Statistical Inference in Sensor Networks

1. Statistical Inference in Sensor Networks Rob Nowak Waheed Bajwa, Paul Barford (CS), Rui Castro, Jarvis Haupt, Aline Martin, Mike Rabbat, Akbar Sayeed (ECE), Aarti Singh, Yolanda Tsang, Rebecca Willett Supported by the NSF and ONR

2. What is a Sensor Network? • Comm between army units was crucial • Signal towers built on hilltops • Wireless comm and coding consisted of smoke signals, fires, flags, cannon fire • e.g., during Ming Dynasty a single column of smoke plus a single gun shot would indicate the approach of a hundred enemy soldiers

3. Internet Boom • performance of Internet is very difficult to predict • too complex to measure everywhere, all the time • traffic measurements expensive (hardware, bandwidth) 1993 1969

4. Internet “Sensor Networks” network sensor • map Internet performance over space and time • discover topology and connectivity • detect malicious activities • locate sources of congestion, failures and attacks

5. Wireless Sensor Networks Goal: Measure, estimate and convey a physical field Ex. Temperature, light, pressure, moisture, vibration, sound, gas concentration, position

6. What is a Sensor Network? A network of sensors spatially distributed over - imperial border - forest - Internet - cropland - manufacturing facility - urban environment For monitoring spatially distributed processes - enemy soldiers - fires - spread of computer viruses - temperature, light, moisture - biological and chemical processes

7. Information Processing in Sensor Networks Extracting, Manipulating, and Communicating salient features from raw sensor data and delivering them to a destination Features: - location/magnitude of sources - summary statistics - signals, maps & images - decisions

8. Information Processing in Sensor Networks This talk will focus on wireless sensing of spatially distributed phenomena, and cover 1. Wireless sensor networks 2. Information processing “in” them 3. And wireless sensing without networks Wireless sensor ensembles

9. Wireless Sensors solar cell battery radio mproc sensor GPS module Each node equipped with power source(s) sensor(s) modest computing capabilities radio transmitter/receiver

10. The trade-off • Low density network • low bandwidth/energy • consumption • low spatial resolution • High density network • high bandwidth/energy • consumption • high spatial resolution

11. But… Physical fields are spatially correlated, so information does not grow linearly with network density Knowledge of correlation (e.g., Slepian-Wolf coding) or in-network processing can significantly reduce number of bits that must be transmitted Gastpar, Neuhoff, Pradhan,Ramchandran,Servetto,Vetterli, RN…

12. R(u,v) v u Key idea: As network density increases, correlation between sensor measurements increases, which offsets communication requirements (more communications, but each is shorter) The upshot Basic Trade-off: Field is more accurately characterized with higher density sampling, but data rate increases as density increases “data rate” grows linearly with network density, but “information rate” grows is sub-linearly

13. Data Compression 458 x 300 pixels - coded using only 37 kB Uncompressed : (458 x 300) x (3 colors x 1 byte) = 412 kB Compression factor 11:1

14. A simple field model Pseudocolor depiction of smooth spatial process • moisture or pesticide • over cropland • chemical distribution • in lake or sea • biochemical agent • in urban environment Assume that field is k-times continuously differentiable i.e., the field is “smooth”

15. Taylor Series Approximation Example: Smooth functions can be locally approximated very well by simple polynomial functions

16. Approximating Smooth Fields

17. Approximating Smooth Fields

18. Approximating Smooth Fields log error slope -k log(1/sidelength)

19. Approximation and Distortion

20. Approximation and Information

21. Rate-Distortion Analysis Encoding polyfit parameters of each cell requires a fixed number of bits, thus number bits required is proportional to m k=1 log distortion k=2 k=3 log bits “Information” content of field is inversely proportional to smoothness (i.e., increasing k)

22. wireless sensors at random locations Wireless Sensing Wireless sensors randomly distributed over field Sensor i makes a noisy measurement Yi of field at its location Xi

23. Wireless Sensing and Information 1. Divide into m cells 2. Fit poly to sensor data in each cell 3. Transmit data or polyfits* to remote destination * This is a form of “data compression”

24. Noisy Wireless Sensing How many cells should we use if we are basing our field approximation on noisy sensor data, instead of perfect Taylor polynomial fits ? What choice of m is best ?

