1. 1 Distributed Online Simultaneous Fault Detection for Multiple Sensors Ram Rajagopal, Xuanlong Nguyen, Sinem Ergen, Pravin Varaiya EECS, University of California, Berkeley

2. 2 Plan 1.Introduction 2.Problem Statement 3.Proposed Solution 4.Analysis and Implementation 5.Experiments 6.Conclusions

3. 3 Building distributed sensing for physical systems  Hierarchical: sensor management, data collection/cleansing, application  Statistically driven  Performance guarantees  Tradeoff computation vs communication/noise Cost of uncertainty Cost of decentralization Feasibility of computations Lifetime of sensing

4. 4 Application: Freeway Traffic Management ACCIDENT ! MeasurementBackhaulProcessing Interne t Control & Info Cellula r Traffic Management Center Traffic Control PeMS http://pems.eecs.berkeley.edu

5. 5 Sensor System State large oscillations

6. 6 Mean days before failure or working continuously (D4)  55% of loops work continuously for fewer than 20 days; none works for more than 50 days in 2004 vs. 20% in 2005.

7. 7 Motivation: freeway monitoring sensors  One sensor per lane every 2 miles  Measures flow, occupancy every 30 seconds  Sensor failures are frequent  Non-stationary environment  Events: onset of traffic jam, accidents, sudden slowdowns

8. 8 Problem statement  Detect faulty sensors that report plausible values  Distinguish events from faults –Events  temporary sudden changes in measurements –Faults  lasting sudden changes in measurements  Real time detection  Each sensor uses only local data

9. 9 Proposed approach Sensor Network Fault GraphChange Point Model  Score S is correlation with block length T samples  Change times have some known priors

10. 10 Model details  Change times have priors  Scores have joint change distributions  Link information strength

11. 11 Preview of results  Accounting for average time scale of physical events  Combining multiple sources of weak evidence  Importance of feedback for detection algorithms  Statistical modeling = feasible implementations

12. 12 Does it make sense?  Empirical distributions from highway deployment WorkingFaulty

13. 13 Does it make sense?  Empirical distributions from highway deployment  Use Box-Cox transformation or conditional normal distribution (Kwon, Rice and Bickel, 03)

14. 14 Selection of block length T  Distinguish events from faults :  Rule: T > Average event duration  Tradeoff: T = minimum waiting time to detect

15. 15 Measuring the performance  Control false alarm:  Minimize Average Detection Delay (ADD): time (n)

16. 16 Single change point review  For minimize ADD  Single change point optimal rule [Shyrayev (1978)]:  Performance [Tartakovsky and Veeravali (2005)]:  Minimum delay achievable for all procedures with false alarm  At time n test:

17. 17 Model for analytic problems  Two sensors:  For each proposed procedure: –Achieved false alarm –Delay  X and Y represent aggregates of many links to working sensors  Among all procedures with false alarm, minimum delay?

18. 18 Delay performance lower bound  Theorem 1: For all procedures with false alarm for each sensor:

19. 19 Multiple sensor posterior rule (no feedback)  Direct extension of single change rule: Common link does not help Z X Y 1 2  Theorem 2:

20. 20 Multiple sensor rule (with one bit feedback)  Use shared link until either sensor thinks it has failed Z X Y 1 2

21. 21 What is procedure doing?  Over time, implicit averaging Z X Y 1 2  Over sensors, 1 bit summarizes other links information

22. 22 False alarm bound  Confusion probabilities  Theorem 3 [Rajagopal et al, 2008]:

23. 23 Confusion probability  Theorem 4 [Rajagopal et al, 2008]:  For example (using some simplifications): and Guarantee that and

24. 24 Delay guarantee  Theorem 5 [Rajagopal et al, 2008]:

25. 25 Implementation issues  T bits per block quantization:  Simple recursive formula for computing test statistics (order constant updates)  Only O(Delay) number of samples required to compute statistics with high accuracy

26. 26 Performance decomposition  Cost of communication/uncertainty:  Cost of decentralization:

27. 27 Delay estimates  Symmetric (X and Y same distribution) method is optimal:  Fully connected i.i.d network:

28. 28 Two sensor network: confusion probability  Theory predicts covariance ratio > 2

29. 29 Two sensor network: ADD vs ratio of uncertainty theorysimulation

30. 30 Fully connected network: false alarm 20 nodes  = 0.12 10 neighbors is a good number

31. 31 Fully connected network: fixed false alarm Small False Alarm (theory is close!)  = 0.1  = 0.0001

32. 32 Torus network: ADD vs number of sensors Local connectivity determines performance

33. 33 Conclusions and future work  Change point framework is good for building algorithms for fault detection  Currently Caltrans collecting data by visiting sensors predicted broken  Developed tools for analysis of multiple change point problems  Simultaneous online multiple event detection

34. 34 Implementation Issues (1)  Efficient correlation computation using dithered quantization (1 bit per sample, T bits per block) and transforms:  Simple recursive implementation for test:

35. 35 Implementation Issues (2)  Finite memory implementation issue : –When a sensor fails, all other sensors that use link recompute stats –Need to remember all samples?  (1) When no change, statistics is close to zero (2) Delay bound is known for all moments  So only need to remember finite number of values: