1 distributed online simultaneous fault detection for multiple sensors ram rajagopal, xuanlong...

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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