Review on Energy Resilience
harsh55
2014_Baroud_Hiba_Dissertation.pdf
UNIVERSITY OF OKLAHOMA
GRADUATE COLLEGE
BAYESIAN KERNEL METHODS FOR THE RISK ANALYSIS AND RESILIENCE
MODELING OF CRITICAL INFRASTRUCTURE SYSTEMS
A DISSERTATION
SUBMITTED TO THE GRADUATE FACULTY
in partial fulfillment of the requirements for the
Degree of
DOCTOR OF PHILOSOPHY
By
HIBA BAROUD
Norman, Oklahoma
2015
BAYESIAN KERNEL METHODS FOR THE RISK ANALYSIS AND RESILIENCE
MODELING OF CRITICAL INFRASTRUCTURE SYSTEMS
A DISSERTATION APPROVED FOR THE
SCHOOL OF INDUSTRIAL AND SYSTEMS ENGINEERING
BY
______________________________
Dr. Kash Barker, Chair
______________________________
Dr. Pakize Simin Pulat
______________________________
Dr. Naiyu Wang
______________________________
Dr. Charles Nicholson
______________________________
Dr. Jose Ramirez-Marquez
© Copyright by HIBA BAROUD 2015
All Rights Reserved.
To Wassim Tabet
And to my family:
My father Sami, my mother Nada, and my brother Charbel
iv
Acknowledgements
First, I would like to give thanks to God the father, Jesus, the Holy Spirit, the
Virgin Mary, my Guardian Angel, and all the Saints. I am grateful for the graces and
blessings I receive every day that helped me accomplish the work done in this
dissertation and throughout my studies.
I am grateful for my family, my father Sami, my mother Nada, and my brother
Charbel. They have supported me, prayed for me, and encouraged me in all my pursuits.
My advisor, Dr. Kash Barker, has been a great educator and mentor. I would not
have been the person I am today nor would I have accomplished this great milestone
without his guidance. He encouraged me to go after every opportunity I had to gain
experience and branch out. He truly went above and beyond to help me succeed.
I want to thank my committee members, Dr. Pakize Simin Pulat, Dr. Naiyu
Wang, Dr. Charles Nicholson, and Dr. Jose Ramirez-Marquez. I want to also thank Dr.
Theodore Trafalis. I greatly appreciate your help and feedback on my work.
To my colleagues at the University of Oklahoma and to my friends in Lebanon,
Canada, and the United States, thank you for believing in me and encouraging me
especially in challenging times.
I appreciate the help and support of the faculty and staff of the School of
Industrial and Systems Engineering, especially the director, Dr. Randa Shehab, thank
you for all the insightful conversations.
To the love of my life and my support system, Wassim Tabet, this achievement
is not possible without your love, patience, and encouragement. I am grateful for your
presence in my life. I love you.
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Table of Contents
Chapter 1 Introduction ...................................................................................................... 1
General Literature Review ......................................................................................... 3
Risk Analysis ........................................................................................................ 3
Resilience Modeling ............................................................................................. 5
Inland Waterways Research ................................................................................. 7
Research Contributions .............................................................................................. 9
Chapter 2 Bayesian Kernel Methods for the Prediction of Risk .................................... 11
Literature Review ..................................................................................................... 11
Bayesian Methods .............................................................................................. 12
Kernel Methods .................................................................................................. 14
Bayesian Kernel Models ..................................................................................... 16
Bayesian Kernel Model for Count Data ................................................................... 19
Count Data Modeling ......................................................................................... 19
Poisson Bayesian Kernel Model ......................................................................... 23
Goodness of Fit Measures .................................................................................. 26
Prediction Accuracy ........................................................................................... 28
Empirical Analysis ............................................................................................. 30
Case Study: Prediction of Inland Waterways Disruptions ....................................... 34
Empirical Analysis of the Inland Waterway Data .............................................. 36
Prior Distribution Implications ........................................................................... 38
Research Outcome .............................................................................................. 42
Concluding Remarks ................................................................................................ 43
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Chapter 3 Managing the Risk of Interdependent Impacts of Infrastructure Disruptions 45
Literature Review ..................................................................................................... 46
Stochastic Decision Tree .................................................................................... 46
Interdependency Model ...................................................................................... 50
Distribution Assumptions ................................................................................... 53
Methodology: Dynamic Multiobjective Inoperability Decision Tree ...................... 55
Infrastructure Preparedness Decision Problem .................................................. 56
Decision Tree Construction and Parameters ...................................................... 58
Dynamic Recovery Model .................................................................................. 59
Simulation of the Decision Tree ......................................................................... 63
Case Study: Inland Waterways Preparedness Strategies Assessment ...................... 64
Motivation .......................................................................................................... 65
Assumptions ....................................................................................................... 65
Decision Tree Solution ....................................................................................... 70
Trade-off Analysis .............................................................................................. 75
Sensitivity Analysis ............................................................................................ 78
Concluding Remarks ................................................................................................ 79
Chapter 4 Interdependent Impacts of Infrastructure Systems Resilience Modeling ...... 81
Literature Review ..................................................................................................... 85
Methodology: Inherent Cost Metrics ........................................................................ 89
Loss of Service Cost ........................................................................................... 89
Total Network Restoration Cost ......................................................................... 90
Interdependent Impacts of Network Resilience ................................................. 92
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Case Study: Interdependent Impacts of Inland Waterways Resilience .................... 93
Parameters and Assumptions .............................................................................. 95
Resilience and Restoration Time Results ........................................................... 97
Inherent Cost Results .......................................................................................... 99
Interdependent Impacts Results ........................................................................ 102
Concluding Remarks .............................................................................................. 111
Chapter 5 Bayesian Kernel Methods for Resilience Importance Measures Prediction 114
Stochastic Analysis of Resilience Importance Measures ....................................... 116
Literature Review ............................................................................................. 116
Methodology: Recovery Strategies Decision Process ...................................... 118
Case Study: Stochastic Analysis of Locks and Dams Resilience Importance
Measures ............................................................................................... 122
Data-Driven Prediction of Resilience-Based Importance Measures ...................... 132
Literature Review ............................................................................................. 133
Methodology: Resilience Worth Data-Driven Prediction ................................ 134
Case Study: Bayesian Kernel Modeling of Locks and Dams Resilience
Importance Measures ............................................................................ 136
Concluding Remarks .............................................................................................. 140
Chapter 6 Conclusions .................................................................................................. 143
Insights and Lessons ............................................................................................... 143
Statistical Modeling for Risk Analysis ............................................................. 143
Economic Impacts of Disruptions .................................................................... 144
Resilience Modeling ......................................................................................... 146
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Future Research ...................................................................................................... 147
Bayesian Kernel Methods ................................................................................. 147
Interdependency Modeling ............................................................................... 148
Resilience Modeling ......................................................................................... 149
Bibliography ................................................................................................................. 150
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List of Tables
Table 2-1: Prediction error measurement metrics .......................................................... 29
Table 2-2. Description of data sets in the Poisson Bayesian kernel model validation
study ............................................................................................................. 31
Table 2-3: Performance metrics results for the empirical analysis ................................ 33
Table 2-4: Sample of the inland waterway disruption data ............................................ 36
Table 2-5: Performance metrics results for the inland waterway data analysis ............. 37
Table 2-6: Locks/dams with highest frequency of closures ........................................... 42
Table 4-1: Annual commodity flow (in tons) across the five links chosen from the
Mississippi River Navigation System .......................................................... 97
Table 4-2: Four recovery sets considered for the restoration of the four disrupted
waterway links .............................................................................................. 97
Table 5-1: Annual commodity flow (in tons) across the five links chosen from the
Mississippi River Navigation System ........................................................ 124
Table 5-2: Lock and dam repair order based on the resilience worth values ............... 139
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List of Figures
Figure 2-1: Binary classification example using a polynomial kernel function
(Schölkopf & Smola, 2002) ........................................................................ 15
Figure 2-2: Test and training error as a function of the model complexity (Hastie et al.,
2001) ........................................................................................................... 28
Figure 2-3: RMSE distribution with perfect prior knowledge ....................................... 39
Figure 2-4: RMSE distribution with imperfect prior knowledge ................................... 40
Figure 2-5: RMSE distribution with no prior knowledge .............................................. 41
Figure 3-1: Depiction of a general stochastic decision tree ........................................... 48
Figure 3-2: Influence diagram describing the infrastructure preparedness investment
decision-making process ............................................................................. 57
Figure 3-3: Multiobjective stochastic inoperability decision tree for infrastructure
preparedness ................................................................................................ 59
Figure 3-4: Frequency distribution of the expected total economic losses across all
regional industries ....................................................................................... 71
Figure 3-5: Pareto frontier for expected total losses versus the amount invested toward
preparedness activities ................................................................................ 72
Figure 3-6: Behavior of standard deviation of the expected total economic loss as
investment increases ................................................................................... 75
Figure 3-7: Trade-off as a function of the cost of investment for a risk-neutral decision
maker ........................................................................................................... 77
Figure 3-8: Trade-off as a function of the cost of investment for a risk-averse decision
maker, 𝑟 = 0.01 .......................................................................................... 77
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Figure 3-9: Sensitivity analysis on the probability of a disruptive event ....................... 79
Figure 4-1: Graphical depiction of state transitions over time with respect to an
increasing system service function, (t) ..................................................... 83
Figure 4-2: Alternative depiction of state transitions describing a decreasing system
service function, (t) ................................................................................... 83
Figure 4-3: Inland waterway network of the Mississippi River Navigation System ..... 94
Figure 4-4: Resilience trajectory based on one realization of the distribution of recovery
time ............................................................................................................. 98
Figure 4-5: Cumulative distribution function for the time to full network restoration .. 99
Figure 4-6: Approximate pdf results for 2000 simulations of (a) the loss of service cost
and (b) the network restoration cost, along with their respective cdfs in (c)
and (d). ...................................................................................................... 100
Figure 4-7: Cumulative distribution function for the network restoration cost ............ 101
Figure 4-8: Resilience (black line, right vertical axis) and sector inoperabilities (gray
lines, left vertical axis) over 50 time periods for the three recovery
strategies ................................................................................................... 103
Figure 4-9: Average economic losses experienced in each of the six primary waterway
industries for the three network recovery strategies ................................. 105
Figure 4-10. Economic losses across the six primary waterway industries (blue curve,
left vertical axis) and network resilience (green curve, right vertical axis)
over time for the three recovery strategies and four disruptive scenarios.
Note that the economic loss axis is held constant for the same disruptive
scenario ..................................................................................................... 107
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Figure 4-11. Total economic loss computed under three recovery activities ............... 108
Figure 4-12. Impact on individual sector inoperability for adopting strategy W2 as
opposed to W1 .......................................................................................... 109
Figure 4-13. Impact on individual sector inoperability for adopting strategy W3 as
opposed to W2 .......................................................................................... 110
Figure 4-14. Sector inoperability over time for the Primary Metal Products industry for
the three recovery strategies ...................................................................... 111
Figure 5-1: Total network-wide vulnerability (in tons) as a function of individual
component vulnerability ........................................................................... 124
Figure 5-2: Cumulative probability distribution for the resilience-based component
importance measure, CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 )........................................................... 125
Figure 5-3: Copeland Score for each link computed based on the resilience-based
component importance measure, CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) ........................................ 127
Figure 5-4: Cumulative probability distribution for the component resilience-worth,
WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) ............................................................................................. 128
Figure 5-5: Copeland Score for each link computed based on the component resilience-
worth, WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) ................................................................................. 129
Figure 5-6: Cumulative probability distribution of the total cost (in thousands of dollars)
of each recovery set................................................................................... 131
Figure 5-7: Distribution of the posterior expected value of the resilience worth ......... 138
Figure 5-8: Posterior cumulative probability distribution of the five most impactful
locks and dams of the navigation system .................................................. 139
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Figure 5-9: Copeland score of the five most impactful locks and dams of the navigation
system ....................................................................................................... 139
xiv
Abstract
The protection of critical infrastructures has recently garnered attention with an
emphasis on analyzing the risk and improving the resilience of such systems. With the
abundance of data, risk managers should be able to better inform preparedness and
recovery decision making under uncertainty. It is important, however, to develop and
utilize the necessary methodologies that bridge between data and decisions. The goal of
this dissertation is to (i) predict the likelihood of risk, (ii) assess the consequences of a
disruption, and (iii) inform preparedness and recovery decision making.
This research presents a data-driven analysis of the risk and resilience of critical
infrastructure systems. First, a new Bayesian kernel model is developed to predict the
frequency of failures and a Beta Bayesian kernel model is deployed to model resilience-
based importance measures. Bayesian kernel models were developed for Gaussian
distributions and later extended to other continuous probability distributions. This
research develops a Poisson Bayesian kernel model to accommodate count data.
Second, interdependency models are integrated with decision analysis and resilience
quantification techniques to assess the multi-industry economic impact of critical
infrastructure resilience and inform preparedness and recovery decision making under
uncertainty.
Examples of critical infrastructure systems are inland waterways, which are
critical elements in the nation’s civil infrastructure and the world’s supply chain. They
allow for a cost-effective flow of approximately $150 billion worth of commodities
annually across industries and geographic locations, which is why they are called
“inland marine highways.” Aging components (i.e., locks and dams) combined with
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adverse weather conditions, affect the reliability and resilience of inland waterways.
Frequent disruptions and lengthy recovery times threaten regional commodity flows,
and more broadly, multiple industries that rely on those commodities. While
policymakers understand the increasing need for inland waterway rehabilitation and
preparedness investment, resources are limited and select projects are funded each year
to improve only certain components of the network. As a result, a number of research
questions arise.
What is the impact of infrastructure systems disruptions, and how to predict
them? What metrics should be used to identify critical components and determine the
system’s resilience? What are the best risk management strategies in terms of
preparedness investment and recovery prioritization?
A Poisson Bayesian kernel model is developed and deployed to predict the
frequency of locks and dams closures. Economic dynamic interdependency models
along with stochastic inoperability multiobjective decision trees and resilience metrics
are used to assess the broader impact of a disruption resulting in the closure of a port or
a link of the river and impacting multiple interdependent industries. Stochastic
resilience-based measures are analyzed to determine the critical waterway components,
more specifically locks and dams, that contribute to the overall waterway system
resilience. A data-driven case study illustrates these methods to describe commodity
flows along the various components of the U.S. Mississippi River Navigation System
and employs them to motivate preparedness and recovery strategies.
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Chapter 1
Introduction
A number of infrastructure systems, both in the U.S. and globally, have been
identified as critical due to their ubiquitous influence on society’s way of life. Among
these critical infrastructures are energy, healthcare, and transportation sectors (DHS,
2009). Their criticality is due to their interconnectedness with other infrastructure
systems, as well as industries and workforces which rely upon them. A disruption to
such a critical infrastructure, whether the result of a terrorist attack, a natural disaster,
an accident, or common failure, could incur widespread losses of functionality that
affect not only the infrastructure itself but all the industries depending on it. Therefore,
to effectively plan for the protection of these infrastructures from failure, and more
importantly the response to and recovery from such failures when they inevitably occur,
an important challenge exists in (i) estimating the likelihood of a disruptive event, (ii)
portraying the widespread economic losses resulting from a disruptive event, as well as
(iii) measuring the efficacy of risk management to determine the appropriate investment
to enable preparedness and recovery.
Of interest in this work are transportation systems, specifically inland waterway
navigation systems, though the methodologies developed in this dissertation have
broader applicability to preparedness in other infrastructure systems. As truck traffic
continues to increase, congestion on highway networks will worsen and become a major
issue for commodity flows (USDOT, 2009, 2011; NCFRP, 2010; GAO, 2011). The
highway and railway networks in many cities near coastal ports are already
experiencing bottlenecks. A viable alternative to these modes for freight transport is the
2
inland waterway navigation system. Vital to commodity flows in the U.S., almost 80%
of all U.S. international trade is transported through coastal ports, with 40% of these
shipments moving inside the U.S. through inland ports before reaching their final
destination (Haveman & Shatz, 2006). However, commodity flows through inland
waterways are a distant third behind highway and rail traffic (USACE, 2010), giving
them an opportunity for expanded use.
Infrastructure networks in today’s global landscape can been characterized as
exhibiting the characteristics of many complex and large-scale systems: (i) a large
number of interacting components and subsystems, (ii) a large number of decision and
state variables, (iii) complicated, complex, and often nonlinear functional relationships,
(iv) uncertainty and variability, (v) hierarchical and/or networked interdependencies,
(vi) multiple and often conflicting performance objectives, (vii) multiple decision
makers, and (viii) dynamic changes, among others (Haimes, 2009b). As such, disruptive
events, whether malevolent attacks, natural disasters, manmade accidents, or common
cause failures, can have devastating, widespread, and often unpredictable, results.
Consider, for example, the August 2003 US blackout which “contributed to at
least 11 deaths and cost an estimated $6 billion” (Minkel, 2008), or the largest blackout
in history experienced by India during August 2012, affecting over 600 million people.
Against the effects of these events, most research efforts have been devoted to
developing traditional measures of protection (hardening) (Ramirez-Marquez, Rocco, &
Levitin, 2011; Levitin & Hausken, 2010; Bier, Haphuriwat, Menoyo, Zimmerman, &
Culpen, 2008) and policies that can be expensive, degrade typical performance, and are
non-reactive. Recent attention has been placed on preparedness, response, and recovery
3
from these events (e.g., in the large-scale homeland security preparedness domain
(DHS, 2009)). A perspective that has recently been collectively referred to in the
systems engineering community as designing for resilience, is considered an essential
component in the design of systems and enterprises (Comfort, Boin, & Demchak,
2010).
The goal of this dissertation is to develop and utilize data-driven methods and
statistical tools as well as stochastic simulation to address the preparedness for,
response to, and recovery from critical infrastructure systems disruptions. The main
objectives of the research are to (i) predict the likelihood of a disruption, (ii) assess the
consequences, and (iii) provide recovery insights.
General Literature Review
While a more specific literature review will be presented in each chapter in the
dissertation, the following is a general overview of the risk analysis, resilience
modeling, and inland waterways research literature. Significant methodological and
theoretical contributions that founded this research will be presented in subsequent
chapters.
Risk Analysis
According to Lowrance (1976), risk is a “measure of the probability and severity
of adverse effects.” The analysis of risk is composed of the (i) risk assessment and (ii)
risk management. The risk assessment would generally consider three questions
(Kaplan & Garrick, 1981):
(i) What can go wrong?
(ii) What are the chances of something going wrong?
4
(iii) What are the consequences if the undesirable event occurs?
Haimes (2009b) complements these questions with another set of three questions
aiming at risk management:
(i) What can be done?
(ii) What options are available and what are the associated tradeoffs in terms of
cost, risks, and benefits?
(iii) What are the impacts of the current management decisions on future
options?
The work in this dissertation addresses the second and third question of the risk
assessment as well as the three questions of the risk management. Statistical methods
are used to estimate the frequency of disruptions occurring in a critical infrastructure
system. Interdependency modeling is used to assess the consequences of disruptions,
and integrated with decision analysis techniques, it is used to identify risk management
strategies and assess their efficacy. Finally, statistical tools and stochastic simulation
techniques are deployed to assess current decisions on the system’s resilience in the
future.
Probabilistic Risk Assessment (PRA) is a common tool used to analyze the risk
as it has been applied in cases where there is little to no data available such as the case
with the estimation of accidents in nuclear power plants (Lewis et al., 1979). The
method relies on the functional form and the physical structure of the system coupled
with other techniques (e.g., fault tree, event tree, among others) that are used to extract
the analysis from a component-level to a system-level. Ezell, Farr, & Wiese (2000)
deploy PRA techniques for the risk analysis of infrastructure systems.
5
Guikema (2009) highlights the need to consider statistical data-driven methods
to address the risk assessment of large-scale critical infrastructure systems facing
natural disasters. The structure of such systems is too complex for PRA methods and
they have an abundance of data for which statistical tools would be more suitable such
as the Generalized Linear Models (GLM). The use of statistical techniques to provide
data-driven analysis of the risk of infrastructure systems has recently emerged with
applications in the prediction of power outages (Liu, Davidson, Rosowsky, & Stedinger,
2005), traffic accidents (Lord, Washington, & Ivan, 2005), and water distribution
systems reliability (Yamijala, Guikema, & Brumbelow, 2009), among others. The key
factor of utilizing such tools is the selection of the best model based on the type of data
available and the outcome variable.
The first part of this dissertation develops a new statistical tool to analyze the
risk in critical infrastructures. Such systems are large and complex in nature that PRA
would be computationally expensive to implement. However, the data available for
some of these systems is not always abundant enough to provide robust estimation
using traditional statistical methods such as GLM. The proposed model has the potential
of addressing risk using data-driven analysis when data is scarce.
Resilience Modeling
The importance of having robust and resilient infrastructure systems has gained
the attention of decision makers and government officials in the past decade. Critical
infrastructure systems, such as power grids and transportation systems have been
vulnerable to numerous disruptive events including natural disasters, willful attacks, and
accidents. DHS announced a set of grant programs targeting different areas prone to
6
willful attacks or natural disasters (DHS, 2012), aiming to provide resources helpful in
supporting the National Preparedness Goal (NPG) in succeeding in its mission of
insuring “a secure and resilient Nation with the capabilities required across the whole
community to prevent, protect against, mitigate, respond to, and recover from the
threats and hazards that pose the greatest risk” (DHS, 2011). Further motivation comes
from Boin, Comfort, & Demchak (2009):
“If we accept that dominant trends such as globalization, increasing
interdependence and complexity, the spread of potentially dangerous
technologies, new forms of terrorism, and climate change create new and
unimaginable threats to modern societies, it is only a small step to recognizing
and accepting the inherent shortcomings of contemporary approaches to
prevention and preparation. If we cannot predict or foresee the urgent threats
we face, prevention and preparation become difficult. The concept of resilience
holds the promise of an answer.”
Resilience is often thought of as the ability exhibited by a system to “bounce
back” following a disturbance. In material science, a modulus of resilience is also
defined to represent the energy absorbed per unit volume of material when stressed to
the proportional limit (Ugural, 2003). In socio-ecological systems, resilience is defined
as the magnitude of disturbance that can be absorbed before the system changes its
structure by changing the processes that control behavior (Holling, 1973; Carpenter,
Walker, Anderies, & Abel, 2001). Regarding enterprise systems, Jackson (2007, 2009)
defines resilience as the ability of organizational, hardware, and software systems to
mitigate the severity and likelihood of failures or losses, to adapt to changing
7
conditions, and to respond appropriately. In business terms, resilience has been defined
as the ability of an organization to sustain the impact of a business interruption and
recover and resume operations to continue to provide minimum services (Hoffman,
2007). With respect to critical infrastructure, the Infrastructure Security Partnership
(2011) noted that a resilient infrastructure sector would “prepare for, prevent, protect
against, respond or mitigate any anticipated or unexpected significant threat or event”
and “rapidly recover and reconstitute critical assets, operations, and services with
minimum damage and disruption.” In an engineering context, Hollnagel, Woods, and
Leveson (2007) promote resilience engineering as a new paradigm for safety
engineering. And many others have defined resilience in various ways (Fiksel, 2003;
Wreathall, 2006; Vogus & Sutcliffe, 2007; Rose, 2009).
This research utilizes the resilience modeling framework developed by Henry
and Ramirez-Marquez (2012) that quantifies resilience based on the system’s service
function. The framework is the basis for the modeling approach developed to analyze
the resilience of critical infrastructure systems and its impact on decision making under
uncertainty.
Inland Waterways Research
“The current system of inland waterways lacks resilience. Waterway usage is
increasing, but facilities are aging and many are well past their design life of 50
years. Recovery from any event of significance would be negatively impacted by
the age and deteriorating condition of the system, posing a direct threat to the
American economy.” 2009 Report Card for America’s Infrastructure, American
Society for Civil Engineers (ASCE) (2009).
8
The multi-modal transportation system plays a vital role in maintaining
commodity flows across multiple industries and multiple regions. As a result of their
critical role, the effects of large-scale disruptive events could result in the closure of key
transportation links and nodes. These critical components in a transportation network
(e.g., inland waterways) are particularly susceptible to disruptions in commodity flows
(Lee, Park, & Lee, 2003; Sacone & Siri, 2009; Lee & Kim, 2010). The recovery of
transportation networks from disruptions has been given some recent attention
(Cadarso, Marín, & Maróti, 2013; Chen & Miller-Hooks, 2012; Zeng, Durach, & Fang,
2012; Zhang & Peeta, 2011).
Although inland ports face many of the same risks as coastal ports, relatively
few studies have developed risk assessments of inland ports (Folga et al. 2009; Pant,
Barker, Grant, & Landers, 2011; MacKenzie, Barker, & Grant, 2012a). Some studies
have focused on forecasting commodity flows in inland waterway networks (Babcock
& Lu, 2002; Beuthe, Jourquin, Geerts, & a Ndjang'Ha, 2001), however, such models do
not capture the effect of uncertain disruptive events and their impacts on the commodity
flows. Pant et al. (2011) provide a simulation model of inland port activities to
parameterize a port disruption within a multi-regional interdependency model, while
MacKenzie et al. (2012a) focus on the multi-regional impacts in functionality and in
economic losses of decision making for shipping alternatives following an inland port
disruption.
During recent ASCE testimony to the U.S. Senate, it was stated that the costs
attributed to delays in U.S. inland waterways were $33 billion in 2010 (rising to $49
billion by 2020), with interdependent impacts cascading to economic sectors that
9
require inland waterway transport (e.g., petroleum, coal) (ASCE, 2013a). As such, the
study of the risk and resilience of inland waterway networks is an important area of
focus.
The case study for inland waterways is analyzed for each of the methods
developed. In each chapter, an aspect of the risk or resilience of the network is analyzed
such as the prediction of the frequency of disruptions (Chapter 2), the assessment of
interdependent economic impacts of a port closure and the corresponding risk
management efficacy analysis (Chapter 3), the impact of the resilience of the waterway
network on the interdependent economic impacts of a disruption (Chapter 4), the
prioritization of recovery activities sets after a disruption occurred (Chapter 5), and
finally, the identification of resilience-based critical components of the inland waterway
(Chapter 5).
Research Contributions
Each chapter in the dissertation outlines the methodology and modeling
approach for the risk and resilience analysis with an application to the inland waterway
network. Thus, each chapter will address a specific literature review of the tools
discussed.
There are three levels of contributions in this research: theory, methodology, and
application. A contribution is made to the theory of statistics and machine learning
through a new model, the Poisson Bayesian kernel model, which expands on the class
of Bayesian kernel methods to accommodate count data (Chapter 2). The work has
appeared in Floyd, Baroud, & Barker (2014). A methodological contribution is made to
the field of risk analysis through the deployment of the Poisson Bayesian kernel model
10
in the risk analysis of critical infrastructure systems, in particular, inland waterways
(Baroud, Barker, & Lurvey, 2013a). Other methodological contributions include the
integration of the economic interdependency model (i) with stochastic decision trees to
analyze the impact of port closures and assess risk management strategies (Chapter 3),
the work has appeared in Baroud, Barker, & Grant (2014a), and (ii) with resilience
metrics to quantify the impact of the resilience of the disrupted infrastructure system on
the multi-industry economic impacts (Chapter 4), the work appears in Baroud, Barker,
Ramirez‐Marquez, & Rocco (2013b, 2014c). Finally, the third type of contribution is at
the application level where Bayesian kernel methods are used to model the resilience of
critical infrastructures. In particular, the Beta Bayesian kernel model is used to analyze
resilience-based importance measures to identify critical components of the inland
waterway network. Data-driven and statistical tools have not been previously deployed
to analyze the resilience of critical infrastructure systems. This work has appeared in
Baroud and Barker (2014).
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Chapter 2
Bayesian Kernel Methods for the Prediction of Risk
In many situations, the likelihood of an event is found using the average rate at
which the event occurs. And often that rate is a function of characteristics surrounding
the event. To integrate the impacts of both the component characteristics and any prior
failure information, a Bayesian kernel model is proposed as an approach to a more
accurate estimation of the rate of occurrence of an event. More specifically, an extended
version of this method is developed, the Poisson Bayesian kernel model to
accommodate count data and estimate the rate of occurrence. This chapter includes an
extensive literature review of Bayesian kernel methods and count data models. The
Poisson Bayesian kernel model is introduced as a new approach to predict the
likelihood of occurrence of an event. An empirical analysis of this model using different
types of data sets and measures of goodness of fit and prediction accuracy is used to
validate the model in comparison to classical approaches. Finally, the Poisson Bayesian
kernel model is deployed in a case study to analyze the rate of disruptions along an
inland waterway network, the Mississippi River Navigation System.
Literature Review
Kernel methods, first introduced in a pattern recognition setting several decades
ago (Aizerman, Braverman, & Rozonoer, 1964), have found popularity across a number
of data mining domains, including bioinformatics (Schölkopf, Guyon, & Weston, 2003;
Ben-Hur & Noble, 2005), sensing (Arias, Randall, & Sapiro, 2007; Camps-Valls, Rojo-
Alvarez, & Martinez-Ramon, 2006), and financial risk management and forecasting
(Wang & Zhu, 2010; Mitschele, Chalup, Schlottmann, & Seese, 2006), among many
12
others. Kernel functions are used to map input data, for which no pattern can be
recognized, to a higher dimensional space, where patterns are more readily detected.
Such functions enable algorithms designed to detect relationships among data in the
higher dimensional space, including least squares regression and support vector
machines (SVM) classification (Cherkassky & Mulier, 1998; Cristianini & Shawe-
Taylor, 2000; Hastie, Tibshirani, & Friedman, 2001). Integrating Bayesian methods
with kernel methods has recently garnered attention (Seeger, 2000; Bishop & Tipping,
2003, Mallick, Ghosh, & Ghosh, 2005; Zhang, Dai, & Jordan, 2011), as Bayesian
methods make use of historical data to estimate posterior probability distributions of the
parameter of interest given that it follows a specific prior distribution.
The integration of Bayesian and kernel methods enables a classification
algorithm which provides probabilistic outcomes as opposed to deterministic outcomes
(i.e., such as those resulting from SVM classification). That is, rather than assigning a
class to a data point, Bayesian kernel methods assign a probability that the data point
belongs to a particular class. Several extensions to Bayesian kernel models have
appeared, including (i) the relevance vector machine (RVM) which assumes a Gaussian
distribution for the probability to be estimated (Tipping, 2001; Schölkopf & Smola,
2002), and (ii) non-Gaussian distributions for binary problems (Montesano & Lopes,
2009; Mason & Lopes, 2011; MacKenzie, Trafalis, & Barker, 2014b). However, there
has been no Bayesian kernel model developed for count data.
Bayesian Methods
The classic Bayes rule assumes that a prior probability for an event of interest,
A, is given as 𝑃(𝐴), and a likelihood of event 𝐵 conditioned on the occurrence of 𝐴 is
13
given as 𝑃(𝐵|𝐴). With these probabilities, along with 𝑃(𝐵), one can calculate the
posterior distribution for the event of interest given knowledge of 𝐵, or 𝑃(𝐴|𝐵), shown
in Eq. (2-1) (Bayes & Price, 1763).
𝑃(𝐴|𝐵) = 𝑃(𝐵|𝐴)𝑃(𝐴)
𝑃(𝐵) (2-1)
This manifests itself, for example, when one wants to develop a posterior
distribution for a parameter of interest, 𝑡, from (i) the prior distribution for that
parameter, 𝑃(𝑡), and (ii) the data describing that parameter in the form of a likelihood
function, 𝑃(𝑥|𝑡), which is a conditional likelihood of obtaining the data given the
parameter. In such a case, the denominator does not depend on the parameter of interest
and can be excluded from the Bayes rule equation when maximum likelihood
calculations are performed. More specifically, the posterior probability distribution for
this parameter t can be estimated as being proportional to its prior distribution
multiplied by the likelihood function, as depicted in Eq. (2-2).
𝑃(𝑡|𝑥) ∝ 𝑃(𝑡)𝑃(𝑥|𝑡) (2-2)
An important concept used in the Bayesian framework is the notion of conjugate
priors, which assumes that the posterior, P(t | x), and the prior, P(t), distributions are
from the same family of distributions. Having the prior and posterior follow the same
distribution insures that the overall data properties are kept while modifying the details
of the distribution such as the parameters to better explain the trends. In addition,
conjugate priors provide analytical solutions for the posterior distribution, allowing for
a much faster computation time than other Bayesian tools that require simulation or
optimization techniques. MacKenzie et al. (2014b) use the Beta-Bernoulli conjugate
prior to build non-Gaussian Bayesian kernel models for binary classification. The
14
research in this dissertation uses the Gamma conjugate prior to construct a Poisson
Bayesian kernel method to model count data.
Kernel Methods
Algorithms for detecting non-linear relationships have started to emerge with
heuristic tools such as decision trees (Breiman, Friedman, Olshen, & Stone, 1984;
Quinlan, 1983, 1986) and artificial neural networks (ANN) (Yegnanarayana, 2009) with
applications across many disciplines such as healthcare (Baxt, 1995; Fonarow et al.,
2005), geotechnical engineering (Shahin, Jaksa, & Maier, 2001), and atmospheric
sciences (Gardner & Dorling, 1998), among many others. These techniques lack
theoretical foundation and robustness, and they have a tendency to overfit the data. In
addition, ANN can be computationally expensive (Broussard, Kennell, Ives, & Rakvic,
2008).
Kernel methods introduced a new class of machine learning for non-linear
classification that was more flexible, capable of accommodating different types of data
and detecting different types of relations (Vapnik, 2013; Vapnik & Vapnik, 1998). As
mentioned earlier, a kernel function maps the data to a higher dimensional space, called
the feature space, and detects linear classifiers in that space. Such technique provides
simple and easy detection to nonlinear and complex relations among the data. For
example, Figure 2-1 is a graphical representation of a simple binary classification
problem. The kernel function is used to map the input data in R2 (plot on the left hand
side) into the feature space R3 (plot on the right hand side) in which the inner product is
computed then integrated into the classification algorithm to find a linear classifier in
the feature space that would be equivalent to an ellipse in the input space.
15
Figure 2-1: Binary classification example using a polynomial kernel function
(Schölkopf & Smola, 2002)
The kernel function used to map and compute the inner product in the figure above, is
the polynomial kernel function of degree 𝑑 = 2, Eq. (2-3), where 𝜙 is the mapping
function that maps a two-dimensional data point, (𝑥1, 𝑥2), to a three-dimensional point
in the feature space, 𝜙(𝑥) = (𝑥1 2, 𝑥2
2, √2𝑥1𝑥2), then the kernel function computes the
dot product of the mapped data points.
𝐾(𝑥, 𝑦) =< 𝜙(𝑥), 𝜙(𝑦) >
= 𝑥1 2𝑦1
2 + 2𝑥1𝑥2𝑦1𝑦2 + 𝑥2 2𝑦2
2
= (𝑥1𝑦1 + 𝑥2𝑦2) 2
= (< 𝑥, 𝑦 >)2
(2-3)
Note in the equation above, the kernel is able to implicitly perform both the mapping
and the dot product computation in the feature space in a single function. This is one of
the most attractive properties of kernel methods, called the kernel trick, which saves
computation time and complexity. The kernel matrix, 𝐊, resulting from such a
16
computation is symmetric and positive definite where each entry is a similarity measure
corresponding to the kernel between two data points.
𝐊 = [ 𝐾(𝑥1, 𝑥1) ⋯ 𝐾(𝑥1, 𝑥𝑚)
⋮ ⋱ ⋮ 𝐾(𝑥𝑚, 𝑥1) ⋯ 𝐾(𝑥𝑚, 𝑥𝑚)
] (2-4)
Algorithms that use such technique include but are not limited to Parzen’s Windows
(Schölkopf & Smola, 2002), Support Vector Machines (SVM) (Cristianini & Shawe-
Taylor, 2000; Shawe-Taylor & Cristianini, 2004), Ridge Regression (Saunders,
Gammerman, & Vovk, 1998), Fisher Linear Discriminant Analysis (LDA) (Schölkopft
& Mullert, 1999), and Principal Component Analysis (PCA) (Schölkopf, Smola, &
Müller, 1997).
While kernel methods have a wide range of uses in different applications, this
research employs kernel functions as a similarity measure to improve the prediction
accuracy of Bayesian methods for count data and introduce probabilistic predictions to
traditional classification tools. In particular, the kernel matrix produced by the implicit
mapping and the computation of the inner product is used to integrate information from
the covariates in the input data into the Bayesian computation of the parameters of the
posterior distribution.
Bayesian Kernel Models
Bayesian kernel methods have recently been introduced to the machine learning
literature providing probabilistic solutions as opposed to deterministic solutions. Most
Bayesian kernel methods are developed with Gaussian prior distributions (Figueiredo,
2001; Tipping, 2001; Zhang et al., 2011). For an 𝑚 × 𝑑 data matrix 𝐗 with rows
corresponding to 𝑚 data points each with 𝑑 attributes, the function 𝐭(𝐗) is considered to
17
be a random vector of length 𝑚 mapping input data 𝐗 into a class. Gaussian Bayesian
kernel models assume the vector-valued function 𝐭 follows a multivariate normal
distribution with mean 𝐸[𝐭(𝐗)] = 𝟎 and covariance matrix Cov(𝐭(𝐗)) = 𝐊, where
matrix 𝐊 is positive definite and matrix element K𝑖𝑗 is the kernel function 𝑘(𝑥𝑖 , 𝑥𝑗 ),
between the 𝑖𝑡ℎ and 𝑗𝑡ℎ data points. The multivariate normal distribution for the
realization of 𝐭 is found in Eq. (2-5).
𝑃(𝐭) = 1
√(2𝜋)𝑚 (𝑑𝑒𝑡 𝐊)−1 2⁄ 𝑒𝑥𝑝 (−
1
2 𝐭𝑇 𝐊−1𝐭) (2-5)
As computing the inverse of the kernel matrix in the probability density function of 𝐭
can be cumbersome, Schölkopf and Smola (2002) introduce a new vector-valued
variable of length 𝑚, 𝛚, such that 𝑡(𝐱𝑖 ) = 𝐤(𝐱𝑖 , 𝐗)𝛚, whose prior is also a multivariate
normal distribution, Eq. (2-6).
𝑃(𝛚) = 1
√(2𝜋)𝑚 (𝑑𝑒𝑡 𝐊)−1 2⁄ 𝑒𝑥𝑝 (−
1
2 𝛚𝑇 𝐊𝛚) (2-6)
Since the first term in the probability density function, 1
√(2𝜋)𝑚 (det 𝐊)−1 2⁄ , does not
depend on the parameter 𝛚, the prior distribution can be reduced to Eq. (2-7).