25. Noiseless vs. Noisy Sensing Noiseless Sensing Noisy Sensing What choice of m is best ?

26. Approximation and Estimation Errors Partition sensor field into m square cells Distortion due to partition-based approximation Distortion due estimating polyfits from noisy data Estimated polyfits fluctuate about optimal Taylor approximations due to noise

27. Total Distortion Optimal # of cells increases (slowly) with # of sensors Distortion decreases with # of sensors

28. What about communications ? • Delivering optimal field reconstruction to destination can be accomplished in several ways. Here are two obvious methods: • Transmit raw sensor data to destination using digital communications; destination computes field reconstruction • Local polyfits are computed “in-network” and only the estimated parameters are transmitted “out-of-network” to destination, both via digital communications

29. destination Digital Transmission of Raw Data Transmit raw sensor data to destination using digital comm; destination computes field reconstruction There are n sensors, so this requires n digital transmissions; i.e., the number of bits that must be transmitted is O(n) Communication requirements grow linearly with n

30. Nodes in each cell self-configure into a wireless network, and cooperatively exchange information to compute polyfit In-Network Processing & Communication Partition sensor field into cells; local polyfits are computed “in-network” using digital communications fixed number of parameters computed per cell

31. destination Out-of-Network Communication Polyfit parameters are transmitted “out-of-network” to destination via digital communications Number of parameters (bits) that must be transmitted to the destination is O(m) = O(n1/(k+1) ) Communication requirements grow sublinearly with n

32. BUT… overhead of forming wireless cooperative networks in each cell consumes a dominant fraction of the system resources As sensor density increases, we move from a network for sensing to a network for networking !!! In-Network Processing & Communication In-network processing and communications drastically reduce resources (bandwidth, power) required to transmit the field information to a remote destination Reduction of out-of-network communication resource demands from O(n) to O(n1/(k+1))

33. Wireless Sensor Ensembles (Waheed Bajwa, Akbar Sayeed and RN ’05) An attractive alternative to the conventional sensor network paradigm is a wireless sensor ensemble destination • each sensor transmits its value via amplitude modulation • no cooperative processing or communications required • transmissions synchronized in each cell to arrive in-phase processing (averaging) implicitly computed by receive antenna

34. Ensemble Beamforming transmitted signals from one cell received signal phase-coherent sum of transmitted signals Phase-coherency “beams” energy to receive antenna received energy is proportional to energy expended by sensors

35. Ensemble Communications and Distortion

36. The Complete Picture approx error (bias) sensor noise variance comm noise variance

37. Minimum Distortion minimum distortion when fixed, finite energy per sensor

38. Distortion vs. Energy Distortion Energy This trade-off cannot be improved on by any scheme

39. : Magnetic Resonance Imaging Sensors = hydrogen atoms Coherent ensemble communications = external EM excitation causes hydrogen atoms to produce coherent externally measurable RF signal proportional to hydrogen density MRI “senses” spatial distribution of hydrogen atoms in my head Proof of Concept reconstruction of Rob’s brain structure using a wireless sensor ensemble

40. destination sends random seed to sensors destination After p transmissions, destination has p random projections of sensor readings Sensors modulate readings by pseudorandom binary variables and coherently transmit to destination each sensor modifies seed according to a local attribute (e.g., location, address) ANY “compressible” field can be very accurately reconstructed from these projections Extension to “Compressible” Fields (Jarvis Haupt and RN ’05)

41. Noisy Random Projection Theorem Random projection sampling is “optimal” and allows us to use entire ensemble as a coherent beamforming array

42. Distortion vs. Energy Distortion Energy

43. Conclusions Complexity (entropy) of field grows far more slowly than volume of raw data information rate data rate wireless sensor ensemblesoffer a promising new architecture for dense wireless sensing nowak@engr.wisc.edu www.ece.wisc.edu/~nowak

44. Papers Matched Source-Channel Communication for Field Estimation in Wireless Sensor Networks, W. Bajwa, A. Sayeed and R. Nowak, IPSN 2005, Los Angeles, CA. Signal Reconstruction from Noisy Random Projections, J. Haupt and R. Nowak, submitted to IEEE Trans. Info. Th., 2005 (also to appear in Proceedings of 2005 IEEE Statistical Signal Processing Workshop) www.ece.wisc.edu/~nowak/pubs.html