three implementations of non-linear Results regression were compared: Discussion (NLR) Standard NLR using studentized residual analysis to Future Work remove outliers. Conclusion (NLR with angles) Angle of Arrival (AOA) measurements were also used as observations (NLR no elimination) Studentized residual analysis was not performed. 13/39
- 14. A Comparison of MDS-MAP(P) - Refinement Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Steps of the algorithm: Algorithms 1. Measure pseudo ranges (or other metric) for local MDS-MAP MDS-MAP(P) neighbourhood. Non-linear Regression 1.1 Perform MDS-MAP on local dissimilarity matrix. Experimental Setup 1.2 Refine with least squares non-linear regression Results (LSQNONLIN in MATLAB, mrqmin() in Numerical Discussion Recipes[3]). Future Work Conclusion 2. Stitch together with other local maps 3. (Optional) Refine global map with least squares non-linear regression. 4. Translate to absolute coordinate system using anchors, if available. [3] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical recipes in C: the art of scientific computing. Cambridge Univ Pr, 1992. 14/39
- 15. A Comparison of MDS-MAP(P) - Refinement Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Results Discussion Future Work Conclusion 15/39
- 16. A Comparison of MDS-MAP(P) - Refinement Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) MDS-MAP(P) mandates non-linear regression for a Non-linear Regression refinement stage. Experimental Setup Performing a comparison of MDS-MAP(P) vs NLR would not Results be valid. Discussion Future Work The output of MDS-MAP(P) is the output of NLR with the Conclusion dissimilarity matrix as observations, MDS-MAP output as initial coordinates. Essentially comparing NLR with different forms of inputs. Therefore MDS-MAP used in the comparisons instead. Network size is not large enough to warrant MDS-MAP(P) anyway. 16/39
- 17. A Comparison of Overview Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Background Algorithms MDS-MAP MDS-MAP(P) Algorithms Non-linear Regression MDS-MAP Experimental Setup MDS-MAP(P) Results Non-linear Regression Discussion Future Work Experimental Setup Conclusion Results Discussion Future Work Conclusion 17/39
- 18. A Comparison of Experimental Setup Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms Used experimental data from cameras MDS-MAP static nodes in a small arena MDS-MAP(P) Non-linear Regression Each node contains 4 Experimental Setup ultrasound sensors and can Results obtain ranging and AOA Discussion static nodes Future Work measures mobile nodes Conclusion Arena size of 2.75m × 2.00m Mixture of 15 node and 5 node experiments 10 layouts, 5 experiments for each layout 10k - 16k measurements in each experiment 18/39
- 19. A Comparison of Experimental Setup Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Three Scenarios: Background Single-hop network Algorithms MDS-MAP Using datasets from the above experiments. MDS-MAP(P) Non-linear Regression Multi-hop network Experimental Setup Using simulated datasets created using the real data’s node Results link error distributions. Discussion Future Work Single-hop network with consistently over ranging link. Conclusion Using a real dataset and artificially over ranging a certain node-node measure. To test robustness in the presence of systematic errors common in WSN. As highlighted by Whitehouse and Culler[4] in their experiments investigating systematic error in WSN. [4] K. Whitehouse and D. Culler. A robustness analysis of multi-hop ranging-based localization approximations. In Proceedings of the 5th international conference on Information processing in sensor networks. ACM, 2006. 19/39
- 20. A Comparison of Experimental Setup Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Five algorithms to be compared Experimental Setup Results MDS-MAP. Discussion MDS-MAP No SPM (Dissimilarity matrix not filled in with Future Work shortest-path measure). Conclusion NLR NLR With Angles NLR No Elimination 20/39