𝑃(𝛚) = 𝑒𝑥𝑝 (− 1
2 𝛚𝑇 𝐊𝛚) (2-7)
In the case of a binary classification, an appropriate likelihood function would be the
logit function shown in Eqs. (2-8) and (2-9).
𝑃(𝑦 = 1|𝑡(𝐱)) = 1
1 + 𝑒𝑥𝑝(−𝑡(𝐱)) (2-8)
18
𝑃(𝑦 = −1|𝑡(𝐱)) = 1
1 + 𝑒𝑥𝑝(𝑡(𝐱)) (2-9)
The posterior distribution is then the product of the likelihood function and the prior
distribution for a data set of 𝑚 data points, found in Eq. (2-10) .
𝑃(𝝎|𝒚) α ∏ ( 1
1 + 𝑒𝑥𝑝(−𝐤(𝐱𝑖 , 𝐗)𝝎) )
0.5+0.5𝑦𝑖 𝑚
𝑖=1
× ∏ ( 1
1 + 𝑒𝑥𝑝(𝐤(𝐱𝑖 , 𝐗)𝝎) )
0.5−0.5𝑦𝑖 𝑚
𝑖=1
× 𝑒𝑥𝑝 (− 1
2 𝝎𝑇 𝐊𝝎)
(2-10)
To estimate the parameter of interest, 𝝎, Eq. (2-10) is maximized (or its negative log is
minimized) using any of several optimization algorithms (e.g., the Newton-Raphson
method).
An important extension to the basic Bayesian kernel model is the non-Gaussian
Bayesian kernel model (Montesano & Lopes, 2009; Mason & Lopes, 2011; MacKenzie
et al., 2014b), which can improve the predictive accuracy for certain problems where a
Gaussian distribution for model parameters should not realistically be assumed.
MacKenzie et al. (2014b) highlight some of the drawbacks of using the Gaussian
distribution for binary classification problems, use a Beta conjugate prior, and offer an
alternative likelihood function to the logit. The research expands on previous work done
on Non-Gaussian kernel models (Montesano & Lopes, 2009; Mason & Lopes, 2011) by
introducing a more generalized model based on the Beta conjugate prior which provides
much faster computation time than traditional Bayesian kernel methods that rely on
19
either optimization or simulation to find the solutions. Beta Bayesian kernel methods
will be revisited in details in Chapter 5 to model the resilience of infrastructure systems.
Bayesian Kernel Model for Count Data
This dissertation proposes a new Bayesian kernel method that can accommodate
count data. The purpose of such model is to integrate prior information in the form of
experts’ knowledge with historical data and attribute information to produce a
probability distribution of the rate of occurrence of a particular event. Using this
distribution, the frequency of events such as disruptions can be predicted. This section
first reviews count data modeling approaches in the literature then discusses the
structure of the Poisson Bayesian kernel model. The model is empirically tested using
sample data and deployed in a case study of a critical infrastructure system disruptions
prediction.
Count Data Modeling
One of the classical approaches used to analyze count data are Generalized
Linear Models (GLM) (Agresti, 2002; Cameron & Trivedi, 2013; Nedler &
Wedderburn, 1972). The Poisson GLM is most commonly used to model count data.
The method assumes that the rate to be estimated has an exponential relationship with a
set of covariates representing coefficients for the different attributes, shown in Eq.
(2-11).
�̂� = 𝑒𝛽𝑖𝑋 (2-11)
Under the Poisson GLM, the response follows a Poisson distribution, Eq. (2-12), and
the log function is the link function that relates the set of covariates and coefficients to
the response variable.
20
𝑃(𝑦) = 𝜆𝑦 𝑒−𝜆
𝑦! (2-12)
Another type of GLM for modeling count data is the Negative Binomial GLM
which relaxes the constraints of homoscedasticity imposed by the Poisson GLM (Cox,
1983; Lawless, 1987). The Negative Binomial GLM assumes that the marginal
distribution of the response follows a Negative Binomial distribution, Eq. (2-13), where
𝑘 is the overdispersion parameter and λ is assumed to follow a Gamma distribution.
𝑃(𝑦) = Γ (𝑦 +
1 𝑘
)
Γ(𝑦 + 1)Γ ( 1 𝑘
) (
𝑘λ
1 + 𝑘λ )
𝑦
( 1
1 + 𝑘λ )
1 𝑘 (2-13)
The Negative Binomial GLM also assumes a log function for the link function and as a
result, the response variable has an exponential relationship with the covariates. While
the Negative Binomial GLM is a formal way of handling overdispersion in count data,
other approaches developed extensions to the classical Poisson regression (Breslow,
1984; Paul & Placket, 1978; Johnson, Kotz, & Kemp, 2005; Yip, 1991). A well-known
approach is to add a dispersion parameter such as the case with the Quasi-Poisson
regression. Many other count data regression models have been developed to introduce
further flexibility and complexity into the modeling approach.
One example of added complexity to the Poisson GLM is the Zero-Inflated
Poisson (ZIP) model which assumes a form of mixture modeling to account for a
specific type of data expressing a large number of occurrences that are equal to zero.
Such an approach has shown improved goodness of fit compared to Poisson and
Negative Binomial GLM, but it is a much more complicated approach, it is difficult to
interpret, and it is only applicable to particular cases where the majority of the response
21
variable observations are equal to zero. Examples of such applications are insurance
claims (Mouatassim & Ezzahid, 2012), dental caries prevention methods (Hall & Shen,
2010), and defects in manufacturing (Lambert, 1992), among others. Also, since the
likelihood is constructed for a mixture of models, there is no analytical solution for the
estimation of parameters, instead, algorithms such as Newton-Raphson or EM (Fong &
Yip, 1993) are used to maximize the log-likelihood function and compute estimates of
the coefficients.
An example of added flexibility to count data modeling is the Conway-Maxwell
Poisson GLM (Conway & Maxwell, 1962). The model can address both overdispersion
and underdispersion in one framework and has been applied in the risk and reliability
context using a Bayesian framework to account for the uncertainty in the regression
parameters and improve on their accurate estimation (Guikema & Goffelt, 2008). Since
such Bayesian techniques employ simulation, the higher accuracy comes at the cost of a
longer computation time.
Other approaches of analyzing count data using a Bayesian framework are
conjugate priors. These methods are quite attractive as they offer the benefit of
uncertainty modeling using Bayesian techniques without adding any computational
cost. Given a specific prior distribution and a specific likelihood function, the posterior
distribution will have the same form as the prior distribution but with updated posterior
parameters. Different forms of conjugate priors are discussed in this dissertation, one of
which is the Gamma conjugate prior used to model count data. The method assumes
that the rate of occurrence follows a Gamma prior and updates the distribution using
information represented by a Poisson likelihood. The Gamma conjugate prior is the
22
foundation of the Poisson Bayesian kernel model proposed in this research and will be
further discussed in the following section. Sophisticated extensions to this conjugate
prior include the analysis of the parameters of the gamma prior distribution
(Winkelmann, 2008). Other extensions to Bayesian Poisson methods consider
hierarchical models (Tunaru, 2002). The model is based on the multivariate Poisson-log
normal distribution with a hierarchical Bayesian application. This multivariate
distribution is used to model discrete multiple count data and is shown in Eq. (2-14).
𝑌𝑘𝑖 |𝜆𝑘𝑖 ~ ind Pois(𝜆𝑘𝑖 )
(log(𝜆𝑘𝑖 ))𝑖=1,…,4|𝜇, 𝑇~ iid𝑁4(𝜇, 𝑇)
𝜇𝑖 ~ iid𝑁(0,0.0001)
𝑇~Wishart(𝑅, 4)
(2-14)
𝑁𝑀(𝜇, 𝑇) is the M-dimensional multivariate normal distribution. The mean vector is
represented by 𝜇, and 𝑇 is the inverse of the covariance matrix. The hyperprior
parameters 𝑅 and 𝜋 = 𝑀 are known. The model is advantageous in that it can model
joint responses and can detect relationships among the categories of count variables.
However, Markov Chain Monte Carlo methods are utilized to make inferences about
the model parameters, for which computation can oftentimes be complex and lengthy.
The Poisson Bayesian kernel model developed in this research is simple enough
to avoid expensive computations but detailed enough to overcome issues in basic
Bayesian modeling approaches, such as the Gamma conjugate prior, and in count data
regression models, such as the GLM.
23
Poisson Bayesian Kernel Model
Poisson Bayesian kernel methods estimate the rate of occurrence of the event
rather than estimating a deterministic value for the number of times the event is
estimated to occur. A common distribution to model count data within a Bayesian
framework is the Gamma-Poisson conjugate prior. The development of the Poisson
Bayesian kernel method discussed can be found in Baroud et al. (2013a) and Floyd et
al. (2014). The approach uses the Gamma conjugate prior as the basis of the model.
It is assumed that the parameter to be estimated is the rate of occurrence, 𝜆 > 0,
which follows a Gamma prior distribution with parameters 𝛼 > 0 and 𝛽 > 0, as shown
in Eq. (2-15).
𝑃(𝜆) = 𝛽𝛼
Γ(𝛼) 𝜆𝛼−1e(−𝛽𝜆) (2-15)
For the likelihood function, the product of the Poisson density function, shown in Eq.
(2-16), is used, since this is a Gamma-Poisson conjugate prior approach.
𝐿 = ∏ 𝑃(𝑦𝑖 )
𝑚
𝑖=1
= ∏ (𝜆
𝑖 𝑦𝑖 𝑒−𝜆𝑖 )
𝑦𝑖 !
𝑚
𝑖=1
= 𝜆𝑖
∑ 𝑦𝑖 𝑚 𝑖=1 𝑒−𝑚𝜆𝑖
∏ 𝑦𝑖 ! 𝑚 𝑖=1
(2-16)
Thus, the posterior distribution is the product of Eqs. (2-15) and (2-16). Rearranging the
product of the likelihood function and the prior distribution function results in a Gamma
posterior distribution where 𝛼∗ = ∑ 𝑥𝑖 𝑚 𝑖=1 + 𝛼 and 𝛽
∗ = 𝑚 + 𝛽.
𝑃(𝜆|𝑥) = ( 𝛽𝛼
Γ(𝛼) 𝜆𝛼−1𝑒−𝛽𝜆) (𝜆∑ 𝑦𝑖
𝑚 𝑖=1 𝑒−𝑚𝜆 )
= 𝜆(∑ 𝑦𝑖
𝑚 𝑖=1 +𝛼−1) 𝑒−𝜆(𝑚+𝛽)(𝑛 + 𝛽)∑ 𝑦𝑖
𝑚 𝑖=1 +𝛼
Γ(∑ 𝑦𝑖 𝑚 𝑖=1 + 𝛼)
= Gamma (𝛼∗, 𝛽∗)
(2-17)
24
This result is the basic Gamma conjugate prior approach used in Bayesian analysis. This
approach assumes the notion of exchangeability meaning that for different sets of
training and testing data, the resulting posterior parameter will be similar since they are
a function of the prior parameter, the size of the dataset, and the summation of all the
data points. The characteristics of each outcome are not taken into consideration in this
case, but rather the overall property of the dataset (Mackenzie et al., 2014b).
The Poisson Bayesian kernel approach extends the notion of the conjugate prior
such that the posterior parameters computation not only depends on the prior parameters
and the historical data but also on the attributes through the kernel matrix. The
parameters for the Bayesian kernel model for counts are expressed in Eqs. (2-18) and
(2-19). 𝐊 is the 𝑚 × 𝑚 kernel matrix, 𝐘 is an 𝑚 × 1 vector containing the output data
associated with the 𝑚 observations of 𝐗, and 𝐕 is an 𝑚 × 1 vector containing ones.
Each entry in the kernel matrix represents the similarity measure between the attributes
of the testing set and the training set. As such, the new data point is compared with the
training set and according to the similarities of the attributes, new values for the
parameter of the posterior distribution are computed. Note that in this case, the training
and testing sets are assumed to have the same size, 𝑚. However, when the model is
deployed, the sets can be of different sizes, and in some cases, the testing set could
include only one data point such as in a leave-one-out analysis that will be illustrated in
the case study.
𝛼∗ = 𝐊𝐘 + 𝛼 (2-18)
𝛽∗ = 𝐊𝐕 + 𝛽 (2-19)
25
As with other statistical and mathematical models, there are a few assumptions
underlying the deployment of such modeling approach. Even though the form of the
prior distribution is known from the conjugate prior, the model user would still need to
identify the values of the prior parameters. While there are formal ways to determine
the prior parameters (Kass & Wasserman, 1996), the selection of such parameters might
not always be considered (Montesano & Lopes, 2009; Mason & Lopes, 2011).
Oftentimes, the priors are either assumed to be known or are assigned such that the prior
distribution is non informative. In other cases, these parameters are estimated using
data and prior knowledge by matching the sample mean and variance to those of the
prior distribution (MacKenzie et. al, 2014b; Carlin & Louis, 2008). Further discussion
on the choice and impact of prior parameters is provided in the case study of this
chapter. Another assumption to consider is the choice of the kernel function which
depends on the application and the model user. This research uses the most popular
kernel function, the radial basis function (RBF) in Eq. (2-20), where 𝑘(𝐱𝑖 , 𝐱𝑗 ) is one
entry in the matrix 𝐊 representing the kernel function between the attributes of the 𝑖𝑡ℎ
and 𝑗𝑡ℎ data points.
𝑘(𝐱𝑖 , 𝐱𝑗 ) = exp (− ‖𝐱𝑖 − 𝐱𝑗 ‖
2
2𝜎2 ) (2-20)
In addition to being commonly used in kernel methods, RBF has nice properties. The
function has only one parameter, 𝜎, to be tuned to an optimal value. This reduces
computation efforts significantly in comparison to other kernel functions with two or
more parameters requiring a grid search to estimate them. Also, the structure of the
function is based on the Euclidean distance, whereby similar data points are closer to
26
each other in the feature space. Finally, the kernel matrix of the RBF has full rank and
the entries fall between zero and one resulting in kernel functions of the data points
acting as weights in the computation of the posterior parameters (Schölkopf & Smola,
2002). More discussion on the tuning process of the RBF parameter, 𝜎, will follow in
the empirical analysis of this chapter.
The rate for the new data point follows then a Gamma distribution with
parameters 𝛼∗ and 𝛽∗. As a point estimate for this parameter, the expected value of the
posterior distribution is considered, shown in Eq. (2-21) as the ratio of the Gamma
distribution parameters 𝛼∗ and 𝛽∗.
�̂� = 𝛼∗
𝛽∗ (2-21)
Note that a different point estimate for the rate can be used such as the median, the
mode, or the variance, depending on the type of problem and the model users.
Goodness of Fit Measures
In order to assess the performance of the model, goodness of fit measures are
analyzed to identify the capability of the model to capture data patterns. The empirical
analysis and the case study compare the Poisson Bayesian kernel (PBK) model to other
classical methods for modeling count data, the Poisson generalized linear model (GLM)
and the Negative Binomial GLM (Cameron & Trivedi, 1986, 2013). The Poisson and
Negative Binomial GLM, presented above, assume that the rate to be estimated has an
exponential relationship with a set of covariates representing coefficients for the
different attributes, �̂�𝑃𝐺𝐿𝑀 = 𝑒 𝛽𝑖𝑋, while the predicted rate for the PBK is equal to the
expected value of the posterior probability distribution, �̂�𝑃𝐵𝐾 = 𝐊𝐘+α
𝐊𝐕+β .
27
The functional values of two metrics are used to compare how well the models
fit the data and are able to explain the variance. The first metric is the deviance, which
computes the difference in the log-likelihood function between the fitted model and the
saturated model, Eq. (2-22), where 𝑦𝑖 is the true value of the data point and �̂� is the
estimated rate for the particular data point.
𝐷 = 2 × (𝑙(𝒚|𝒚) − 𝑙(�̂�|𝒚)) (2-22)
The deviance is the generalized form of the sum of squared errors used in the linear
regression model, it is a metric that analyzes the discrepancy between the observed and
estimated values and it is the most commonly used goodness of fit measure by GLM
users (McCullagh & Nelder, 1989). The deviance for a Poisson regression model is
represented in Eq. (2-23), where 𝑦𝑖 log(𝑦𝑖 �̂�𝑖⁄ ) = 0 when 𝑦𝑖 = 0. 𝐷𝑃 is used to assess
how well the fitted values are representing the observed rate of occurrences in the
Poisson Bayesian kernel model in comparison to the Poisson and Negative Binomial
GLM.
𝐷𝑃 = 2 × ∑ 𝑦𝑖 log ( 𝑦𝑖
�̂�𝑖 ) − (𝑦𝑖 − �̂�𝑖 )
𝑚
𝑖=1
(2-23)
The second metric used is the functional value of the log-likelihood, shown in
Eq. (2-24), which is to be maximized. The log-likelihood function represents the joint
probability of the observed data as a function of the parameter of interest which is �̂� in
this case. The larger the value of this function, the better the model is able to capture the
data patterns using the estimated parameters.
𝑙(�̂�|𝒚) = ∑[𝑦𝑖 ln (�̂�𝑖
𝑚
𝑖=1
) − �̂�𝑖 − ln (𝑦𝑖 !)] (2-24)
28
Prediction Accuracy
The ultimate objective of building the Poisson Bayesian kernel model is to
deploy it in risk analysis problems, such as predicting the frequency of disruptions in a
particular network system. While the goodness of fit is important to assess whether the
model is capturing the pattern and variability in the data, is it equally important to
analyze the prediction power of a statistical model if it is going to be used for
forecasting purposes. Prediction accuracy is assessed by the out-of-sample error, which
accounts for the discrepancy between the estimated parameter and the actual
observation of data points that were not in the set used to train the model. Figure 2-2 is
a representation of the error in the training sample (in-sample error) and the test sample
(out-of-sample error) as a function of the model complexity.
Figure 2-2: Test and training error as a function of the model complexity (Hastie et al.,
2001)
29
Adding complexity to the model will decrease the training error but may cause
overfitting at some point resulting in a poor prediction accuracy when the model is
applied to an independent data set. Depending on the ultimate application of the
method, a model can be selected based on either the test or the training error. The model
developed in this dissertation is aimed at ultimately predicting the frequency of
disruptions to inform preparedness and recovery decision making strategies and
investment. An accurate prediction represented by a small test error is desired.
In order to validate the prediction power of the models, several metrics are
evaluated to assess the out-of-sample error, and they are summarized in Table 2-1.
