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QUANTIFYING RESILIENCE-BASED
IMPORTANCE MEASURES
USING BAYESIAN KERNEL METHODS
Hiba Baroud, Ph.D.
Civil and Environmental Engineering
Vanderbilt University
Thursday, May 19, 2016
Photo: Marco Monetti
WHAT IS RESILIENCE?
Hosseini, S., Barker, K. and Ramirez-Marquez, J.E., 2016. A review of definitions and
measures of system resilience. Reliability Engineering & System Safety, 145, pp.47-61.
HOW DID IT START?
WHY IS IT IMPORTANT?
4
WHY IS IT IMPORTANT?
Hosseini, S., Barker, K. and Ramirez-Marquez, J.E., 2016. A review of definitions and
measures of system resilience. Reliability Engineering & System Safety, 145, pp.47-61.
A resil ient infrastructure sector would “rapidly recover and
reconstitute crit ical assets, operations, and services with minimum
damage and disruption.”
WHAT IS THE DEFINITION OF RESILIENCE?
6
In frastructure Security Par tnership
There are almost as many definitions of
resilience as there are people defining it.
Congressional Research Service Repor t for Congress
Resil ience is the abil ity of assets, networks and systems to anticipate,
absorb, adapt to and/or rapidly recover from a disruptive event .
Cabinet Of f ice , UK
Infrastructure resil ience is the abil ity to reduce the magnitude and/or
duration of disruptive eventsNational In frastructure Advisory Council
HOW DO WE MEASURE IT?
Hosseini, S., Barker, K. and Ramirez-Marquez, J.E., 2016. A review of definitions and
measures of system resilience. Reliability Engineering & System Safety, 145, pp.47-61.
BEFORE A DISRUPTION
8Henry, D. and J.E. Ramirez-Marquez. 2012. Generic Metrics and Quantitative Approaches for System
Resilience as a Function of Time. Reliability Engineering and System Safety, 99(1): 114-122.
Risk Management
Planning and
preparedness decision
making
Risk mitigation
AFTER A DISRUPTION
9Henry, D. and J.E. Ramirez-Marquez. 2012. Generic Metrics and Quantitative Approaches for System
Resilience as a Function of Time. Reliability Engineering and System Safety, 99(1): 114-122.
Recovery management
Post-disaster strategies
Stochastic behavior of
recovery
Component importance measures (CIM)
Commonly found in risk and reliability
engineering
Extended to resilience analysis
Impact of a component on the resilience of the
system
How is the recovery of the entire system impacted
by the recovery of a component?
RESILIENCE-BASED CIM
10
Barker, K., J.E. Ramirez-Marquez, and C.M. Rocco. 2013. Resilience-Based Network Component
Importance Measures. Reliability Engineering and System Safety, 117(1), 89-97.
Baroud, H., K. Barker, J.E. Ramirez-Marquez, and C.M. Rocco. 2013. Importance Measures for Inland
Waterway Network Resilience. Transportation Research Part E: Logistics and Transportation, 62(1): 55-67.
RESILIENCE-BASED CIM
11
CIЯ𝜑,𝑖 𝑡r 𝑒𝑗 =
𝜑 𝐱 𝑡0 − 𝜑 𝐱 𝑡0 , 𝑥𝑖 𝑡𝑑 𝑉𝑖𝑗
max𝑖 𝜑 𝐱 𝑡0 − 𝜑 𝐱 𝑡0 , 𝑥𝑖 𝑡𝑑 𝑉𝑖𝑗𝑇𝜑 𝐱 𝑡0 𝑉𝑖
𝑗
Network
performance
loss due to
disruption of
component 𝑖
Maximum loss
among all the
components
Time to full
network
restoration
RESILIENCE WORTH INDEX
12
WЯ𝜑,𝑖 𝑡𝑟 𝑒𝑗 =
𝑇𝜑 𝐱 𝑡0 𝑉𝑖
𝑗 − 𝑇𝜑 𝐱 𝑡0 𝑉𝑖
𝑗=0
𝑇𝜑 𝐱 𝑡0 𝑉𝑖
𝑗
0 < WЯ𝜑,𝑖 𝑡𝑟 𝑒𝑗 < 1
Time to total
network
recovery
Time to total
network recovery
when component 𝑖is invulnerable
ASCE REPORT CARD ON AMERICA’S INFRASTRUCTURE
13
The average age of
the 84,000 dams in
the country is 52
years old
By 2020, 70% of
the total dams in
the United States
will be over 50
years old
A S C E ’ s R e p o r t C a r d f o r A m e r i c a ’ s I n f r a s t r u c t u r e [ 2 0 1 3 ]
WHY INLAND WATERWAYS?
14
15-barge grain tow, hauling
approximately 22,500 tons of export
grain, exits Lock & Dam 13
Over 200 lock chambers
Over 566 million tons of freight
(~51 million truck trips)
Over $152 billion equivalence of
goods
Low-cost and fuel-efficient
freight mode
“The dam safety engineering practice is moving
towards a risk-based decision-making process
for the design, rehabilitation, and operation of
dams. Risk-based decisions enable the dam
owner to better utilize limited funding, and
prioritize projects, by focusing on repairs and
operational changes that reduce risk to
acceptable levels, thus improving community
resilience.”