- 21. A Comparison of Overview Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Background Algorithms MDS-MAP MDS-MAP(P) Algorithms Non-linear Regression MDS-MAP Experimental Setup MDS-MAP(P) Results Non-linear Regression Discussion Future Work Experimental Setup Conclusion Results Discussion Future Work Conclusion 21/39
- 22. A Comparison of Results - Single Hop Non-linear Regression and MDS-MAP C. Ellis, M. Hazas CDF of algorithms being run upon single hop networks Background 1 Algorithms MDS-MAP 0.9 MDS-MAP(P) Non-linear Regression 0.8 Experimental Setup Results 0.7 Discussion Percentile Error 0.6 Future Work Conclusion 0.5 0.4 0.3 MDS−MAP 0.2 MDS−MAP No SPM NLR With angles 0.1 NLR NLR No elimination 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Distance Error (m) 22/39
- 23. A Comparison of Results - Single Hop Non-linear Regression and MDS-MAP C. Ellis, M. Hazas CDF of algorithms being run upon single hop networks Background 1 Algorithms MDS-MAP 0.9 MDS-MAP(P) Non-linear Regression 0.8 Experimental Setup Results 0.7 MDS-MAP Discussion Percentile Error 0.6 Future Work Conclusion 0.5 0.4 NLR 0.3 MDS−MAP 0.2 MDS−MAP No SPM NLR With angles 0.1 NLR NLR No elimination 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Distance Error (m) 22/39
- 24. A Comparison of Results - Multi Hop Non-linear Regression and MDS-MAP C. Ellis, M. Hazas CDF of algorithms being run upon multi hop networks Background 1 Algorithms MDS-MAP 0.9 MDS-MAP(P) Non-linear Regression 0.8 Experimental Setup Results 0.7 Discussion Percentile Error 0.6 Future Work Conclusion 0.5 0.4 0.3 MDS−MAP 0.2 MDS−MAP No SPM NLR With angles 0.1 NLR NLR No elimination 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance Error (m) 23/39
- 25. A Comparison of Results - Multi Hop Non-linear Regression and MDS-MAP C. Ellis, M. Hazas CDF of algorithms being run upon multi hop networks Background 1 Algorithms MDS-MAP 0.9 MDS-MAP(P) Non-linear Regression 0.8 Experimental Setup NLR Results 0.7 Discussion Percentile Error 0.6 Future Work Conclusion 0.5 0.4 MDS-MAP 0.3 MDS−MAP 0.2 MDS−MAP No SPM NLR With angles 0.1 NLR NLR No elimination 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance Error (m) 23/39
- 26. A Comparison of Results - Erroneous Link Non-linear Regression and MDS-MAP C. Ellis, M. Hazas CDF of algorithms being run upon single hop networks with one link consistently over estimating Background 1 Algorithms MDS-MAP 0.9 MDS-MAP(P) Non-linear Regression 0.8 Experimental Setup Results 0.7 Discussion Percentile Error 0.6 Future Work Conclusion 0.5 0.4 0.3 MDS−MAP 0.2 MDS−MAP No SPM NLR With angles 0.1 NLR NLR No elimination 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance Error (m) 24/39
- 27. A Comparison of Results - Erroneous Link Non-linear Regression and MDS-MAP C. Ellis, M. Hazas CDF of algorithms being run upon single hop networks with one link consistently over estimating Background 1 Algorithms MDS-MAP 0.9 MDS-MAP(P) Non-linear Regression 0.8 Experimental Setup MDS-MAP Results 0.7 Discussion Percentile Error 0.6 Future Work Conclusion 0.5 0.4 0.3 NLR MDS−MAP 0.2 MDS−MAP No SPM NLR With angles 0.1 NLR NLR No elimination 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance Error (m) 24/39
- 28. A Comparison of Overview Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Background Algorithms MDS-MAP MDS-MAP(P) Algorithms Non-linear Regression MDS-MAP Experimental Setup MDS-MAP(P) Results Non-linear Regression Discussion Future Work Experimental Setup Conclusion Results Discussion Future Work Conclusion 25/39