Table 2-1: Prediction error measurement metrics
Prediction accuracy metrics Formula
Root Mean Square Error 𝑅𝑀𝑆𝐸 = 1
𝑛 √∑ (𝑌𝑖 − �̂�𝑖 )
2 𝑛𝑖=1
Normalized Root Mean Square Error
𝑁𝑅𝑀𝑆𝐸𝐷 = 1
𝑛 √∑ (𝑌𝑖−�̂�𝑖)
2 𝑛𝑖=1
𝑠𝑑(𝑌𝑖)
𝑁𝑅𝑀𝑆𝐸𝑀 = 1
𝑛 √∑ (𝑌𝑖−�̂�𝑖)
2 𝑛𝑖=1
𝑌𝑚𝑎𝑥𝑖𝑚𝑢𝑚−𝑌𝑚𝑖𝑛𝑖𝑚𝑢𝑚
Mean Absolute Error 𝑀𝐴𝐸 = 1
𝑛 ∑ |𝑌𝑖 − �̂�𝑖 |
𝑛 𝑖=1
While RMSE and MAE are the most commonly used measurements of error, the
normalized RMSE (NRMSE) is also considered to account for the variability across the
different data sets evaluated in the empirical study. NRMSE can either be normalized
based on the standard deviation of the observed values, 𝑠𝑑(𝑌𝑖 ), or the range of values in
the testing set, 𝑌𝑚𝑎𝑥𝑖𝑚𝑢𝑚 − 𝑌𝑚𝑖𝑛𝑖𝑚𝑢𝑚, and both cases are considered as the different
30
data sets exhibit different variability patterns in the observed values of the outcome
variable.
Empirical Analysis
The PBK is applied to several data sets (Agresti & Finlay, 2008; Kutner,
Nachtsheim, Neter, & Li, 2005; Lovett & Flowerdew, 1989; Roberts & Foppa, 2006;
Sepkoski & Rex, 1974), and its performance is compared to the Poisson and Negative
Binomial GLM using the goodness of fit and prediction accuracy metrics discussed in
the previous section. A brief description of the data sets is found in Table 2-2. Most of
the data sets are similar in terms of the number of predictors and the size of the data.
One of the sets has a larger number of predictors for a small data set, and another is a
large data set with a small number of predictors. Note that the number of predictors in
all the models is held constant across the data sets to ensure consistency in the
comparison. Also, the parameters of the prior distribution for all data sets are assumed
to be 𝛼 = β = 1, also in order to maintain consistency in the evaluation of the
performance of all models.
31
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32
A holdout analysis is performed where each data set is randomly split into
training and testing sets for 100 trials. Traditional holdout analyses would train the
model on a portion of the data and deploy it on the testing set to make predictions and
compare them to the actual observations. With the PBK, an intermediate step in the
training process is added to tune the unknown parameter, 𝜎, in the kernel function. This
parameter is optimized based on the minimum mean square error. As a result, 30% of
the data was used for testing the model, with 50% of the data used as a training set and
20% as a tuning set. The training and the tuning sets were then combined into one
training set to perform the testing. For each of the three models, the estimated rate of
occurrence is computed for the testing test and used to evaluate the deviance, the log-
likelihood functional value, and the four out-of-sample error measurements given the
observed values. This process is repeated 100 times where, at each iteration, random
samples of training, tuning, and testing sets are chosen. Table 2-3 contains a summary
of the analysis. The performance metrics values presented in the table below are the
average values of the performance measures evaluated over 100 trials. PBK refers to the
Poisson Bayesian kernel model and PGLM refers to the Poisson GLM, and NBGLM
refers to the Negative Binomial GLM. Recall that the model with a smaller deviance
and errors and a larger log-likelihood functional value is a better model.
33
Table 2-3: Performance metrics results for the empirical analysis
Data Metrics PBK PGLM NBGLM
Crime
LL -276.25 -256.56 -155.61 DEV 352.87 313.49 343.14
RMSE 26.47 33.15 37.69
NRMSEM 0.28 0.35 0.39
NRMSED 0.89 1.13 1.29
MAE 21.26 21.97 23.19
Murder
LL -120.96 -77.79 -77.79 DEV 107.69 21.38 21.38 RMSE 9.81 3.85 3.86 NRMSEM 0.28 0.17 0.17 NRMSED 0.98 0.58 0.58 MAE 4.68 2.59 2.59
Mussels
LL -97.33 -78.91 -78.72 DEV 66.55 29.71 26.43
RMSE 5.60 5.84 5.83
NRMSEM 0.27 0.31 0.31
NRMSED 0.96 1.08 1.07
MAE 4.00 4.32 4.31
Customer
LL -230.02 -194.46 -194.46 DEV 149.15 78.04 77.69 RMSE 5.13 3.58 3.58 NRMSEM 0.18 0.13 0.13 NRMSED 0.77 0.55 0.55 MAE 3.78 2.75 2.75
West Nile virus in birds
LL -135.35 -103.94 -80.76 DEV 181.74 118.93 36.14 RMSE 7.78 8.44 9.14
NRMSEM 28.85 33.47 36.44
NRMSED 98.22 113.65 123.84
MAE 4.91 5.09 5.23
West Nile virus in equines
LL -40.12 -40.47 -39.42 DEV 43.42 44.12 32.57 RMSE 1.75 2.08 2.05
NRMSEM 0.30 0.41 0.41
NRMSED 0.95 1.24 1.25
MAE 1.17 1.33 1.29
Apprentices Migration to
Edinburgh
LL -127.98 -106.03 -64.46 DEV 442.9 146.62 25.78 RMSE 31.23 32.29 32.45
NRMSEM 0.43 0.53 0.51
NRMSED 1.32 1.61 1.55
MAE 15.98 14.71 15.86
34
Overall, there are five out of seven data sets for which the Poisson Bayesian
kernel model outperforms the Poisson and Negative Binomial GLM in terms of the
predictive accuracy. In particular, those five cases are all among the six small data sets.
The RMSE, NRMSEM, NRMSED, and MAE all behave similarly for all the datasets
and lead to the same conclusion of the model performance, except for a minor
difference in the Apprentices Migration to Edinburgh where the PBK performs
similarly to the NBGLM and slightly worse than the Poisson GLM in terms of MAE
values. With respect to goodness of fit measures, the GLMs perform better than the
PBK. Overall, the Negative Binomial fits the best. PGLM and NBGLM perform
similarly in the two data sets for which the GLM outperforms the PBK in the predictive
accuracy, Customer and Murder. The Poisson Bayesian kernel model appears to be a
good model for prediction purposes when the data set is small with a small number of
predictors, a situation known to cause issues with regression modeling (Cameron &
Trivedi, 1986, 2013).
Case Study: Prediction of Inland Waterways Disruptions
With over 200 lock chambers and more than $150 million worth of goods
flowing yearly (US Army Corps of Engineers, 2011), the inland waterway system plays
an important role in the nation’s economy. Unfortunately, the system’s reliability is
declining due to the aging components of the network (Grier, 2009). According to the
American Society of Civil Engineers’ most recent report card on America’s
infrastructure, inland waterways received a grade of “𝐷−“ while dams received a grade
of “𝐷”. Among the most common causes for the degrading status of inland waterways
are aging components. On average, dams in the United States are 52 years old, and by
35
the year 2020, 70% of the dams will be over 50 years old (ASCE report card, 2013b).
As a result, locks and dams are frequently closed for unscheduled or scheduled
maintenance which causes delays in the flow of commodities and incurs large economic
losses across the nation. In 2009, 90% of locks and dams in the US experienced service
interruption resulting in an average of 52 delays a day.
The Poisson Bayesian kernel model is applied to analyze the frequency of lock
closure due to disruptive events on the Mississippi River transportation network. The
network has 29 locks acting as key connectors between different ports nationwide. The
navigation system reflects 9,000 miles of navigable waterway with 70.5% of the U.S.
inland waterway commodity flowing through the network (Clark, Henrickson, Thoma,
2005).
The data, retrieved from the database collected by the U.S. Army Corps of
Engineers (2011), contains detailed information on each lock’s characteristics including
the river mile, the total number of vessels passing by the lock, the total tonnage, and the
frequency and average delay for the vessels and tows experiencing delay time due to the
lock’s closure. In addition to that, data is available on the yearly frequency of closure
for each lock which is considered in this case the outcome to be estimated. A sample of
the data is represented in Table 2-4.
36
Table 2-4: Sample of the inland waterway disruption data
𝒀 𝑿𝟏 𝑿𝟐 𝑿𝟑 𝑿𝟒 …
Lock &
Dam
Closure
Frequency River 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 . . .
... .. .
.. .
.. .
.. .
.. .
. . .
Empirical Analysis of the Inland Waterway Data
The goal of deploying the Bayesian kernel model is to obtain an accurate
prediction of the frequency of disruptions to inform preparedness strategies and
investment decision making. Using the Poisson Bayesian kernel model, decision makers
are able to produce a probability distribution of the number of times a particular lock
and dam will close each year. The distribution can be used to improve risk management
along the inland waterways and make them a more reliable transportation system.
As a first step, the prediction accuracy of the PBK model is tested in comparison
with the PGLM and the NBGLM. Similarly to the analysis done in the empirical study,
a holdout analysis is performed to assess the goodness of fit and prediction accuracy of
PBK for the inland waterway, and the results are summarized in Table 2-5.
37
Table 2-5: Performance metrics results for the inland waterway data analysis
Metrics PBK PGLM NBGLM
Full model
LL -285.06 -148.06 -75.09 DEV 486.03 211.65 23.89 RMSE 32.82 63.94 131.03
NRMSEM 0.34 0.66 1.24
NRMSED 0.94 1.88 3.45
MAE 21.53 33.23 57.15
Best model - PGLM
LL -252.25 -146.01 DEV 420.53 208.06 RMSE 32.60 42.37 NRMSEM 0.34 0.46 NRMSED 0.95 1.34 MAE 20.76 25.08
Best model - NBGLM
LL -238.07 -78.13
DEV 391.33 24.15
RMSE 28.46 46.74
NRMSEM 0.30 0.54
NRMSED 0.87 1.56
MAE 18.00 26.01
According to the values of the average out-of-sample error expressed in the four
metrics, RMSE, NRMSEM, NRMSED, and MAE, PBK does a better job at making
accurate predictions of the average frequency of lock and dam closures even though
based on the values of the log-likelihood and deviance, GLM, more specifically the
Negative Binomial GLM is better at fitting the data. One of the reasons a GLM might
not be providing good prediction errors is overfitting. In order to check whether the
results obtained, after fitting a full model that includes all covariates, are due to the
GLM overfitting the data, the analysis is performed given the best version of the GLMs.
The selection of covariates for each of the Poisson and Negative Binomial GLM is
based on Akaike’s Information Criterion (AIC) (Akaike, 1970) that penalizes additional
parameters contributing to the model complexity. In terms of goodness of fit measures,
both reduced GLMs did not express any change in the values of the log-likelihood and
38
deviance from the full model. However, the prediction accuracy improved significantly
for the reduced model with about 60% decrease in the values of RMSE, NRMSEM,
NRMSED, and MAE for NBGLM and about 30% decrease for PGLM error
measurement values. The covariates selected for the best models were also included in
the PBK evaluation for consistency. The PBK still performed better than the best
version of both GLMs and maintained better predictive error measures even though the
reduced versions of GLMs significantly improved their prediction accuracy.
Prior Distribution Implications
One of the advantages of using Bayesian methods in risk analysis is the
flexibility of the approach in (i) establishing assumptions, and (ii) interpreting the
results. Any prior belief about the risk measure to be estimated can be embedded in the
prior distribution. Determining the prior parameters can be challenging and can result in
significant implications on the posterior parameters’ estimation. Not much work has
been devoted to analyzing priors, but discussions on selecting priors can be found in
MacKenzie et al. (2014b), Carlin & Louis (2008), and Guikema (2007). So far, the
analysis considered the same prior distribution with prior parameters 𝛼 = 𝛽 = 1 to
insure consistency in the empirical study across the different data sets and models. This
section examines the implications of changing the priors on the posterior parameters.
In risk analysis problems, experts in the field can help in assessing any prior
knowledge about the parameter to be estimated. Ideally, risk managers are interviewed,
and using probability elicitation techniques (Spetzler & Stael von Holstein, 1975), a
prior probability distribution is defined. Three levels of knowledge are considered in
this case that influence the estimation of the priors. For each case scenario, the posterior
39
frequency of disruptions is computed and compared to results from fitting a PGLM and
a NBGLM. The distribution of the RMSE across the three models under each case
scenario is used to assess the impact of the priors.
The first approach assumes the experts have a perfect knowledge about the
frequency of disruptions and the prior parameters are estimated from the data using the
method of moments, Eq. (2-25) , where �̅� and 𝑠2 are respectively the mean and variance
of the historical data.
𝛼 = �̅�2
𝑠2
𝛽 = �̅�
𝑠2
(2-25)
The plot in Figure 2-3 shows that the distribution of RMSE values is skewed towards
the smaller values (around 25), while the PGLM and NBGLM distributions of RMSE
values are spread across larger values with thicker tails. The dashed lines correspond to
the mean RMSE showing that PBK performs the best in terms of prediction accuracy.
Figure 2-3: RMSE distribution with perfect prior knowledge
40
The second case scenario assumes that the risk managers have some prior
knowledge but it is not perfect like in the first approach. The bias introduced by the risk
managers is modeled with a random noise and the distribution of RMSE for the three
models is depicted in Figure 2-4. The expected value of RMSE and the overall
distribution of the values are both quite similar to the case where the knowledge is
assumed to be perfect. Note that the added noise in this case is not very significant. If
the risk manager expressed a stronger bias, more noise would be added which could
impact the prior parameters estimation and ultimately the posterior distribution and
predicted values of the frequency of disruption.
Figure 2-4: RMSE distribution with imperfect prior knowledge
For the third approach, it is assumed that the risk managers have no prior
knowledge, or there is no access to a reliable source of information to estimate informed
prior parameters. Therefore, the priors are arbitrarily determined and the distribution of
the RMSE values is plotted in Figure 2-5. The smaller values of RMSE still have the
41
highest frequency in the distribution of RMSE for PBK; however this peak is now
centered around values equal to 50 as opposed to 25 in the first two approaches. The
overall distribution shifted to the right, towards larger values of the RMSE, and the
distribution is overlapping with PGLM and NBGLM RMSE distributions. In addition,
the expected value of RMSE for PBK increased and is approaching the RMSE expected
value of the PGLM, although it is still significantly smaller.
Figure 2-5: RMSE distribution with no prior knowledge
The selection of prior parameters has an implication on the form of the posterior
distribution. A poorly formulated prior distribution can impact the performance of the
PBK in predicting the frequency of lock and dam closures. On the other hand, a perfect
prior distribution relying solely on the historical data is unrealistic. Therefore, the model
user must carefully formulate the prior distribution and prior parameters to ensure
accurate posterior inferences.
42
Research Outcome
The model developed in this chapter can impact the decision making process for
the protection and rehabilitation of the U.S. inland waterways. As mentioned earlier,
this critical infrastructure system is suffering from aging components resulting in
frequent disruptions of the flow of commodities across the nation. The Department of
Homeland Security announced a set of grant programs to protect and rehabilitate critical
infrastructure systems. These grants are normally assigned based on the priority of the
rehabilitation project due to the limited availability of resources. Using the hold-one-out
analysis approach, the PBK is used to produce a rank of the locks and dams of the
inland waterway network. Such a ranking of an infrastructure system’s components is
one way to implement data-driven risk analysis into real world decision making. Table
2-6 contains the top five locks & dams with the highest predicted frequency of closures
per year.
Table 2-6: Locks/dams with highest frequency of closures
Ranking Lock/Dam ID
1 L&D 27
2 Mel Price L&D
3 L&D 19
4 L&D 21
5 L&D 22
Frequent disruptions might be one indication of a component’s reliability and
urgent need for rehabilitation. As such, the ranking produced in the table above can
either be used to allocate grants accordingly or it can be integrated into a multiobjective
decision tool that incorporates other factors into the assessment of a rehabilitation
43
project. In fact, the methodologies developed throughout this dissertation yield different
ways to prioritize the inland waterway recovery and repair strategies. Other factors are
considered in the following chapters such as the economic interdependent impacts of a
disruption and cost of preparedness strategies (Chapter 3), the resilience interdependent
effects and cost of repair (Chapter 4), and the resilience importance measures of each
component (Chapter 5).
Concluding Remarks
Bayesian kernel methods are powerful tools in forecasting data. These models
make use of the Bayesian property by relying on historical data and experts’ knowledge,
but they also add more specificity to the model by using the kernel function. Gaussian
Bayesian kernel models became very popular recently and were extended and applied to
a number of classification problems. An important extension to those models is the non-
Gaussian model which gives more flexibility in applying this methodology to all types
of data set, however, there has been no Bayesian kernel model in the literature that
addresses count data.
This chapter introduced count data modeling to the class of Bayesian kernel
methods. Using the notion of the conjugate prior, the rate of occurrence is assumed to
follow a Gamma prior and posterior distribution using the Poisson likelihood function.
The parameters of the posterior distribution are constructed using results from the
classical Bayesian Gamma conjugate prior and the exchangeability argument.
The Poisson Bayesian kernel model presented in this chapter is empirically
tested and compared with the classical Poisson and Negative Binomial GLM. The three
models were used to fit several datasets having similar characteristics in terms of the
44
size of the data and the number of predictors. The evaluation of the performance of each
model is based on the values of metrics corresponding to the goodness of fit and
prediction accuracy. Based on the results obtained, the Poisson Bayesian kernel model
outperforms the Poisson and Negative Binomial GLM in the majority of the sets for
most of the performance metrics representing the out-of-sample error. Also, the Poisson
Bayesian kernel model is potentially a better model for small-sized data sets having few
predictors. Such a result can be very useful in risk analysis applications to estimate the
rate of occurrence of a certain disruption in transportation systems or power grids. In
such cases, data can be limited due to the lack of occurrence of the event and the
possible factors that might cause a disruption. The need for a more accurate estimation
of the rate of disruption can help save lives and lead to more efficient preparedness and
recovery investment and allocation.
The Poisson Bayesian kernel model is illustrated using waterway transportation
network data of the frequency of lock closure along the Mississippi river, and compared
to the classical Poisson and Negative Binomial GLM for the six metrics used in the
empirical study. While GLMs exhibit a better fit of the data, the Bayesian kernel model
produces a smaller out-of-sample error suggesting a better prediction power.
Accurate predictions of the frequency of disruptions are used to rank the locks
and dams and allocate rehabilitation resources accordingly. Realistically, the rank
would be one of many criteria used in the decision making process. This chapter
addresses the prediction of risk of infrastructure disruptions, the second step would be
to understand and quantify the interdependent economic impacts of a disruptive event
and how they influence preparedness decision making accordingly.
45
Chapter 3
Managing the Risk of Interdependent Impacts of Infrastructure
Disruptions
Decision making for managing risks to critical infrastructure systems requires
accounting for (i) the uncertain behavior of disruptive events which was addressed in
Chapter 2, and (ii) the interdependent nature of such systems that lead to large-scale
inoperability which will be addressed in this chapter. This work integrates a dynamic
risk-based interdependency model, the Dynamic Inoperability Input-Output Model, with
a multiobjective decision tree to analyze preparedness decisions. The use of a dynamic
model allows for resilience and recovery decisions to be incorporated in the decision
making framework, and uncertainty is accounted for using probability distributions. The
multiobjective inoperability decision tree is applied to the study of transportation
infrastructure disruptions, namely closures of an inland waterway port.
Research focusing on economic impact analyses of disruptions to transportation
systems has primarily been devoted to highway and railway transport systems (Sohn,
Hewings, Kim, Lee, & Jang, 2004; Gordon et al. 2004; Ham, Kim, & Boyce, 2005a,b)
and coastal ports (Rosoff & von Winterfeldt, 2007; Park, 2008; Jung, Santos, &
Haimes, 2009). Little work is done to understand the impacts of inland waterway port
and network closures: Pant et al. (2011) provide a simulation model of inland port
activities to parameterize a port disruption within a multi-regional interdependency
model, while MacKenzie et al. (2012a) focus on the multi-regional impacts in
46
functionality and in economic losses of decision making for shipping alternatives
following an inland port disruption.
