Localization is Sensor Data Dimensionality ReductionNeal Patwari, Jessica Croft, and Piyush Agrawal
Dept. of Electrical & Computer EngineeringUniversity of Utah, Salt Lake City, UT
Sensing and Processing Across Networks
S P A N
at the University of Utah
MotivationImprove Sensor Localization for large-scale environmentaldeployments.Sensor Location Needed tomake sensor data meaningful,for greedy routing. Key limita-tions:
1. Low Device Cost
2. Low Configuration
3. Distributed Algorithm
Sensor DataPair-wiseEnvironmental
••
Sensor Location• Relative
Absolute•
Dimension Reduction
EnvironmentalField Knowledge
Key Insight: Location is essentially data dimension reduction!
Data: Both Pairwise and Environmental
What do we mean by data?
• Ambient field data: e.g.,Temp., Chemistry, Humidity,Sunlight, Acoustic, Seismic,RF Space-time
• Active pair-wise meas’ts: Sig-nal strength, Propagation de-lay.
Figure: Sensor deployed inRed Butte Canyon, Utah (aprotected watershed).
Red Butte Canyon Deployment Applications:Water Balance is key to understanding, reducing developmentimpacts on water supply in Western U.S. Test deployment:
• Stream water temp (above) indicates water absorption intoground. Space-time data provides more detailed stream wa-ter balance than previously attempted.
Data Contains Location Information
When a sensor data field has isotropic spatially-decaying cor-relation, Sensor measurements contain spatial information.
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• Above: Correlation of daily precipitation totals with (a) Merri-man, NE, and (b) Highmore, SD.
Use meas’ts {vi} to find ‘data distances’:
δi,j = ‖vi − vj‖, ∀i 6= j
with some appropriate distance metric, e.g., Euclidean, lp, . . ..
What is Data Dimension Reduction?Nonlinear Dimensionality Reduction: Pre-serve nearest-neighbor relationships in alower dimension:
1. Measure M data points over time,mode.
2. Compute weights and/or distances btwnneighboring points.
3. Non-linear dimensionality reduction (i.e.,Isomap, Laplacian Eigenmaps, dwMDS)to generate 2-D or 3-D coords.
4. Rotate, translate, and scale to match.
{ }vi iData Vectors
Calc NeighborDistances, Weights
Reduce Dimensionto 2D or 3D
Rotate, Scale, &Translate
{ }dij ij,w ij
{ }xi i
{ }xi i
Distributed Weighted Multi-dimensionalScaling
Implementation of a robust distributed sensor localization al-gorithm on a network of wireless sensors running TinyOS withNO CENTRALIZED COMPUTATION. Features [Costa 06]:
• Fully distributed measurement, commun., and calculation;
• Constant per-node complexity: O(k) for k neighbors;
• Robustness to poor pair-wise distance estimates;
• Convergence: Cost is non-increasing in each round.
Distance Estimation from Averaged RSS
δMLEi,j = d010
P0−Pi,j10np
• np: Path-loss exponent,
• P0: Received power (dBm) at distanced0 (1 m).
• First measurement set used to estimate {np, Π0}.
• Frequency Averaging: Hop & Average Pi,j across band.
• Time Averaging. Cons: Latency, non-ergodic signal.
• Reciprocal Averaging: Average Pi,j with Pj,i.
dwMDS Algorithm CalculationGlobal cost S =
∑i Si, a sum of Local cost functions:
Si =∑
i
wi,j
(‖zi − zj‖ − δMLE
i,j
)2+ ri‖zi − zi‖2
Constants wi,j from LOESS. Majorization-based optimization.
Si is minimized by a simple weighted average of coordinates ofsensor i’s neighbors.
z(m+1)i = biz
(m)i +
∑
j∈H(i)
bjz(m)j
Requires O(k) multiplies and adds in each round, where k isthe number of neighbors.
Experiment: Sensor Data MeasurementsSetup: Use US historical climatology weather stations data1221 stations collect daily
• Total precipitation
• Total snowfall
• Minimum and maximum temperature
From 66 Sensors in Nebraska and South Dakota, Test Isomap[Tenenbaum 00] and dwMDS [Costa 06] algorithms usingtemp. midpoint, i.e., 1
2(max + min)
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Figure: Actual (•) and estimated (—-x) coordinates ofunknown-location nodes for (a) dwMDS, and (b) Isomap,
algorithms. Rotated for best match. Achieved RMS location errorof (a) 0.69 and (b) 1.07 degrees.
Future Directions
• Apply with short-term, small-scale field meas’t sets.
• Eg. App: RF meast’s for dynamic spectrum access.
• Use sensor data coordinates for routing.
• RF Tomographic Imaging.
Experiment: Direct PairwiseMeasurements
Setup: Sensors (Crossbow mica2) in grass, in 6 by 6 grid, in a6.7 m by 6.7 m. Four known-location nodes (in corners)
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Figure: Actual (•) and estimated (—x) coordinates ofunknown-location nodes, along with reference coordinates (x).
Achieved RMSE of 55.3 cm.
Conclusion
• Sensor data can help map sensor locations
• Use both RF pairwise and environmental field meas’ts
• Dimension reduction provides the general framework
TinyOS Module Available: Contact [email protected].
References
• N. Patwari and A. O. Hero III, “Manifold learning algorithms forlocalization in wireless sensor networks,” in IEEE Intl. Conf. onAcoustic, Speech, & Signal Processing (ICASSP’04), vol. 3,May 2004, pp. 857–860.
• J. A. Costa, N. Patwari, and A. O. Hero III, “Distributed multidi-mensional scaling with adaptive weighting for node localizationin sensor networks,” ACM/IEEE Trans. Sensor Networks, vol. 2,no. 1, pp. 39–64, Feb. 2006.
• Y. Baryshnikov and J. Tan, “Localization for Anchoritic SensorNetworks,” arXiv:cs/0608014v1 [cs.NI], Aug. 2, 2006.
• J. B. Tenenbaum, V. De Silva and J. C. Langford, “A global ge-ometric framework for nonlinear dimensionality reduction,” Sci-ence 290 (5500), pp. 2319-2323, 2000.
• C. N. Williams Jr., M. J. Menne, R. S. Vose, andD. R. Easterling, “United States Historical Climatol-ogy Network Daily Temperature, Precipitation, andSnow Data,” ORNL/CDIAC-118, NDP-070, 2006,http://cdiac.ornl.gov/epubs/ndp/ushcn/usa.html.
