Localization is Sensor Data Dimensionality Reduction Neal Patwari, Jessica Croft, and Piyush Agrawal Dept. of Electrical & Computer Engineering University of Utah, Salt Lake City, UT Sensing and Processing Across Networks S P A N at the University of Utah Motivation Improve Sensor Localization for large-scale environmental deployments. Sensor Location Needed to make sensor data meaningful, for greedy routing. Key limita- tions: 1. Low Device Cost 2. Low Configuration 3. Distributed Algorithm Sensor Data Pair-wise Environmental • • Sensor Location • Relative Absolute • Dimension Reduction Environmental Field 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 in Red Butte Canyon, Utah (a protected watershed). Red Butte Canyon Deployment Applications: Water Balance is key to understanding, reducing development impacts on water supply in Western U.S. Test deployment: • Stream water temp (above) indicates water absorption into ground. 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. (a) -105 -100 -95 39 40 41 42 43 44 45 46 Degrees Longitude Degrees Latitude -1 -0.5 0 0.5 1 (b) -105 -100 -95 39 40 41 42 43 44 45 46 Degrees Longitude Degrees Latitude -1 -0.5 0 0.5 1 • Above: Correlation of daily precipitation totals with (a) Merri- man, NE, and (b) Highmore, SD. Use meas’ts {v i } to find ‘data distances’: δ i,j = kv i - v j k, ∀i 6= j with some appropriate distance metric, e.g., Euclidean, l p , .... What is Data Dimension Reduction? Nonlinear Dimensionality Reduction: Pre- serve nearest-neighbor relationships in a lower dimension: 1. Measure M data points over time, mode. 2. Compute weights and/or distances btwn neighboring 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. {} v i i Data Vectors Calc Neighbor Distances, Weights Reduce Dimension to 2D or 3D Rotate, Scale, & Translate { } d ij ij ,w ij {} x i i {} x i i Distributed Weighted Multi-dimensional Scaling Implementation of a robust distributed sensor localization al- gorithm on a network of wireless sensors running TinyOS with NO 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 δ MLE i,j = d 0 10 P 0 -P i,j 10n p • n p : Path-loss exponent, • P 0 : Received power (dBm) at distance d 0 (1 m). • First measurement set used to estimate {n p , Π 0 }. • Frequency Averaging: Hop & Average P i,j across band. • Time Averaging. Cons: Latency, non-ergodic signal. • Reciprocal Averaging: Average P i,j with P j,i . dwMDS Algorithm Calculation Global cost S = ∑ i S i , a sum of Local cost functions: S i = X i w i,j ‡ kz i - z j k- δ MLE i,j · 2 + r i kz i - z i k 2 Constants w i,j from LOESS. Majorization-based optimization. S i is minimized by a simple weighted average of coordinates of sensor i’s neighbors. z (m+1) i = b i z (m) i + X j ∈H (i) b j z (m) j Requires O (k ) multiplies and adds in each round, where k is the number of neighbors. Experiment: Sensor Data Measurements Setup: Use US historical climatology weather stations data 1221 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 using temp. midpoint, i.e., 1 2 (max + min) (a) -105 -100 -95 38 39 40 41 42 43 44 45 46 47 Degrees Longitude Degrees Latitude (b) -105 -100 -95 39 40 41 42 43 44 45 46 47 Degrees Longitude Degrees Latitude Figure: Actual (•) and estimated (—-x) coordinates of unknown-location nodes for (a) dwMDS, and (b) Isomap, algorithms. Rotated for best match. Achieved RMS location error of (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 Pairwise Measurements Setup: Sensors (Crossbow mica2) in grass, in 6 by 6 grid, in a 6.7 m by 6.7 m. Four known-location nodes (in corners) 10 12 14 16 18 9 10 11 12 13 14 15 16 17 18 X Position (m) Y Position (m) Figure: Actual (•) and estimated (—x) coordinates of unknown-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 for localization in wireless sensor networks,” in IEEE Intl. Conf. on Acoustic, 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 localization in sensor networks,” ACM/IEEE Trans. Sensor Networks, vol. 2, no. 1, pp. 39–64, Feb. 2006. • Y. Baryshnikov and J. Tan, “Localization for Anchoritic Sensor Networks,” 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, and D. R. Easterling, “United States Historical Climatol- ogy Network Daily Temperature, Precipitation, and Snow Data,” ORNL/CDIAC-118, NDP-070, 2006, http://cdiac.ornl.gov/epubs/ndp/ushcn/usa.html.