This work integrates a dynamic interdependency modeling methodology with a
stochastic decision analysis methodology to assess investment strategies for
infrastructure preparedness, guided by prior work by Santos, Barker, & Zelinke Iv
(2008), who developed a static formulation for the interdependent effects of biofuel
subsidies and the widespread adoption of biofuels. The dynamic interdependency model
and the stochastic decision tree have both been separately used in risk analysis and
decision making problems. However, the integration of both approaches with the
addition of uncertainty analysis through probabilistic measures is a novel idea
constituting the main contribution of this work. The sections of this chapter, in order,
provide: the methodological background of the dynamic multiobjective inoperability
decision tree (MOIDT); an extension of the MOIDT to address stochastic decision tree
problems; a case study of a disruption of the Port of Catoosa, an inland waterway port
on the Mississippi River Navigation System near Tulsa, Oklahoma; and concluding
remarks. This research has appeared in Baroud et al. (2014a).
Literature Review
This section provides a discussion of several of the components that will make
up the dynamic stochastic multiobjective inoperability decision tree deployed for
infrastructure preparedness.
Stochastic Decision Tree
A common tool for aiding the decision making process through the graphical
depiction of a sequence of decisions and uncertain events is the decision tree (Raiffa,
47
1968). A simple one-period decision tree is depicted in Figure 3-1, with notation from
Santos et al. (2008) and Barker and Wilson (2012). Decisions are made at decision
nodes, designated by squares in the decision tree, and branches extending from a
decision node represent actions, options, or alternatives from which the decision maker
has to choose. The notation in Figure 3-1 for the 𝑙th decision alternative for the 𝑘th
time period in which that decision is made is alternative 𝑎𝑙 𝑘. Chance nodes are
designated with a circle, and branches emanating from a chance node represent the
states of nature, the occurrences which occur with some known probability. The 𝑗th
state of nature in the 𝑘th time period in which the chance event may occur is
represented by 𝑠𝑗 𝑘 . The probability of occurrence of a state of nature is represented by
𝑝(𝑠𝑗 𝑘 ), where ∑ 𝑝(𝑠𝑗
𝑘 ) 𝑗∈𝐽𝑖
𝑘 = 1 and set 𝐽𝑙 𝑘 refers to the set of state of nature subscripts
which follow the 𝑙th alternative in period k and which themselves occur in period k.
Stochastic decision trees, like that depicted in Figure 3-1, can better incorporate
uncertainty by using probability distributions for the likelihood of the occurrence of
states of nature instead of probability point estimates (Hespos & Strassman, 1965). The
use of probability distributions in stochastic decision trees, as opposed to the point
estimates in the traditional decision trees, provides a more comprehensive way to model
uncertainty and a wider selection of decision key parameters. For example, some
decision makers might be interested in the mean of the distribution while more risk
averse decision makers base their analysis on the 70th or higher percentile. The point
estimate alone does not offer sufficient information in the selection of parameters
representing the outcome function.
48
Figure 3-1: Depiction of a general stochastic decision tree
The outcome of a particular path of alternatives and states of nature is quantified
with the function 𝑟(∙). Alternatives are compared by the rollback method, which
consists of computing the expected value of each outcome resulting from a specific path
followed in the tree. For a v-period tree, the expected value for the lth alternative at the
𝑣th decision is 𝐸[𝑎𝑙 𝑣 ] = ∑ 𝑝(𝑠𝑗
𝑣 )𝑗∈𝐽𝑙 𝑣 𝑟(𝑎𝑖
𝑣 , 𝑠𝑗 𝑣 ), and in prior periods, the expected value
of the 𝑙th alternative at the 𝑘th decision is 𝐸[𝑎𝑙 𝑘 ] = ∑ 𝑝(𝑠𝑗
𝑘 ) 𝑗∈𝐽𝑙
𝑘 𝐸[𝑎𝑗 𝑘−1]. If outcome
𝑟(∙) represents an adverse outcome (e.g., risk), the alternative which minimizes 𝐸[𝑎𝑙 𝑘 ]
would be chosen at time 𝑘. Likewise, if 𝑟(∙) represents a beneficial outcome (e.g.,
profit), the alternative that maximizes 𝐸[𝑎𝑙 𝑘 ] would be chosen. Folding back stochastic
decision trees is done using Monte Carlo simulation, as the point estimates 𝑝(𝑠𝑗 𝑘 ) are
replaced with distributions (Dicdican & Haimes, 2005), and consequently 𝑟(∙) functions
1
1 a
1
2 a
11 2 ,
j sar
11 1 ,
j sar
1 j
sP
1 j
sP
49
follow a distribution function. The decision is then based on the probability distribution
of the expected value.
There are typically a finite number of alternatives and states of nature. The
drawback of using a decision tree is that the number of branches emanating from the
nodes must be small, otherwise computations can easily become cumbersome. In
addition, it might not be possible to sufficiently specify all the alternatives necessary to
represent the uncertainty of the event.
Many real-world decisions often require the trade-off of multiple objectives.
Haimes, Li, and Tulsiani (1990) provide an approach for dealing with multiple
objectives in sequential decision making with the multiobjective decision tree (MODT).
The single outcome function 𝑟(∙) corresponding to a sequential set of alternatives is
replaced with a vector-valued outcome function or a vector of 𝑚 outcomes, (𝑟1(∙
), . . . , 𝑟𝑚(∙)). The rollback procedure for MODTs is similar to that for decision trees
with a single objective. Each sequential path of decisions and chances results in a
vector-valued outcome function at the final period v, generally represented as
[𝑟1(𝑎𝑖 𝑣 , 𝑠𝑗
𝑣 ), . . . , 𝑟𝑚(𝑎𝑖 𝑣 , 𝑠𝑗
𝑣 )], noting again that the value of the vector-valued outcome
function is found for a specific path of alternatives and states of nature from periods 1 to
𝑣. When each of the 𝑚 outcomes in (𝑟1(∙), . . . , 𝑟𝑚(∙)) represents an adverse outcome, a
minimum to the vector-valued function is sought at each decision node. However, there
may exist no single optimal alternative to roll back to the previous period but more
likely a set of noninferior, or Pareto-optimal, solutions. A noninferior solution is
defined as a solution to a multiobjective problem where any improvement in one
objective comes only at the expense of another objective (Chankong & Haimes, 2008).
50
Trade-offs among the 𝑚 competing objectives can be calculated between each string of
alternatives.
Interdependency Model
A widely accepted model for describing the interconnected relationships among
infrastructure systems and industry sectors is the Nobel Prize-winning economic input-
output model (Leontief, 1966), as shown in Eq. (3-1). For a set of 𝑛 infrastructure and
industry sectors, 𝑛 1 vector 𝐱 quantifies production outputs in each sector, 𝑛 𝑛
matrix 𝐀 represents the proportional interdependence among sectors (that is, 𝐀𝐱
represents intermediate demand resulting from the production of 𝐱), and 𝑛 1 vector
𝒄 provides final exogenous consumer demand. As such, Eq. (3-1) describes how
changes in consumer demand lead to widespread changes in sector production.
𝐱 = 𝐀𝐱 + 𝐜 ⟹ 𝐱 = [𝐈 − 𝐀]−1𝐜 (3-1)
The extensive usage of input-output models is due, in part, to the availability of
economic interdependency data describing the interconnected nature of infrastructures
and industries in a number of countries (OECD, 2011), including an extensive data
collection effort by the US Bureau of Economic Analysis (BEA), which maintains
input-output tables at different levels of aggregations (BEA 2010). The input-output
framework has also been used in modeling interdependent systems that are not
economic in nature (e.g., by Setola and De Porcellinis (2008)).
The input-output model was extended to describe the propagation of
inoperability, or the proportional extent to which sectors are not performing in an as-
planned manner (e.g., reduced production capability), through several interdependent
51
infrastructure and industry sectors (Santos & Haimes, 2004b). This model, the
Inoperability Input-Output Model (IIM), is expressed in Eq. (3-2).
𝐪 = 𝐀⋆𝐪 + 𝐜⋆ ⟹ 𝐪 = [𝐈 − 𝐀⋆]−1𝐜⋆ (3-2)
Vector 𝐪 is a vector of infrastructure and industry inoperabilities, proportional
reductions in production, describing the extent to which ideal functionality is not
realized following a disruptive event. Inoperability for sector 𝑖 is defined in Eq. (3-3),
where as-planned total output is represented with �̂�𝑖 and degraded total output resulting
from a disruption is represented with �̃�𝑖 .
𝑞𝑖 = (�̂�𝑖 − �̃�𝑖 ) �̂�𝑖⁄ ⟺ 𝐪 = [diag(�̂�)] −1(�̂� − �̃�) (3-3)
An inoperability of 0 suggests that an industry is operating at normal production
levels, while an inoperability of 1 means that the industry is not producing at all.
Normalized interdependency matrix 𝐀∗ is a modified version of the original 𝐀 matrix
describing the extent of economic interdependence among a set of infrastructure and
industry sectors. Shown in Eq. (3-4), 𝐀∗ is an interdependency matrix in which every
entry represents how much inoperability is contributed by the column industry to the
corresponding row industry.
𝑎𝑖𝑗 ⋆ = 𝑎𝑖𝑗 (�̂�𝑗 �̂�𝑖⁄ ) ⟺ 𝐀
⋆ = [diag(�̂�)]−1𝐀[diag(�̂�)] (3-4)
Such a matrix would translate the relationship between industries and the impact
caused to an industry by the disruption of another industry sector. If, for example, a
disruption affects petroleum production, electric power and transportation industries
would also be adversely impacted by the disruption. Note that industry sectors are
impacted differently by the disruption of one particular industry, and in some cases, the
impact is infinitesimal or does not exist, as a result, the matrix entry would be null. Eq.
52
(3-5) calculates 𝐜⋆, a vector of normalized demand reduction, that drives inoperability
in the demand-reduction IIM (Santos, 2006).
𝑐𝑖 ⋆ = (�̂�𝑖 − �̃�𝑖 ) �̂�𝑖⁄ ⟺ 𝐜
⋆ = [diag(�̂�)]−1(�̂� − �̃�) (3-5)
The elements of 𝐜⋆ represent the difference in as-planned demand �̂�𝑖 and perturbed
demand �̃�𝑖 divided by as-planned production, quantifying the reduced final demand for
sector 𝑖 as a proportion of total as-planned output. The effects of inoperability across
multiple sectors can be expressed with total economic losses, Q = xTq, or the amount of
production made inoperable due to a disruption.
A dynamic version of this model, the Dynamic Inoperability Input-Output
Model, or DIIM, calculates the inoperability at any point in time using the recursive
formula in Eq. (3-6), quantifying the temporal nature of how inoperability propagates
across sectors then dissipates with recovery (Lian & Haimes, 2006). The inoperability
vector, 𝐪(𝑡), as well as the vector of demand perturbation, 𝐜⋆(𝑡), change in time. An
𝑛 𝑛 resilience matrix, 𝐊, represents the capability of a certain sector to recover from
the disruptive event and reach a desired performance state.
𝐪(𝑡 + 1) = (𝐈 − 𝐊)𝐪(𝑡) + 𝐊[𝐀⋆𝐪(𝑡) + 𝐜⋆(𝑡)] (3-6)
One way to estimate the entries in matrix 𝐊 is in Eq. (3-7), a result of the
dynamic version of Eq. (3-6) with no temporal demand perturbations (Lian & Haimes,
2006). Value 𝑞𝑖 (0) is the initial inoperability experienced in sector 𝑖 following a
disruptive event, 𝑞𝑖 (𝑇𝑖 ) is the desired inoperability state after recovery (assumed to be
small but nonzero), which requires 𝑇𝑖 time periods to achieve, and 𝑎𝑖𝑖 ∗ is the diagonal
entry in the interdependency matrix.
53
𝑘𝑖 =
ln ( 𝑞𝑖 (0) 𝑞𝑖 (𝑇𝑖 )
)
𝑇𝑖 (1 − 𝑎𝑖𝑖 ∗ )
(3-7)
The IIM and its extensions have been used in a number of risk-based
applications, including inventory decision making (MacKenzie, Santos, & Barker,
2012b; Barker & Santos, 2010a), workforce disruptions (Barker & Santos, 2010b; Orsi
& Santos, 2010), and electric power outages (MacKenzie & Barker, 2012; Anderson,
Santos, & Haimes, 2007), among others.
Distribution Assumptions
Two sources of uncertainty enter the analysis of infrastructure preparedness as
described here: (i) the uncertainty associated with the states of nature that may occur, as
depicted in the decision tree, and (ii) the uncertainty associated with how sequences of
decisions and states of nature will impact demand perturbations in the interdependency
model (thereby driving the calculation of inoperability across sectors). These two
representations of uncertainty are described subsequently.
Discussed in more detail in the methodological development section, the states
of nature describe the magnitude of a disruptive event at the inland port, where such an
event may be rare in nature with large impacts. Originally introduced through the idea
of the scale invariance (Richardson, 1948), which is an inverse power scaling between
independent and dependent variables, the power-law distribution has been used to
model different social, political, and financial patterns in history. Studies aimed at
determining the probability distribution of the severity of a terrorist attack agree that,
according to empirical data on worldwide terrorist events from 1968 to 2008, the
54
probability of a terrorist event claiming x deaths follows a power-law distribution
(Clauset & Young, 2005; Johnson et al., 2006; Clauset, Young, & Gleditsch, 2007).
𝑓(𝑑) = 𝐶𝑑−𝜆 (3-8)
This model, which is provided in Eq. (3-8), for scale parameter 𝜆 and normalizing
constant 𝐶, has been extended to a more general form to describe other covariates such
as the size and experience of the terrorist organization and the type of weapons used in
the attack (Clauset & Gleditsch, 2012; Clauset & Wiegel, 2009). Power-law
relationships are also used to model the severity of natural disasters such as tornadoes
(Malamud & Turcotte, 2012), other severe weather conditions (Dessai & Walter, 2000),
and large earthquakes (Mega et al., 2003), among others. Statistical methods such as the
least square method, the maximum likelihood function, or an extended combination of
these with other statistical tests are used to estimate the parameters of this distribution
(Clauset, Shalizi, & Newman, 2009): , 𝐶, and minimum value of the random variable
𝑑𝑚𝑖𝑛.
The manner in which the disruptive event affects changes in demand is itself
represented with a probability distribution. Like Santos et al. (2008) and Santos (2008),
the beta distribution is used here to describe uncertainty in 𝐜⋆. An advantage of using
this distribution, whose probability density function is shown in Eq. (3-9), is that its
support is on [0,1]. This is beneficial in describing a parameter such as 𝑐𝑖 ⋆, the minimum
and maximum are controllable, and the range of values of its parameters determine
several shapes for the probability distribution.
𝑓(𝑥) = Γ(𝛼 + 𝛽)
Γ(𝛼)Γ(𝛽) 𝑥𝛼−1(1 − 𝑥)𝛽−1 (3-9)
55
The mean and variance of the beta distribution are shown in Eqs. (3-10) and (3-11),
respectively.
𝜇 = 𝛼
𝛼 + 𝛽 (3-10)
𝜎2 = 𝛼𝛽
(𝛼 + 𝛽)2(𝛼 + 𝛽 + 1) (3-11)
The parameters of this distribution are estimated using the method of moments. Given
the values of the sample mean, �̅�, and sample variance, 𝑠2, the parameters are estimated
with Eqs. (3-12) and (3-13).
�̂� = �̅� ( �̅�(1 − �̅�)
𝑠 − 1) (3-12)
�̂� = (1 − �̅�) ( �̅�(1 − �̅�)
𝑠 − 1) (3-13)
Methodology: Dynamic Multiobjective Inoperability Decision Tree
Extending a static version of the multiobjective inoperability decision tree
(MOIDT) (Santos et al., 2008), this section describes a dynamic and stochastic MOIDT
to model the (i) investment in infrastructure preparedness, and (ii) the interdependent
benefits of such investments (or adverse impacts of a lack of investment).
The use of the DIIM considers the cumulative effect of the total economic loss,
a risk measure of interest to decision makers. Also, the problem is solved here by taking
into account the uncertainty in the parameters of the DIIM as well as in the states of
nature in the decision tree. As a consequence, the parameters of the beta distribution for
56
each entry in the demand perturbation vector, 𝐜⋆, depend on the state of nature as well
as on the point estimates of each entry.
Infrastructure Preparedness Decision Problem
An influence diagram is a useful graphical tool that helps to visualize the
decision problem in terms of the decision, uncertainties, objectives, and the influential
relationship each has on the others. For the infrastructure preparedness decision
problem in Figure 3-2, the decision to make, represented by a rectangle, is a certain
amount of dollars invested in preparedness activities. This decision will directly affect
one of the objective functions, represented by a diamond shape, which is the cost. It will
also indirectly affect the other objective function which is the expected total economic
loss as it will impact two uncertainties, represented by a circular shape: (i) the
probability of a disruptive event occurring (in case of a manmade attack), and (ii) the
severity of the disruptive event. Hence, the problem presented is a multiobjective
decision problem in which the decision maker is interested in minimizing both the cost
of preparedness and the expected total economic loss in case of a disruptive event. Both
objectives are competing (as increased investment in preparedness would expectedly
lead to fewer losses). And while both are measured in dollar terms, the objectives are
noncommensurable in that investment budgets would likely come primarily from
government sources, while losses would be experienced across a wide number of
infrastructure and industry sectors.
57
Figure 3-2: Influence diagram describing the infrastructure preparedness investment
decision-making process
From Figure 3-2, the preparedness investment decision directly affects the cost, and this
investment has an effect on the likelihood and severity of a disruptive event. The way in
which the investment affects these others quantities is through a factor of influence, 𝜃𝐼 ,
linearly related to the investment amount, 𝐼, and the maximum amount to be invested,
𝐼𝑚𝑎𝑥, as shown in Eq. (3-14).
𝜃𝐼 = 𝐼
𝐼𝑚𝑎𝑥 (3-14)
The factor of influence is used as a translation of the amount invested from
dollars to fraction terms expressing how large is the investment compared to other
possible investments. In particular, the amount invested is compared to the maximum
investment that can be made and is then expressed as a value between 0 and 1,
providing the factor of influence. The impact of the investment on the likelihood and
severity of the disruption can then be modeled using this factor. A simple linear
relationship between the investment and the factor of influence is more than enough to
58
express how large the investment is in proportion terms. A more sophisticated
relationship would make unnecessary complications to the overall model. Note that the
impact of investment through that factor of influence may not be as simple as a linear
relationship and is thoroughly discussed in the following section.
Decision Tree Construction and Parameters
A dynamic MOIDT is constructed in Figure 3-3 to address the investment in
infrastructure preparedness, where the decision addresses the amount of investment in
period 𝑘, 𝐼𝑗 𝑘 . Two chance nodes are assumed: the first concerns the occurrence of a
disruptive event, and the second concerns the severity of the event if it occurs. The
occurrence of the event is a deterministic chance node where the likelihood of
occurrence is known to be 𝑝. The second chance node representing the severity has
infinitely many states of nature to represent the severity of the disruption (thus, a
stochastic chance node), and consequently the demand perturbation and ultimate total
economic losses across all sectors. The stochastic nature of the severity of the disruption
is modeled with the power law distribution, and the demand perturbation is modeled
with the beta distribution. While multiple planning periods could be explored with this
problem, a one-period tree is addressed here.