PLAN OF ACTION
15
ASCE’s Repor t Card for America’s In frastructure [2013]
SIMULATION APPROACH
Probability distribution
for the magnitude
of disruption and speed of recovery
Simulation
Probability distribution
for the resilience –based CIM
𝑃 𝑎 < 𝑉𝑖𝑗≤ 𝑏 =
𝑎
𝑏
𝑓 𝑣𝑖𝑗𝑑𝑣𝑖𝑗
𝑃 𝑡𝑠 < 𝑈𝑖𝑗𝑉𝑖𝑗≤ 𝑡𝑟
= 𝑡𝑠
𝑡𝑟
𝑓 𝑢𝑖𝑗𝑉𝑖𝑗𝑑𝑣𝑖𝑗
𝑓 WЯ𝐹,𝑖 𝑡r 𝑒𝑗
STOCHASTIC RANKING OF COMPONENTS
17
WЯ𝜑,𝑖CIЯ𝜑,𝑖
Baroud, H., K. Barker, J.E. Ramirez-Marquez, and C.M. Rocco. 2013. Importance Measures for Inland
Waterway Network Resilience. Transportation Research Part E: Logistics and Transportation, 62(1): 55-67.
DATA-DRIVEN APPROACH
18
Lock &
Dam
Closure
FrequencyRiver Mile Vessels Tonnage Lockages . . .
L&D 3 0 797 9,397 6,747 4,406
L&D 13 6 523 2,810 14,545 3,155
L&D 2 0 815 4,478 6,735 2,893
L&D 20 23 343 2,508 20,828 3,582
L&D 22 40 301 2,280 22,476 3,486
L&D 8 6 679 4,333 10,277 2,620...
US Army Corps of Engineers. 2011. Interactive access to website.
http://www.ndc.iwr.usace.army.mil//lpms/lpms.htm.
DATA-DRIVEN APPROACH
Prior distribution
(prior knowledge, expertise)
Bayesian kernel model
(binary historical
data, attributes)
Posterior distribution
of the resilience
worth
𝑓 WЯ𝐹,𝑖 𝑡r 𝑒𝑗
BETA BAYESIAN KERNEL MODEL
Prior
Posterior
𝑚− = number of
negative labels in
training set
𝑚+ = number of
positive labels in
training set
𝑚 = size of training
set
𝑘 = kernel function
of 𝑥𝑖 and 𝑥𝑗
20MacKenzie, C.A., T.B. Trafalis, and K. Barker. 2014. A Bayesian Beta Kernel Model for Binary
Classification and Online Learning Problems Statistical Analysis and Data Mining, 7(6), 434-449.
𝜃𝑖 𝒚~beta(𝛼, 𝛽)
𝛼∗ = 𝛼 +𝑚−𝑚
{𝑗 𝑦𝑗=1}
𝑘(𝑥𝑖 , 𝑥𝑗)
𝛽∗ = 𝛽 +𝑚+
𝑚 {𝑗 𝑦𝑗=−1}𝑘(𝑥𝑖 , 𝑥𝑗)
Expected value of the posterior distribution
WЯ𝜑,𝑖 𝑡𝑟 𝑒𝑗 = 𝜃𝑖 =
𝛼∗
𝛼∗ + 𝛽∗
Posterior probability distribution
𝑓 WЯ𝜑,𝑖 𝑡𝑟 𝑒𝑗 =
WЯ𝛼∗−1 1 − 𝑊Я 𝛽
∗−1
Β α∗ , β∗
RISK ANALYSIS USING THE RESILIENCE WORTH
21
PREDICTION ACCURACY – UNWEIGHTED MODEL
22
TP=0.89 TN=0.48 ACC=0.55 TP=0.79 TN=0.79 ACC=0.70
TP=0.21 TN=0.96 ACC=0.26 TP=0.99 TN=0.12 ACC=0.16
PREDICTION ACCURACY – WEIGHTED MODEL
23
TP=0.85 TN=0.75 ACC=0.76 TP=0.75 TN=0.86 ACC=0.68
TP=0.21 TN=0.96 ACC=0.01 TP=0.96 TN=0.27 ACC=0.35
INTERPRETABILITY
24
• Rank components
based on their
resilience
• Identify critical
components for
resource allocation
of preparedness
and recovery
strategies
• Incorporate
uncertainty into the
decision
• Integrate the
decision maker’s
expertise and risk
attitude
INTERPRETABILITY
25
ACC=0.80
ACC=0.68
ACC=0.65
POSTERIOR CUMULATIVE DISTRIBUTION
26
Posterior cumulative distribution for the 5
most impactful locks and dams according to
the posterior expected value
Copeland score method: a multi-criteria decision analysis technique
Comparing discrete objects 𝑎 and 𝑏
𝐶𝑘 𝑎, 𝑏 =
𝐶𝑘−1 𝑎, 𝑏 + 1 𝑞𝑘 𝑎 < 𝑞𝑘 𝑏
𝐶𝑘−1 𝑎, 𝑏 − 1 𝑞𝑘 𝑎 > 𝑞𝑘 𝑏
𝐶𝑘−1 𝑎, 𝑏 𝑞𝑘 𝑎 = 𝑞𝑘 𝑏
Copeland Score of object 𝑎
CS 𝑎 =
𝑏≠𝑎
𝐶Ω 𝑎, 𝑏
COPELAND SCORE METHOD
27
RANKING OF COMPONENTS
28
𝑊Я rankingPosterior
expected value
Posterior
Copeland score
1 L&D 24 L&D 22
2 L&D 5 L&D 27
3 L&D 27 L&D 20
4 L&D 20 L&D 5
5 L&D 22 L&D 24
Different ranking of components when the entire
probability distribution is considered
Conclusions
Data-driven and Bayesian methods integrate historical information with the decision maker’s opinion
Posterior probability distributions are more flexible, comprehensive, and informative for risk-based decision making
Prediction accuracy and interpretability of results are highly sensitive to the definition of the prior distribution
A better prediction accuracy does not necessarily mean a better interpretability
Future Research
Investigate a more realistic identification of the prior based on prior knowledge and empirical estimation
Study the tradeoff between prediction accuracy and interpretability and decision making
CONCLUDING REMARKS
29
