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1
Distributed Online Simultaneous Fault Detection for Multiple Sensors
Ram Rajagopal, Xuanlong Nguyen, Sinem Ergen, Pravin Varaiya
EECS, University of California, Berkeley
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Plan
1. Introduction
2. Problem Statement
3. Proposed Solution
4. Analysis and Implementation
5. Experiments
6. Conclusions
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Application: Freeway Traffic Management
ACCIDENT!
Measurement Backhaul Processing
Internet
Internet
Control & Info
Cellular
Cellular
Traffic Management
Center
TrafficControl
PeMShttp://pems.eecs.berkeley.edu
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Sensor System State
large oscillations
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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.
0 20 40 60 80 100 120 140 160 1800
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20
30
40
50
60
70
80
90
100
Days
% F
ails
Befo
reMean Days to Fail (per Sensor) Distribution
2004
2005
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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
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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
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Proposed approach
Sensor Network Fault Graph Change Point Model
Score S is correlation with block length T samples
Change times have some known priors
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Model details
Change times have priors
Scores have joint change distributions
Link information strength
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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
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Does it make sense?Empirical distributions from highway deployment
Working Faulty
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Does it make sense?Empirical distributions from highway deployment
Use Box-Cox transformation or conditional normal distribution (Kwon, Rice and Bickel, 03)
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Selection of block length T
Distinguish events from faults :
Rule: T > Average event duration
Tradeoff: T = minimum waiting time to detect
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Measuring the performance
Control false alarm:
Minimize Average Detection Delay (ADD):
time (n)
time (n)
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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:
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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?
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Delay performance lower bound
Theorem 1: For all procedures with false alarm for each sensor:
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Multiple sensor posterior rule (no feedback)Direct extension of single change rule:
Common link does not help
ZX Y1 2
Theorem 2:
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Multiple sensor rule (with one bit feedback)
Use shared link until either sensor thinks it has failed
ZX Y1 2
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What is procedure doing?
Over time, implicit averaging
ZX Y1 2
Over sensors, 1 bit summarizes other links information
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False alarm boundConfusion probabilities
Theorem 3 [Rajagopal et al, 2008]:
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Confusion probabilityTheorem 4 [Rajagopal et al, 2008]:
For example (using some simplifications):
and
Guarantee that
and
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Delay guarantee
Theorem 5 [Rajagopal et al, 2008]:
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Delay estimates
Symmetric (X and Y same distribution) method is optimal:
Fully connected i.i.d network:
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Two sensor network: confusion probability
Theory predicts covariance ratio > 2
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Fully connected network: fixed false alarm
Small False Alarm (theory is close!)
= 0.1
= 0.0001
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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