59
Figure 3-3: Multiobjective stochastic inoperability decision tree for infrastructure
preparedness
The outcome of each sequential decision and state of nature is a (i) cost of
investment and (ii) total economic loss across all sectors resulting from the disruptive
event corresponding to the chance node. The relationships below describe how the
investment decision, through the factor of influence, 𝜃𝐼 , from Eq. (3-14), alters the
different parameters in the model. The probability of a disruptive event occurring is
decreased with increasing investment, as governed by Eq. (3-15).
𝑝𝐼 = 𝑒 −𝜃𝐼 𝑝 (3-15)
That is, depending on the nature of the investment, some amount of protection is
assumed in preventing the event (or reducing its likelihood of occurrence). The
probability of the occurrence of the event, 𝑝, is reduced to 𝑝𝐼 .
Dynamic Recovery Model
It is assumed that a disruptive event befalling the infrastructure system impacts
demand for commodities whose flow or production is enabled by the infrastructure
(e.g., commodities that flow through a port would be impacted by the port’s closure).
60
Such a demand perturbation for some commodities could be the result of a supply
reduction in other commodities. Given the magnitude of disruptive events in
infrastructure systems and their large economic impacts, industries and consumers
overall tend to become more conservative in demand activity due to the instability of
the economy after the disruption. In addition, supply or production could also be
impacted by the disruption. In fact, the inoperability vector in the DIIM is
representative of the supply reduction following disruptive events. To account for the
uncertainty in the demand reduction, the values of the entries in the 𝐜⋆ vector are drawn
from the beta distribution whose parameters depend on the severity of the disruptive
event. The severity is generated from the stochastic chance node in the decision tree
following a power-law distribution.
The severity of the event and the amount invested in preparedness activities are
two factors having opposite impacts on several distributional parameters as well as
other parameters in the model. As noted earlier, the shape and scale parameters of the
beta distribution for 𝐜⋆ are estimated using the method of moments according to Eqs.
(3-12) and (3-13). This method considers the sample mean and variance to be known.
The sample mean of the demand reduction, �̅�, is directly related to the maximum
demand perturbation 𝐜max ⋆ , with the latter representing the worst case scenario that
could occur. For example, this could describe a one-year closure of a transportation
facility (e.g., inland waterway port) or any type of prolonged dysfunction of an
infrastructure affecting the economy. Such scenarios are generally assessed by means of
expert solicitation.
61
The sample mean of the demand perturbation is in fact a proportion of the
maximum perturbation, and this proportion, 𝛾, depends on the severity of the event. For
example, if severity is measured in terms of the number of months a certain facility is
closed, and the worst case scenario assumes a full year of closure, then 𝛾 = 0.75 for a
nine-month closure. Further, the sample mean has a negative exponential relationship
with the factor of influence, 𝜃𝐼 , computed in Eq. (3-14). This relationship ensures that
as the amount of investment is increased, the reduction in demand after a disruptive
event is exponentially decreased. These relationships are represented in vector form in
Eq. (3-16), where �̅� increases as the proportion of the maximum demand perturbation
increases, depending on the severity of the event, and it decreases exponentially as the
factor influence (or equivalently the amount invested) increases.
�̅� = 𝛾𝑒−𝜃𝐼 𝐜max ⋆ (3-16)
Likewise, standard deviation 𝐬 is computed by assuming that the maximum
demand perturbation lies three standard deviations above the mean, shown in vector
form in Eq. (3-17). No further restrictions or assumptions are applied to the sample
variance, which results in larger variations when the sample mean is smaller.
𝐬 = 1
3 (𝐜max
⋆ − �̅�) (3-17)
Similar to the sample mean, the initial inoperability is a function of both the
investment strategy, 𝜃𝐼 , and the severity of the event, 𝛾, as shown in Eq. (3-18), where
𝐪max(0) represents the vector of initial inoperability under the most extreme
circumstances. The values of 𝐪max(0) could either be determined by experts or
estimated depending on the system considered in the application.
62
𝐪(0) = 𝛾𝑒−𝜃𝐼 𝐪max(0) (3-18)
The concept of resilience will be revisited in much more details and in-depth
analysis in Chapter 4 and Chapter 5. For the purpose of this analysis, resilience is
defined in two aspects: (i) reducing the impact of a disruptive event, and (ii) improving
the speed with which recovery occurs (Henry & Ramirez-Marquez, 2012; Zobel, 2011).
The effect of an investment in improving the first aspect is found in Eqs. (3-16) and
(3-18), where the initial inoperability and the initial demand perturbation experienced
after a disruptive event is lessened with an investment in infrastructure preparedness.
The second aspect is addressed in Eq. (3-19), where preparedness investments can also
decrease the time to full recovery of industry 𝑖, 𝑇𝑖 . Like the previous relationships,
recovery time is reduced according to 𝑒−𝜃𝐼 𝑇𝑖, an exponentially decreasing function of
the factor of influence, 𝜃𝐼 . As recovery commences, inoperability decreases over time,
consequently impacting total economic losses across all sectors over time, 𝑄(𝑡) =
𝐱(𝑡)𝑇 𝒒(𝑡).
𝑘𝑖 =
ln ( 𝑞𝑖 (0)
𝑞𝑖 (𝑒 −𝜃𝐼 𝑇𝑖 )
)
(𝑒−𝜃𝐼 𝑇𝑖 )(1 − 𝑎𝑖𝑖 ⋆ )
(3-19)
The amount invested in port security and system hardening, in general having an
exponential impact on the effect of a disruption, is a more realistic representation of the
real life behavior of improvements in the risk management of infrastructure systems.
The idea is deemed useful in such as way that is not too simplistic, such as the case of a
linear relationship, and not too complicated in a manner to make it problem-specific.
Similar ideas have been implemented in other scenarios, such as resource allocation
problems (MacKenzie, Baroud, & Barker, 2014a).
63
Note the double effect on the input parameter from the severity of an event
leading to more perturbation, more inoperability, and less resilience; while risk
management helps in decreasing the initial impact and increasing the recovery rate by
decreasing the time required to full recovery.
It is also worth noting that the demand perturbation, which is a function of time,
starts to decrease as recovery commences and is updated at discrete time points from
𝑡 = 1 to 𝑡 = 𝑇 = max(𝑇1, 𝑇2, … , 𝑇𝑛) . Updating the demand reduction during the
recovery process depends on the case study and will be further developed in the case
study section. The decision is then based on the aggregate economic loss computed at
the time, 𝑇, when all industries are fully recovered, and since the decision tree is a one-
period, more precisely one-year decision tree, there is one investment to be made.
Simulation of the Decision Tree
Alluded to previously, stochastic decision trees require the use of Monte Carlo
simulation in the folding back process. For each alternative emanating from the first
decision node, a fixed cost representing the amount invested is assumed, and 𝜃𝐼 for an
investment of I can be computed. The following steps guide the simulation for a
sufficiently large number of iterations.
1. Generate a random variable from the power law distribution within the
bounds of the minimum and maximum severity, guided by Eq. (3-8).
2. Given the severity, factor of influence, 𝜃𝐼 , and the elicited values 𝐜max ⋆ and
𝐪max(0), calculate the sample mean, �̅�, and standard deviation, 𝐬, of the
demand perturbation from Eqs. (3-16) and (3-17), the parameters of the
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DIIM from Eqs. (3-18) and (3-19), and the probability, 𝑝𝐼 , of a disruptive
event occurring from Eq. (3-15).
3. Using the method of moments, compute estimates for the parameters of the
beta distribution from Eqs. (3-12) and (3-13) and draw random variables
from this distribution for each industry element in the demand perturbation
vector. It is assumed that the number 𝑛′ of initially perturbed industries can
represent a subset of all industries, 𝑛′ ≤ 𝑛.
4. Compute the inoperability vector at each point in time from Eq. (3-6).
5. Compute the total economic loss, 𝑄 = 𝑄(𝑇) = 𝐱𝑇 ∑ 𝐪(𝑡)𝜏𝑡=1 .
6. Compute the expected value of the total economic loss as the product of the
loss and the probability 𝑝𝐼 of a disruptive event, 𝐸𝑄 = 𝐸𝑄(𝑇) = 𝑄(𝑇)𝑝𝐼
Repeating the above steps for a number of iterations, 𝑁, for one particular investment
cost results in a distribution for EQ under this specific investment. The mean values of
the EQ distributions for each investment could be compared to determine the
appropriate investment level. Further, the conditional expected value (upper-tail value)
provides an idea of how each investment performs in extreme conditions (Asbeck &
Haimes, 1984).
Case Study: Inland Waterways Preparedness Strategies Assessment
The methodology developed is illustrated with a decision problem regarding
inland waterway port security and resilience. This data-driven case study involves the
Port of Catoosa in Tulsa, Oklahoma, the largest inland port in the U.S. in terms of area.
Located on McClellan-Kerr Arkansas River, the port is part of the Mississippi River
Navigation System and roughly two million tons of commodities flow annually through
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the port. As such, preparing for disruptive events is crucial to the port itself as well as to
the economic system depending on the trading activity at the port.
Motivation
The Department of Homeland Security announced a set of grant programs
targeting different areas prone to willful attacks or natural disasters (DHS 2012).
Among this set of grants is the Port Security Grant Program (PSGP) whose purpose is
“to support increased port-wide risk management; enhanced domain awareness; training
and exercises; expansion of port recovery and resiliency capabilities; and further
capabilities to prevent, detect, respond to, and recover from attacks…and…assist ports
in obtaining the resources required to support the NPG’s associated mission areas and
core capabilities” (DHS, 2012). According to the grant overview report (DHS 2012),
the ports in the U.S. were categorized into three different groups, and DHS divided the
grant of the PSGP among the three groups. However, the ports within each category
were to compete for the funding available in their group. According to that report, the
group in which the Tulsa Port of Catoosa is placed contains a total of 22 ports and
received a grant of $4,875,000. The approach developed in this research could provide a
helpful means to determine the appropriate investment amount allocated to each port
based on the port’s size, its mix of annual commodity flows, and its location, among
other factors impacting the probability of a disruptive event occurring as well as its
consequences in terms of economic losses.
Assumptions
The probability of a (rare) event causing a disruption of an extreme nature at the
port is assumed to be similar to the probability of a terrorist attack since this type of
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event would incur large impacts and rarely occurs. This assumption is merely for
illustration purposes, a sensitivity analysis will follow to address scenarios with
different probabilities. Bun (2006) developed a mathematical model to measure the
probability of a nuclear terrorist attack, finding that, with a set of plausible parameters,
the probability of a nuclear terrorist attack in the next 10 years is 0.29. Extending from
this result, the probability associated with a terrorist event occurring in a one-year
period is assumed to be 0.29/10 = 0.029 which corresponds to one period of a decision
tree. This computation assumes independence between probabilities of occurrence of a
terrorist attack over the years and generalizes the likelihood of a nuclear terrorist attack
to any broadly-defined disruptive event. Therefore, a baseline probability for a
disruptive event occurring in one year is assumed to be 0.02 for this particular
application. Given that the result used is specific to a nuclear attack, it might seem
unrealistic to model any type of disruptive event occurring at the port of Catoosa.
However, a baseline probability is set to be slightly less than the given result and
followed up by a sensitivity analysis considering other values for the probabilities. The
choice of this value serves solely for illustration purposes, though motivated from
previous work.
Generally extreme events that are modeled with a power-law distribution result
in an estimated scale parameter ranging between 2 and 3, for further discussion on the
distribution of disruptive events, please refer to the Distribution Assumptions section
above. It is then assumed that λ = 2.5. In addition, it is assumed that the severity of the
disruptive event is bounded with a minimum and a maximum number of days during
which the port is closed, between 15 and 60 days.
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The cost of investment, 𝐼, ranges from $0 to $1,000,000 and the proportion
parameter of the severity of the event, 𝛾, is the number of days the port is closed
divided by 365, meaning that the maximum perturbation and initial inoperability
represent a one year closure of the port. Since this case study is motivated by the
amount of grant from PSGP allocated to the group of ports to which the Port of Catoosa
belongs, the maximum amount of investment is determined accordingly. It is also
assumed that all the industries require the same time, 𝑇 = 𝑇𝑖, to fully recovery, where 𝑇
depends on the duration of the closure of the port. In this particular example it is
assumed that 𝜅 = 0.5, and that if the port was closed for 𝑑 days, the industries relying
on the port as well as the interdependent industries will all recover in 𝑇 = 𝜅𝑑 days. This
assumes that while the port was closed, the products that were supposed to be exported
were held at the port and those that were supposed to be imported were held at their
original port, and after the port reopens, the port needs 𝑇 days to ship all the products
held and hence the industries would be fully recovered. This would mostly affect the
manner in which the demand perturbation is decreasing in the recovery period, and
instead of adding 𝐜⋆(𝑡), in Eq. (3-6), the change in the demand perturbation, Δ𝐜⋆ , Eq.
(3-20), is added, which considers that once the recovery starts, the commodity flows
through the port at a rate of 1/κ. In this case the flow is twice as much as it usually is to
make up for the loss and the time during which the commodity had to wait at the port.
During the recovery, the daily demand perturbation is subtracted from the initial
demand perturbation and updated accordingly every day until Δ𝐜⋆ (𝑇) = 0. Note that
while the port was closed for d days, the recovery process does not start until the port
reopens. This suggests that the port and associated industries are not functioning for d
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days, after which the recovery process commences and spans 𝑇 = 𝜅𝑑 days before the
port and industries are fully recovered. However, since the DIIM models the recovery
process, the model is implemented for 𝑇 = 𝜅𝑑 days and not the full period of time
required to fully recover, 𝑑 + 𝜅𝑑.
Δ𝐜⋆ (1) = 𝐜 ⋆(0) −
1
𝑒−𝜃𝐼 𝑇 𝐜⋆(0)
Δ𝐜⋆ (𝑡) = Δ𝐜⋆ (𝑡 − 1) − 1
𝑒−𝜃𝐼 𝑇 𝐜⋆(0)
(3-20)
Finally, the demand perturbation comes from the loss in exports and is
computed as the ratio of exports of industry 𝑖 to the total production output of that
industry at a specific time, 𝑡, as shown in Eq. (3-21).
𝑐𝑖 ⋆(𝑡) =
𝑒𝑖 (𝑡)
𝑥𝑖 (𝑡) (3-21)
Similarly, the inoperability is the result of imports not reaching the port and causing a
shortage in the material needed to produce in the industries relying on such commodity,
it is then the ratio of imports to the total production output of industry 𝑖, provided in
Eq.(3-22) (MacKenzie et al., 2012a; Pant et al., 2011).
�̂�𝑖 (𝑡) = ( 𝑚𝑖 (𝑡)
𝑥𝑖 (𝑡) ) (3-22)
If both the numerator and the denominator in Eqs. (3-21) and (3-22) are yearly
estimates, the above metrics can serve as the maximum demand perturbation and the
maximum inoperability since the worst case scenario is considered to be a full year of
port closure and they are used to compute the initial demand perturbation and
inoperability. The elements of the sample mean and initial inoperability vectors in Eqs.
(3-16) and (3-18) are then expressed in Eqs. (3-23) and (3-24), where 𝑒𝑖 (𝑡), 𝑚𝑖 (𝑡), and
69
𝑥𝑖 (𝑡) respectively represent the exports, imports and total production output for industry
𝑖 at time 𝑡 of one year.
�̅�𝑖 = 𝛾𝑒 −𝜃𝐼 (
𝑒𝑖 (𝑡)
𝑥𝑖 (𝑡) ) (3-23)
𝑞𝑖 (0) = 𝛾𝑒 −𝜃𝐼 (
𝑚𝑖 (𝑡)
𝑥𝑖 (𝑡) ) (3-24)
The input data described above relies on either (i) parameter assumptions that
could be altered depending on the application, or (ii) port-specific data. Several data
sources from the Tulsa Port of Catoosa, the US Army Corps of Engineers, and the
Bureau of Transportation Statistics were used by MacKenzie et al. (2012a) to derive
estimates of the dollar amount of commodities flowing through the port of Catoosa.
These estimates are inputs to Eqs. (3-21) and (3-22). The interdependencies between the
critical infrastructure and the rest of industries are expressed by two matrices in the
DIIM model: (i) the 𝐀∗ matrix describes how inoperability propagates among industries
when the infrastructure is disrupted, and (ii) the 𝐊 matrix governs the recovery of
infrastructure and industry sectors. The entries in the 𝐀∗ matrix are computed using data
from the Bureau of Economic Analysis, and the formula for the entries of the 𝐊 matrix
is discussed the Interdependency Model background section, Eq. (3-7). The idea of
expressing initial inoperability and demand perturbation as a fraction of the worst case
scenario has been used in a different study on port disruptions (Pant et al., 2011). The
length of closure of the port is chosen to be anywhere between two weeks and two
months, a reasonable assumption in the case of any disaster (Pant et al., 2011). And the
time to full recovery controlled by 𝜅, is randomly chosen. These parameters can be
70
altered according to each case and largely depend on the decision makers and the type
of risk they seek to mitigate.
Decision Tree Solution
A simulation of 100,000 iterations is performed, and several metrics related to
inoperability and economic loss are computed for each investment amount using the
aggregate inoperability throughout the recovery period.
The distribution of total economic loss without preparedness is depicted in
Figure 3-4. The property of the power-law distribution can be easily seen, as disruptive
events with larger impacts have a smaller chance of occurring while less impactful
events are more common. An important feature of this distribution is that it does not
rule out extreme events as outliers but rather considers them as events with an
infinitesimal likelihood of occurrence, important for a risk averse decision maker who is
interested in minimizing extreme risks.
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Figure 3-4: Frequency distribution of the expected total economic losses across all
regional industries
Recall that this MOIDT is constructed for two minimization objectives: (i)
expected total economic loss across all industries, and (ii) investment amount. While
both of these objectives are measured in dollars, they are not necessarily commensurate
as the preparedness investment would likely come from a port authority or DHS
funding program while economic losses would be shared across multiple industries.
The resulting Pareto-optimal frontier plotting expected total economic loss, 𝐸𝑄(𝑇),
computed once the industries are fully recovered versus investment is shown in Figure
3-5. Note that, as Figure 3-4 suggests, there exists a probability distribution for expected
total economic loss for each realization of investment. Shown in Figure 3-5 are the
mean, 𝑀, of these distributions, as well as conditional means 𝐶𝑀0.01 and 𝐶𝑀0.05.
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Although the Pareto-optimal frontiers in Figure 3-5 are not completely smooth, it is
concluded that none of the investment strategies is dominant due to the overall shape of
the curve. Additional iterations would likely result in smoother curves.
Figure 3-5: Pareto frontier for expected total losses versus the amount invested toward
preparedness activities
The conditional expected total economic loss values are upper-tail or extreme values
associated with higher consequence, but lower probability, events (Asbeck & Haimes,
1984). The conditional expected value is found with Eq. (3-25) and indicator function in
Eq. (3-26).