- 29. A Comparison of Discussion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Non-linear regression, even without the observation rejection, Algorithms MDS-MAP has a 90th percentile solution error which is lower than MDS-MAP(P) Non-linear Regression MDS-MAP by a factor of up to 10. Experimental Setup Results Experimental solution error (cm) Discussion Single Multi Overranging Future Work MDS-MAP 51.6 179.2 92.8 Conclusion MDS-MAP No SPM 44.9 609.5 70.1 NLR With angles 3.6 6.3 3.8 NLR 4.1 9.4 8.9 NLR No elimination 9.2 35.6 52.2 Table: Comparison of 90th percentile errors of each protocol for each experiment 26/39
- 30. A Comparison of Discussion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Non-linear regression, even without the observation rejection, Algorithms MDS-MAP has a 90th percentile solution error which is lower than MDS-MAP(P) Non-linear Regression MDS-MAP by a factor of up to 10. Experimental Setup Results Experimental solution error (cm) Discussion Single Multi Overranging Future Work MDS-MAP 51.6 179.2 92.8 Conclusion MDS-MAP No SPM 44.9 609.5 70.1 NLR With angles 3.6 6.3 3.8 NLR 4.1 9.4 8.9 NLR No elimination 9.2 35.6 52.2 Table: Comparison of 90th percentile errors of each protocol for each experiment 26/39
- 31. A Comparison of Discussion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Non-linear regression, even without the observation rejection, Algorithms MDS-MAP has a 90th percentile solution error which is lower than MDS-MAP(P) Non-linear Regression MDS-MAP by a factor of up to 10. Experimental Setup Results Experimental solution error (cm) Discussion Single Multi Overranging Future Work MDS-MAP 51.6 179.2 92.8 Conclusion MDS-MAP No SPM 44.9 609.5 70.1 NLR With angles 3.6 6.3 3.8 NLR 4.1 9.4 8.9 NLR No elimination 9.2 35.6 52.2 Table: Comparison of 90th percentile errors of each protocol for each experiment 26/39
- 32. A Comparison of Discussion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Non-linear regression, even without the observation rejection, Algorithms MDS-MAP has a 90th percentile solution error which is lower than MDS-MAP(P) Non-linear Regression MDS-MAP by a factor of up to 10. Experimental Setup Results Experimental solution error (cm) Discussion Single Multi Overranging Future Work MDS-MAP 51.6 179.2 92.8 Conclusion MDS-MAP No SPM 44.9 609.5 70.1 NLR With angles 3.6 6.3 3.8 NLR 4.1 9.4 8.9 NLR No elimination 9.2 35.6 52.2 Table: Comparison of 90th percentile errors of each protocol for each experiment 26/39
- 33. A Comparison of Discussion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Non-linear regression, even without the observation rejection, Algorithms MDS-MAP has a 90th percentile solution error which is lower than MDS-MAP(P) Non-linear Regression MDS-MAP by a factor of up to 10. Experimental Setup Results Experimental solution error (cm) Discussion Single Multi Overranging Future Work MDS-MAP 51.6 179.2 92.8 Conclusion MDS-MAP No SPM 44.9 609.5 70.1 NLR With angles 3.6 6.3 3.8 NLR 4.1 9.4 8.9 NLR No elimination 9.2 35.6 52.2 Table: Comparison of 90th percentile errors of each protocol for each experiment 26/39
- 34. A Comparison of Discussion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Non-linear regression’s increase in accuracy comes at the cost Algorithms of algorithmic complexity. MDS-MAP MDS-MAP(P) At least 8x more complex, calculations are compounded by Non-linear Regression the multiple rounds used within elimination. Experimental Setup Results However, all algorithms are of the same order O(n3 ) and will Discussion both scale accordingly to network density. Future Work Conclusion Complexity MDS-MAP 38k 3 + 116k 2 + 120k + 41 NLR no elimination 276k 3 − 460k 2 + 756k − 287 NLR i(276k 3 − 460k 2 + 756k − 287) Table: Comparison of computational complexities for each algorithm. i ∈ Z+ is an unknown variable which represents number of regressions ran. In our experiments it was on average 7. 27/39