𝐶𝑀𝑟 = 𝐸[𝐸𝑄|𝛽 < 𝐸𝑄 < ∞] = ∑ 𝛿(𝑖)𝐸𝑄(𝑖)𝑁𝑖=1
∑ 𝛿(𝑖)𝑁𝑖=1 (3-25)
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𝛿(𝑖) = { 1 if 𝐸𝑄(𝑖) > 𝛽 0 otherwise
(3-26)
The use of an upper-tail conditional expected economic loss calculation is particularly
telling for the power-law distribution, which naturally has high consequence, low
probability values in its upper tail. The indicator function is used to select the
observations falling under the upper-tail of the probability distribution and include them
in the computation of the conditional expectation, a means to determine the extent to
which the extremity of the event is considered. Depending on the decision maker, more
or less extreme case scenarios could be analyzed to determine the amount of
preparedness investment needed (e.g., risk averse decision makers tend to be prepared
for more extreme events). Hence, using the indicator function, it is possible to
determine the degree of risk aversion of the decision maker (Santos & Haimes, 2004b).
The formula in Eq. (3-27) calculates the significance level which is the probability
corresponding to the upper-tail portion of the distribution of EQ.
𝑟 = 1
𝑁 ∑ 𝛿(𝑖)
𝑁
𝑖=1
(3-27)
The larger the 𝑟, the less risk averse is the decision maker since the calculation is
incorporating more points into the expectation and the risk aversion is moving towards
a risk neutral decision making that uses the mean. According to Figure 3-5, all the
metrics decrease as the investment cost increases, suggesting that the more the risk
manager invests, the less impactful is the disruptive event in terms of economic losses.
If the investment is higher than $800,000, both the mean and conditional means provide
very similar estimated values for the expected total economic loss. 𝐶𝑀0.01 and 𝐶𝑀0.05
are two conditional expectation for different levels of risk aversion. Risk averse
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decision makers would have to invest much more to get the same expected total
economic loss as a risk neutral decision maker. For example, a risk neutral decision
maker investing $100,000 would see an expected total economic loss of about $10
million, while a risk averse decision maker aiming to reach the same level of expected
economic loss would need to invest $500,000 to $700,000 depending on their extent of
risk aversion.
Recall that no particular restriction was applied to the variance of the beta
distribution except that 𝐜max ⋆ = �̅� + 3𝑠. Figure 3-6 is a plot of the standard deviation of
the total expected economic loss as the investment changes. Note that the variation in
the expected total economic loss decreases as the investment increases, suggesting that
no further restriction is required for the variance of the beta distribution as the model
accounts for the behavior of the variance of the expected total economic loss, ensuring
that it decreases with larger investments. Noted previously, additional iterations would
result in smoother curves and hence the small jumps along the curve should be
considered insignificant.
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Figure 3-6: Behavior of standard deviation of the expected total economic loss as
investment increases
Trade-off Analysis
A trade-off analysis is necessary to study the additional effect on the total
economic loss of investing an extra dollar in port security. The trade-off analysis is
done using the generalized trade-off function for discrete strategies used in Santos et al.
(2008). The trade-off of a risk function 𝑓𝑚 with respect to another risk function 𝑓𝑛 for
two different risk management strategies 𝑆𝑖 and 𝑆𝑗 is computed in Eq. (3-28).
𝜆 (𝑓𝑚, 𝑓𝑛 |𝑆𝑖 𝑆𝑗 )
= − 𝑓𝑛|𝑆𝑖 − 𝑓𝑛|𝑆𝑗 𝑓𝑚|𝑆𝑖 − 𝑓𝑚|𝑆𝑗
for 𝑖 ≠ 𝑗 (3-28)
The trade-off analysis gives insights on the efficacy of additional risk management
investments. It is true that the more the decision maker invests, the less the expected
economic loss would be, however, how efficient is this additional investment? Consider
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the numerical example in Eq. (3-29), investing $100,000 more would decrease the
expected total economic loss by $5 million. A trade-off of 50 means that the total
expected loss decreases by $50 for each $1 invested in preparedness.
𝜆 = − 30 − 35
0.2 − 0.1 = 50 (3-29)
As long as the trade-off is positive, neither of the two strategies considered
dominates the other. However, the value of that trade-off indicates how efficient the
investment is. Note in Figure 3-7 that the trade-off decreases as the amount invested
increases while it remains positive. This means that although the investment is helping
to decrease the expected total economic loss, the efficiency of that risk management
procedure is decreasing as the investment increases. The same pattern is observed in
Figure 3-8, in which the trade-off is based on the values of 𝐶𝑀0.01 from Figure 3-5.
Although it has a similar pattern of decreasing the efficiency of the investment as more
money is invested, however the value of the trade-off is much larger than the one
computed using the mean value of the expected total economic loss. That is due to the
decision maker being risk averse, any additional investment would highly impact the
risk function the decision maker is trying to minimize.
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Figure 3-7: Trade-off as a function of the cost of investment for a risk-neutral decision
maker
Figure 3-8: Trade-off as a function of the cost of investment for a risk-averse decision
maker, 𝒓 = 𝟎. 𝟎𝟏
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Sensitivity Analysis
A number of input parameters needed to be estimated in this methodology,
therefore a sensitivity analysis is useful in determining the impact of these parameters
on the output of the model. One of the key parameters in this paper is the probability 𝑝
of a disruptive event occurring, an important issue addressed in the probabilistic risk
analysis applications. Determining the probability of an accident, a willful attack or a
natural disaster occurring is challenging and has been a subject of research in the past
decade.
A sensitivity analysis is performed to determine the effect of the probability of a
disruptive event on the expected economic loss as a function of different preparedness
investment strategies. Figure 3-9 depicts the mean of the distribution of the expected
total economic loss, 𝑀, as a function of different preparedness strategies for several
possible probabilities of a disruptive event occurring ranging from 0.01 to 0.05.
Naturally, the higher the probability, the higher is the expected economic loss under any
preparedness strategy. However, the more the decision maker chooses to invest, the less
is the effect of the probability on the economic loss. More precisely, for preparedness
investments of $600,000 or higher, the average expected total economic loss is almost
the same regardless of the estimated value used to express the probability of a disruptive
event.
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Figure 3-9: Sensitivity analysis on the probability of a disruptive event
Concluding Remarks
Preparedness decision making is an important issue that critical infrastructure
systems have been dealing with for the past decade given their vulnerability to
disruptive events and their interdependence with other industries. Therefore, it is crucial
to consider accurate measures of the risk functions and use the appropriate
methodologies to solve such decision problems. This research provides a new
framework for analyzing such infrastructure preparedness problems, by integrating two
well-known methodologies in risk analysis and decision making, a stochastic decision
tree and a dynamic interdependency model to capture the uncertain and widespread
economic impacts of disruptive events. With this framework, more comprehensive
means are contributed to quantify risk and measure the efficacy of risk management
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while accounting for uncertainties in the parameters of the models used in this
integrated approach. Particular distributions, namely the power law and beta
distributions, are used here, though any appropriate distributions could describe the
parameters of the decision problem.
The framework is applied to an inland port preparedness investment problem.
Results suggest that while an increase in the investment is providing better protection to
the port in terms of the average and conditional expectation of the upper tail of the total
economic loss distribution as well as the likelihood of a disruptive event occurring, the
decision maker should be aware of the value this increased investment is adding.
Sometimes investing an additional dollar in port security might not greatly improve the
port security, depending on the risk preference of the decision maker.
The methodologies presented in this chapter aimed at introducing the concept of
interdependent impacts of disruptions and their effect on decision making.
Interdependent economic losses resulting from disruptive events play a key role in risk
management and preparedness strategies. But how do these interdependencies impact
the recovery process? And what are the best ways to account for the economic
interdependent impacts given the resilience of the disrupted critical infrastructures?
These questions are addressed in the following chapter.
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Chapter 4
Interdependent Impacts of Infrastructure Systems Resilience Modeling
Recent studies in system resilience have proposed metrics to understand the
ability of systems to recover from a disruptive event, often offering a qualitative
treatment of resilience. This work provides a quantitative treatment of resilience and
focuses specifically on measuring resilience in infrastructure networks. Inherent cost
metrics are introduced: Loss of Service Cost and Total Network Restoration Cost.
Further, “costs” of network resilience are often shared across multiple infrastructures
and industries that rely upon those networks, particularly when such networks become
inoperable in the face of disruptive events. As such, this work integrates the quantitative
resilience approach with a model describing the regional, multi-industry impacts of a
disruptive event to measure the Interdependent Impacts of Network Resilience. The
approaches discussed are deployed in a case study of an inland waterway transportation
network, the Mississippi River Navigation System.
Several qualitative schema have recently been offered for describing resilience
(Haimes, Crowther, & Horowitz, 2008; Haimes, 2009a; Woods, 2006; Bruneau et al.,
2003), with a quantitative treatment by Henry and Ramirez-Marquez (2012) describing
how system performance is affected by the change of the state of the system in the
presence of a disruption and throughout the recovery process. It is this resilience
paradigm, developed for deterministic (Henry & Ramirez-Marquez, 2012) and
stochastic (Pant, Barker, & Ramirez-Marquez, & Rocco, 2014) analyses that drives the
resilience cost metrics in this research.
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Figure 4-1 and Figure 4-2 illustrate a general approach for visualizing system
performance over time when faced with a disruptive event 𝑒𝑗 . The states of the system
are depicted across the bottom of Figure 4-1: the original system state 𝑆0 transitions to
disrupted state 𝑆d following event 𝑒 𝑗 , and then to recovered state 𝑆f following a
recovery effort. The performance of the system is quantified with the system’s service
function 𝜑(𝑡) (e.g., 𝜑(𝑡) could represent commodity flows along a waterway network).
In Figure 4-1, larger values of 𝜑(𝑡) are preferred (therefore 𝑒𝑗 leads to reduced 𝜑(𝑡)),
with the opposite (smaller values of 𝜑(𝑡) preferred) depicted in Figure 4-2. The
reliability, vulnerability, survivability, and recoverability system descriptors, among
other details of the state transition, can be found in Baroud, Ramirez‐Marquez, Barker,
& Rocco, (2014d). Note that a recovered state of the system need not necessarily be the
same as the initial state prior to the disruptive event. For instance, the state of
infrastructure following the 2010 earthquake in Haiti may be improved over pre-
disruption levels, as the recovery activities aiming at helping the infrastructure system
regain its functionality might at the same time be helping in improving the system.
Further, system performance could fluctuate over time even when no disruption occurs.
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Figure 4-1: Graphical depiction of state transitions over time with respect to an
increasing system service function, (t)
Figure 4-2: Alternative depiction of state transitions describing a decreasing system
service function, (t)
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Resilience is considered as a time-dependent proportional measure of how the
system is performing relative to an as-planned performance level 𝜑(𝑡0), namely in how
different the disrupted performance level 𝜑(𝑡d) is from 𝜑(𝑡0). Given that 𝑅 is
historically reserved for quantifying reliability, resilience is given the notation of Я and
computed at time 𝑡 as the ratio of the network performance recovered by time 𝑡 over the
loss of the performance after the disruption occurred, Eq. (4-1). As such, resilience is a
function of the extent of loss experienced at time 𝑡d (or vulnerability) and the speed at
which the system recovers (or recoverability) (Henry & Ramirez-Marquez, 2012; Pant
et al., 2014; Baroud et al., 2014d).
Я(𝑡) = Recovery(𝑡)/Loss(𝑡𝑑) (4-1)
A thorough analysis of the recovery dynamics in a network are useful to
accurately assess the time (Baroud et al. 2014d) and cost needed for a system to regain
its functionality and improve the risk-informed decision making in regards to allocating
resources after the disruption. Such costs resulting from a disrupted system are due to
inherent costs such as (i) the loss of service, (ii) the cost of restoring the system back to
a functional state, and other losses such as (iii) the interdependent effects on industries
relying on the disrupted system (Baroud et al., 2013b). For example, if the power grid
was disrupted in a certain region, (i) the loss of service could be quantified in terms of
energy not supplied (megawatts-hours) to a number of customers, (ii) the cost of
restoration involves the work crews and equipment needed for repair, and (iii) one of
the interdependent impacts could be the loss of production in industries relying on the
power generation in that area (and broader to the interdependent relationships outside of
the directly impacted area). Other examples include a disruption in certain river links of
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a waterway transportation network impacting the commodity flow throughout the entire
system and in particular, disrupting the functionality of neighboring ports and industries
relying on the import and export of commodities through the port.
This work, which appeared in Baroud et al. (2013b) and Baroud et al. (2014c),
deals with cost and impact metrics aimed at assisting risk managers in accurately
identifying and quantifying the multiple costs of a disruptive event in the context of
building resilience with particular emphasis given to decision making under uncertainty.
The primary contribution of this research is the integration of (i) the resilience modeling
paradigm in Figure 4-1 and Figure 4-2 for a disrupted network with (ii) an economic
interdependency model to more accurately quantify the resilience trajectory of
indirectly impacted sectors and more effectively make network recovery decisions. A
literature review section provides the methodological background with respect to
stochastic resilience metrics. Inherent cost metrics are then introduced in the
methodology section followed by a case study of the application to the Mississippi
River Navigation System.
Literature Review
This section details the resilience metrics derived from the depiction of
resilience in Figure 4-1 and Figure 4-2. For a review on the interdependency model,
please refer to the Literature Review section in Chapter 3. The background on these
methodologies will motivate the development of the inherent cost and interdependent
impact metrics proposed thereafter. The discussion of resilience that follows will be
centered on networks as opposed to general systems.
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Assume that disruptive event 𝑒𝑗 affects the original network state 𝑆0 at time te.
The effect on the network is assessed by quantifying the damage to the network service
function 𝜑(∙). For example, if the network under study is an inland waterway network,
𝜑(∙) could measure commodity flows, noting that lower service function values are
considered undesirable. After a period of degradation of length (𝑡d − 𝑡e) the network
service function is damaged from its original state, 𝑆0 (with corresponding 𝜑(𝑡0)), to a
disrupted state, 𝑆d (with corresponding 𝜑(𝑡d)). That is, the disruptive effect of such an
event is quantified via the analysis of a function 𝜑(𝑡) describing the behavior of the
network as a function of time. After a disrupted state of length (𝑡s − 𝑡d), the network
restoration commences until it reaches a stable system state, 𝑆f, with corresponding
𝜑(𝑡f). Eq. (4-2) provides a more specific quantification of the value of resilience
Я𝜑 (𝑡𝑟 |𝑒 𝑗 ) evaluated at time 𝑡𝑟 ∈ (𝑡d, 𝑡f) (Henry & Ramirez-Marquez, 2012). Set 𝒟 is
the set of possible disruptive events.
Я𝜑 (𝑡𝑟|𝑒 𝑗 ) =
[𝜑(𝑡𝑟 |𝑒 𝑗 ) − 𝜑(𝑡d|𝑒
𝑗 )]
[𝜑(𝑡0) − 𝜑(𝑡d|𝑒 𝑗 )]
∀𝑒𝑗 ∈ 𝒟 (4-2)
This model enables quantifying and tracking the changes in the network state as a
function of time and accurately observing the network response to the recovery
strategies employed. Using this metric, decision makers can dynamically assess their
resilience-building decisions during the aftermath of a disruption. It could also be used
as a preparedness decision tool, whereby risk managers decide on investments in
vulnerability reduction and/or increased recoverability.
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The resilience from Eq. (4-2), Я𝜑 (𝑡𝑟 |𝑒 𝑗 ), is operationalized with a set of three
metrics describing the time required to achieve different resilience and restoration goals
(Ramirez-Marquez & Rocco, 2012; Pant et al., 2014; Baroud et. al 2014d).
First, the metric Time to Total Network Restoration, 𝑇𝑇 (𝑒 𝑗 ), records the total
time spent from the point when recovery activities commence, at time 𝑡s, up to the time
when all recovery activities finalize, 𝑇𝑇 (𝑒 𝑗 ). Since these are stochastic metrics, one can
calculate the probability that total system restoration is finished before mission time 𝑡 as
𝑃𝑅 (𝑡) = 𝑃(𝑇𝑇 (𝑒 𝑗 ) ≤ 𝑡).
The second metric, Time to Full Network Service Resilience, 𝑇𝜑(𝑡0)(𝑒 𝑗 ), records
the total time spent from the point when recovery activities commence, at time 𝑡s, up to
the exact time, 𝑡f, when network service is completely restored. From 𝑇𝜑(𝑡0), one can
define the probability that network service restoration is finished before mission time 𝑡f
as 𝑃𝐹 (𝑡f) = 𝑃(𝑇𝜑(𝑡0)(𝑒 𝑗 ) ≤ 𝑡f). Note that 𝑇𝑇 (𝑒
𝑗 ) ≥ 𝑇𝜑(𝑡0)(𝑒 𝑗 ), or the time at which the
network is fully restored, is at least as lengthy as the time until a desired network
resilience, say Я𝜑 (𝑡𝑟 |𝑒 𝑗 ) = 1 (though a different target, either better or worse than
𝜑(𝑡0), may be desired), is achieved. For example, flows along a network can occur with
full capacity despite not all arcs being restored: full capacity would suggest full network
service resilience without all network components being completely restored.
Finally, the metric Time to α100% Resilience, 𝑇𝛼 (𝑒 𝑗 ), records the total time
spent from the point when recovery activities commence, at time ts, up to the exact time,
𝑡𝛼, when the system service is restored to 𝛼𝜑(𝑡0). From 𝑇𝛼, one can define the
probability that network service is restored by α100%, or 𝛼𝜑(𝑡0), before mission time
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𝑡𝛼 as 𝑃𝛼 (𝑡𝛼 ) = 𝑃(𝑇𝛼 (𝑒 𝑗 ) ≤ 𝑡𝛼 ). This metric provides a means to compare different
recovery strategies, determining which strategy achieves α100% resilience the
quickest holding resilience constant. Similarly, Я𝜑 (𝑡𝑟 |𝑒 𝑗 ) can be found for different
strategies, holding 𝑡𝑟 constant.
A comparison of the different distributions for the different resilience time
metrics can help decision makers choose the best recovery strategy that would lead to
the optimal time to full recovery. For a more detailed discussion on these metrics and a
stochastic analysis of the time to full recovery in inland waterways, more specifically
the Mississippi River Navigation System, please refer to Baroud et al. (2014d).
The work in this chapter utilizes the concept of resilience in Eq. (4-2) to model
cost and economic impacts of resilience by developing resilience-based economic
interdependent metrics of critical infrastructure disruptions. More specifically, the
resilience paradigm of Eq. (4-2) is integrated with a risk-informed interdependency
model to quantify the economic impact of a disruptive event on the interdependent
industries that rely on the directly impacted and disrupted network.
This work is using the DIIM that has been presented and reviewed in Chapter 3.
The interdependency model from Eq. (3-6) uses an 𝑛 𝑛 resilience matrix, K,
representing the capability of a certain sector to recover from the disruptive event and
reach a desired performance state. One means to estimate the entries in matrix K is to
use the formula in Eq. (3-7) (Lian & Haimes, 2006) that assumes a constant recovery
rate over time. In this case, K is a diagonal matrix with zero non-diagonal entries and
𝑘𝑖𝑖 as its diagonal entries. An alternative approach to estimate K that incorporates the
resilience paradigm of Eq. (4-2) is proposed in this chapter.
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Methodology: Inherent Cost Metrics
Alluded to previously, two primary inherent costs of network resilience are (i)
the cost of lost service and (ii) the cost of restoring the system back to a desired state.
Stochastic measures of these two costs are described here.