- 35. A Comparison of Discussion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Computation complexity comparison of MDS-MAP and NLR Algorithms 8e+06 MDS-MAP MDS-MAP MDS-MAP(P) NLR No Elim Non-linear Regression 7e+06 NLR Experimental Setup 6e+06 Results Discussion 5e+06 Future Work Operations Conclusion 4e+06 3e+06 2e+06 1e+06 0 0 5 10 15 20 25 30 Number of neighbours 28/39
- 36. A Comparison of Discussion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Results Whilst non-linear regression is more accurate than Discussion MDS-MAP, it only exists as a centralised option. Future Work Conclusion Adopting the patch and stitch method of MDS-MAP(P) could provide a distributed algorithm. 29/39
- 37. A Comparison of Discussion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Results Discussion Future Work Conclusion 30/39
- 38. A Comparison of Overview Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Background Algorithms MDS-MAP MDS-MAP(P) Algorithms Non-linear Regression MDS-MAP Experimental Setup MDS-MAP(P) Results Non-linear Regression Discussion Future Work Experimental Setup Conclusion Results Discussion Future Work Conclusion 31/39
- 39. A Comparison of Future Work Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Perform an accuracy and complexity comparison of Results Discussion MDS-MAP(P) and NLR(P). Future Work Experiment on the accuracy of MDS-MAP(P) without any Conclusion refinement, and compare results. Investigate the benefit of global refinement. 32/39
- 40. A Comparison of Overview Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Background Algorithms MDS-MAP MDS-MAP(P) Algorithms Non-linear Regression MDS-MAP Experimental Setup MDS-MAP(P) Results Non-linear Regression Discussion Future Work Experimental Setup Conclusion Results Discussion Future Work Conclusion 33/39
- 41. A Comparison of Conclusion Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Non-linear regression shows a 90th percentile error Results improvement of up to 10x over MDS-MAP Discussion Future Work MDS-MAP(P) uses non-linear regression in the refinement Conclusion stage - losing the complexity advantage. Using studentized residual analysis we have shown non-linear regression can effectively combat systematic error. 34/39
- 42. A Comparison of Ending Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Results Discussion Thank you very much for your time. Any questions? Future Work Conclusion 35/39
- 43. A Comparison of Raw measurements aggregated range error Non-linear Regression and MDS-MAP C. Ellis, M. Hazas CDF of raw localisation error Background 1 Algorithms MDS-MAP 0.9 MDS-MAP(P) Non-linear Regression 0.8 Experimental Setup Results 0.7 Discussion Percentile Error 0.6 Future Work Conclusion 0.5 0.4 0.3 0.2 0.1 Raw data 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Distance Error (m) 36/39
- 44. A Comparison of Raw measurements aggregated range error - Non-linear Regression and MDS-MAP extremes C. Ellis, M. Hazas Background Algorithms 2% MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Results 1.5% Discussion Percentage of occurence Future Work Conclusion 1% 0.5% 0% −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Range error (metres) 37/39
- 45. A Comparison of Raw measurements per node range error Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background 1 Algorithms MDS-MAP 0.9 MDS-MAP(P) Non-linear Regression 0.8 Experimental Setup Fraction of errors less than abcissa Results 0.7 Discussion 0.6 Severe over−ranging Future Work (4.5% of links) Conclusion 0.5 0.4 0.3 0.2 Severe under−ranging (2.2% of links) 0.1 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Range error (metres) 38/39
- 46. A Comparison of Sensor Node Internals Non-linear Regression and MDS-MAP C. Ellis, M. Hazas Background Algorithms MDS-MAP MDS-MAP(P) Non-linear Regression Experimental Setup Results Discussion Future Work Conclusion 39/39