Loss of Service Cost
When a disruptive event occurs, a loss in the service of the disrupted network is
expected, and the change in the service function quantifies the extent of loss. For
example, in the case of a road transportation network, the performance measure could
be the traffic flow. Different problems might use different performance measures for the
same network, with relevant performance measures determined by the risk manager or
the decision maker.
A disruptive event is assumed to impact a certain number of components in the
network. For road networks, components would be bridges or roads; for inland
waterway networks, river links, ports, or dams/locks. Impacted components are
assumed to cease functioning for a certain period of time during which the network is in
a disrupted state (Figure 4-1). It is assumed that the length of the disrupted state
depends on the severity of the event. In Baroud et al. (2014a), the intensity of a
disruptive event is assumed to follow a power-law distribution suggesting that more
severe events resulting in longer disrupted state periods of time have a lower probability
of occurring. For more details on the power-law distribution, please refer to the
Distribution Assumptions section of Chapter 3.
The distribution used to model the severity of a disruptive event is presented in
Eq. (3-8) where 𝑑 is the number of days the impacted components are not functional
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(i.e., the period of time during which the system is in a disrupted state). The loss of
service cost is then computed as a function of the severity of the disruptive event. If the
performance measure is the commodity flow, then the loss of service is the aggregate
commodity that was supposed to flow across the disrupted components for the duration
of the disruption. Eq. (4-3) is the performance measure of the network system at the
disrupted state.
𝜑(𝑡𝑑) = 𝜑(𝑡0) − ∑ 𝜑𝑖 (𝑡0) × 𝑑
365
𝑚
𝑖=1
(4-3)
𝜑𝑖 (𝑡0) is the original performance measure of component 𝑖 prior to disruption
𝑒𝑗 . It is assumed that annual flow 𝜑(𝑡0) across the network is the sum of individual
𝜑𝑖 (𝑡0) such that flows are not counted more than once in the sum. If daily commodity
flow are available and if a disruption renders 𝑚 components completely inoperable for
𝑑 days, then the disrupted flow across the network, 𝜑(𝑡𝑑), can be measured with Eq.
(4-3). Since 𝑑 follows a probability distribution, simulation techniques (e.g., Monte
Carlo) can be used to construct the probability distribution of the loss of service.
Total Network Restoration Cost
Restoring a disrupted network is not only time consuming but costly as a
consequence. Careful preparedness strategies and accurate resource allocation should be
made to determine the right amount of resources invested at the right time.
To develop a probability distribution for the cost of network restoration, it is
assumed that the cost of repairing one component is stochastic. More specifically,
𝐶𝑖 (𝑒 𝑗 ) = 𝐶𝑖
𝑗 is introduced to be the cost of repairing link 𝑖 disrupted by event 𝑒𝑗 . 𝐂(𝑒𝑗 )
defines the vector of costs for all the links disrupted by the same event. The individual
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component’s restoration cost probability distribution is described in Eq. (4-4). Note that
this distribution could also be a function of the severity of the event, the relationship
between the component restoration cost distribution and the severity of the event would
be case dependent. One particular example will be explored in the case study.
𝐶𝑖 𝑗
= {𝐶𝑖 𝑗 |𝑃(𝑐𝑠 < 𝐶𝑖
𝑗 ≤ 𝑐𝑟 ) = ∫ 𝑓(𝑐𝑖
𝑗 )𝑑𝑐𝑖
𝑗 𝑐𝑟
𝑐𝑠
} (4-4)
The total network restoration cost would then be the sum of the individual
components’ restoration costs. However, note that taking the sum assumes that
component repair is performed in series. To account for potential parallel recovery
activities, a constant factor 𝜃𝑖 is introduced that would depend on the order in which
components are repaired. This factor would have higher values in cases where more
components are repaired in parallel and would be multiplied by the individual
component’s cost of restoration, shown in Eq. (4-5). This is similar to the idea of a
weighted average, where more weight (in this case higher cost), is given to components
repaired in parallel.
𝐶total(𝑒 𝑗 ) = ∑ 𝜃𝑖 𝐶𝑖
𝑗
𝑖
(4-5)
Similar to the loss of service cost, the total network restoration cost’s probability
distribution is constructed by means of simulation, such as Monte Carlo simulation.
Ultimately, decision makers would be interested in a metric describing the
overall aggregate cost of the entire disruptive event that covers both the loss of service
and the cost of restoration. Such a metric can also be computed using simulation of each
of the other metrics, provided that they are expressed in the same unit, either dollars,
tons of commodity flow, or other units.
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Interdependent Impacts of Network Resilience
A disruptive event impacting an infrastructure network does not only have
impacts on the network itself but also on the surrounding regional infrastructure systems
and industries related to and relying upon it. The costs discussed previously are
considered to be inherent costs related directly to the disrupted network, and indirect
impacts would be losses and costs incurred by infrastructures and industries related to
the disrupted system that were not necessarily directly impacted by the disruptive event.
In order to quantify for those indirect losses, an integration of the resilience
paradigm in Eq. (4-2) with the discrete time dynamic interdependency model in Eq.
(3-6) is developed. The original interdependency model considered a resilience matrix
𝐊 that is held constant throughout the recovery time. That matrix represented the
capability of the economy to restore its functionality without taking into consideration
the resilience of the underlying disrupted physical infrastructure. Hence, the 𝐊 matrix in
Eq. (3-6) assumed Я(𝑡) = 1 for the disrupted sector, and not effectively accounting for
the recovery of the underlying physical infrastructure that caused the economic
perturbation in the first place.
The approach considers a dynamic version of the resilience matrix that governs
the trajectory of interdependent recovery, introducing a matrix whose values are
functions of time, 𝐊(𝑡). Further, the resilience matrix is updated with information
regarding the trajectory of resilience as a function of time, shown in Eq. (4-6). Matrix 𝐊
is considered to be a baseline matrix of recovery trajectory whose entries are computed
according to Eq. (3-7) which updates the resilience matrix at each point in time with the
cumulative resilience of the physically disrupted system. It is also assumed that the
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perturbation is expressed through a production inoperability, hence, 𝐜⋆(𝑡) = 0, ∀ 𝑡. The
new resilience-based dynamic interdependency model is expressed in Eq. (4-7), where
0 < Я𝜑(𝑡|𝑒 𝑗) ≤ 1.
𝐊(𝑡) = 𝐊 𝑒Я𝜑(𝑡|𝑒 𝑗) (4-6)
𝐪(𝑡 + 1) = (𝐈 − 𝐊(𝑡))𝐪(𝑡) + 𝐊(𝑡)[𝐀⋆𝐪(𝑡)] (4-7)
The disrupted system might recover before the rest of the economy does, for which case
Я(𝑡) = 1 for all 𝑡 ≥ 𝑡𝑟, where 𝑡𝑟 is the time at which the physically disrupted
infrastructure network is recovered. In some other instances, the economy’s recovery
does not start until the physically disrupted system is fully recovered, which is the case
of a port closure for example. In other cases in which components of the system (nodes
or links) are disrupted, the recovery of the economy would overlap with the recovery of
the physically disrupted system, the relation in Eq. (4-6) takes into account the
quantitative analysis of such overlap and how the recovery strategy of the physically
disrupted system impacts the recovery trajectory of the entire economy.
Case Study: Interdependent Impacts of Inland Waterways Resilience
The methodology discussed is applied to the Mississippi River Navigation System. This
type of transportation system has a special feature that differentiates it from other
regular network systems. Generally, there is a single point to point access: there is no
redundancy due to the nature of the links being a part of the river.
The nation’s economy depends strongly on this waterway as it carries the
equivalence of 51 truck trips of commodity circulating through the network each year
(ASCE 2013b). The National Waterway Network (NWN) is composed of a large
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number of links and nodes. A link represents either a shipping lane or simply a path in
open water, and a node could be a facility such as a port, lock, dam, or perhaps another
intermodal terminal. The case study analyzes the resilience of a known number of links
that might become completely inoperable due to a disruptive event. The US Army
Corps of Engineers database was used to construct the 3046 links of the Mississippi
River Navigation System network shown in Figure 4-3.
Figure 4-3: Inland waterway network of the Mississippi River Navigation System
The Mississippi River is prone to different types of disruptive events, including
periods of drought (Schwartz, 2012) and flooding (Jevis & Bath, 2012). Closing
sections of the river impacts the nation’s economy by incurring losses to a large number
of industries relying on the shipments that are being delayed, with macro-level,
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interdependent losses becoming quite significant (MacKenzie et al., 2012a; Pant et al.,
2011). As such, resilience planning for inland waterway networks is of high importance.
Parameters and Assumptions
A few assumptions were made in this case study. There is one disruptive event
impacting four specific links of the river, highlighted in red in Figure 4-3. Note that the
four links represent in fact 10 segments of the river according to the data of the Army
Corps of Engineers. Those segments were condensed into four links due to similar
commodity flow capacities and proximity of their location. Hence, the event in this case
is disrupting operations along a stretch of 141.6 miles of the river located in the area
surrounding the port of Catoosa in Oklahoma resulting in delays in the flow of imports
and exports to and from Oklahoma. The individual component restoration cost follows a
uniform distribution, with a multiplicative relationship with the severity of the event,
Eq. (4-8). The severity is expressed in terms of days of disrupted state, 𝑑, and follows a
power-law distribution, Eq. (4-9).
𝐶𝑖 ~UNI(0,1) × 𝑑 (4-8)
𝑓(𝑑) = (1.5 × 𝑑min 1.5 )𝑑−2.5 (4-9)
Generally, extreme events that are modeled with a power-law distribution result
in an estimated scale parameter 𝜆 ranging between 2 and 3 (as discussed in Chapter 3).
It is then also assumed that 𝜆 = 2.5 in this case. In addition, the severity of the disruptive
event is bounded with a minimum and a maximum number of days during which the
links are disrupted, between 𝑑min = 5 and 𝑑max = 30 days. The values chosen for the
parameters can be thought of as starting values, and sensitivity analysis should be done
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to observe the model’s outcome over a specific range for the parameters. For example,
𝜆 controls the spread of the probability distribution, with larger values resulting in a
larger spread of the likelihood of more severe events. Risk averse decision makers
might consider larger values to design preparedness options that account for extreme
events expressed by the upper tail of the distribution. Also, since the application
pertains to a transportation network measured with the number of days transportation
flow capability is reduced, there is a need to specify a threshold for the severity of the
event to avoid unreasonable impacts that alter the preparedness decision and resource
allocation without a significant tradeoff of protection.
In the interdependency model, industry inoperability is assumed to be the result
of imports failing to reach the port and causing a shortage in the material needed to
produce in the industries relying on that commodity. As such, inoperability in industry
𝑖 is then the ratio of its imports to its total production output, as an industry can only be
as productive as its most disrupted supplier (MacKenzie et al., 2012a; Pant et al., 2011).
Eq. (4-10) illustrates this thought process as the maximum initial inoperability
experienced in industry i. The yearly estimate of imports from industry j is 𝑚𝑗 , the total
production output for industry j is 𝑥𝑗 , and the number of industries that typically
circulate on the disrupted links is h. This maximum perturbation corresponds to a
disruption lasting for a year, therefore for a disruption of duration d days, the initial
inoperability is computed in Eq. (4-11). This approach has been used in previous
interdependent impact analyses for disruption of inland waterway ports (MacKenzie et
al., 2012a; Baroud et al., 2014a).
𝑞𝑖
max = max ( 𝑚1 𝑥1
, … , 𝑚𝑗
𝑥𝑗 , … ,
𝑚ℎ 𝑥ℎ
) (4-10)
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𝑞𝑖 (0) = ( 𝑑
365 ) 𝑞𝑖
max (4-11)
Commodity flow data for each link in the Mississippi River Navigation system
is provided by the US Army Corps of Engineers, comprised of the yearly tonnage of
commodity flow, with the commodity flows of five chosen links provided in Table 4-1.
The daily commodity flow is assumed to be the annual flow divided by 365 days.
Table 4-1: Annual commodity flow (in tons) across the five links chosen from the
Mississippi River Navigation System
Link ID Total annual commodity flow
231500 5,007,904
231600 240,925
231709 4,766,979
231810 6,236,462
Three possible recovery activities sets are considered for the four disrupted links
with IDs 500, 600, 709, and 810, the first three numbers of the ID are omitted to
simplify the notation. The recovery sets are described in Table 4-2.
Table 4-2: Four recovery sets considered for the restoration of the four disrupted
waterway links
Recovery set Description
W1 Repair links in series in the order: 500 – 600 – 709
– 810
W2 Repair link 500 first, then 600 and 709 in parallel
in the second order, and 810 in the third order
W3 Repair link 500 first, then links 600, 709, and 810
in parallel in the second order
Resilience and Restoration Time Results
One possible realization for the resilience trajectory over time is first observed.
One observation is drawn from a triangular distribution with parameters randomly
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selected for each link, and the resilience is computed at each point in time using Eq.
(4-2) based on the three strategies in Table 4-2. Note the difference in the time required
to achieve full network resilience, portrayed in Figure 4-4. W1 requires approximately
15 additional time units of recovery activities when compared with W2, and almost 35
additional time units when compared with W3. Also, W2 and W3 differ by 20 time
units. Clearly, as more links are repaired in parallel, the trajectory towards a fully
resilient network tends to be faster leading to a shorter recovery time. Note that the step
function is used here to describe the resilience over time, while other types of linear and
non-linear functions can be investigated (Zobel, 2014).
Figure 4-4: Resilience trajectory based on one realization of the distribution of recovery
time
To more effectively represent the variability in the underlying model
parameters, 2000 scenarios of possible disruptive events were simulated and the
cumulative distribution function (cdf) of time to full restoration under the three
strategies was constructed. The conclusions align with the observations from Figure
4-4, as more links are repaired in parallel, the overall time to full network restoration
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decreases. Figure 4-5 suggests that W3 generally dominates the other two strategies for
much of the length of the cdfs. For example, the probability that full network restoration
occurs before 20 days is approximately 0.2 for W1, 0.60 for W2, and 0.80 for W3. Also,
the average point estimate of the simulated times to full recovery suggests a similar
outcome. There is, on average, a decrease of 9.5 time units between W1 and W2, 15.4
between W1 and W3, and 5.9 between W2 and W3. It is clear that going from strategy
W1 to strategy W2 has a greater impact, almost double, than going from strategy W2 to
strategy W3. To make a better decision regarding which strategy to choose, risk
managers look at other objectives, such as costs and interdependent impacts, to
determine the trade-off between the strategies.
Figure 4-5: Cumulative distribution function for the time to full network restoration
Inherent Cost Results
Mentioned previously, the cost of a disruptive event has several dimensions: the
loss of service cost, the cost of network restoration, and the cost incurred by
interdependent impacts. Using the 2000 simulations of possible scenarios for disruptive
events, the probability distribution function (pdf) and cumulative probability
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distribution (cdf) are constructed for the two cost metrics: the loss of service cost in
Figure 4-6a and Figure 4-6c, and the total restoration cost in Figure 4-6b and Figure
4-6d for recovery strategy W1. Given the nature of the power-law distribution, smaller
values of cost are more likely to occur with extreme values becoming increasingly less
likely. Also, note the difference in the units of the cost between the loss of service cost
and the network restoration cost: the loss of service cost is measured in terms of tons of
commodity flow and the restoration cost is measured in thousands of dollars. To
commensurate costs, the commodity flow in tons would be converted into dollar
amounts (if available) before adding the two random variables and generating a
distribution for the total cost of the disruption.
(a) (b)
(c) (d)
Figure 4-6: Approximate pdf results for 2000 simulations of (a) the loss of service cost
and (b) the network restoration cost, along with their respective cdfs in (c) and (d).
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The loss of service cost depends solely on the severity of the event and is not
impacted by the difference in the recovery strategies. However, from Eq. (4-5), different
strategies have different costs that increase as more links are repaired in parallel, which
impacts the cost incurred by the recovery activities. The individual cost for repairing
each link is multiplied by the number of links that are being repaired in parallel with
this particular link. Given the manner in which this is modeled in Eq. (4-5), Figure 4-7
suggests that W3 incurs the highest cost with three out of the four disrupted links being
repaired in parallel, while W1 has the lowest cost with all links being repaired in series.
Also, note that going from W2 to W3 involves a larger investment than going from W1
to W2. In fact, on average, W2 cost more than W1 by 9.4 cost unit and by 19.2 for W3,
this will help decision makers in choosing a recovery strategy by assessing the trade-off
between the strategies.
Figure 4-7: Cumulative distribution function for the network restoration cost
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Interdependent Impacts Results
This section examines the interdependent impacts of a disruptive event in the
waterway network on the industries relying on the commodities flowing on the network,
in light of the resilience quantification and the three strategies considered above.
Figure 4-8 depicts the relationship among (i) network resilience, Я𝜑 (𝑡|𝑒𝑗 ),
represented as a dark line, and (ii) inoperability of a selected number of industries,
𝑞𝑖 (𝑡), represented with lighter gray lines. The selected industries are those with imports
flowing primarily on the disrupted links: food and beverage and tobacco products
(FBT), petroleum and coal products (PC), chemical products (CH), nonmetallic mineral
products (NMM), primary metal products (PM), and fabricated metal products (FM).
Originally considered over 120 days in Figure 4-5, the number of days is reduced along
the horizontal axis in Figure 4-8 to better illustrate (i) when Я𝜑 (𝑡|𝑒𝑗 ) approaches 1, or
full network resilience, and (ii) when 𝑞𝑖 (𝑡) approaches 0 for the selected industries.
Inoperability is depicted on the vertical axis on the left, while resilience, ranging from 0
to 1, appears on the vertical axis on the right.
Naturally, resilience as measured by 𝜑(𝑡) is increasing with time, while
inoperability is decreasing. For these particular recovery examples, the individual
sectors recover faster with strategies W2 and W3 with steeper decreasing trajectories for
the individual industries’ inoperability. This is due to the parallel recovery activities in
strategies W2 and W3 speeding the recovery of the disrupted system and hence
resulting in a faster recovery for the rest of the economy. Also, note that for recovery
strategies W1 and W2, the full recovery of the economic sectors almost aligns with the
full recovery of the disrupted system, while for strategy W3, the disrupted system
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recovered very fast and the economy fully recovers shortly after that. Finally, for one of
the sectors, the inoperability continues to increase at the beginning, this is mainly due to
the relatively high initial inoperability resulting in an infinitesimal impact of the
resilience matrix on the recovery process. Such observation might trigger a need for
system hardening or extra resources if decision makers want this sector to start to
recover sooner.
(a) W1
(b) W2
(c) W3
Figure 4-8: Resilience (black line, right vertical axis) and sector inoperabilities (gray
lines, left vertical axis) over 50 time periods for the three recovery strategies
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When economic losses are aggregated across time, that is the cumulative effect
of inoperability multiplied by the as-planned output of each industry, the effect of
recovery strategy can impact industries in different ways. The average economic losses
from 2000 simulations are depicted in Figure 4-9 for the three strategies. The same
pattern is seen across all three strategies: the Petroleum and Coal Products industry
experiences the most economic losses by far, with Fabricated Metal Products next, and
the Chemical Products industry affected the least in economic terms. However, the
extent to which these industries are impacted does depend on the strategy, with W3
resulting in the fewest losses across industries. Such a breakdown can point a decision
maker in the appropriate direction when patterns point to key industries.
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(a) W1
(b) W2
(c) W3
Figure 4-9: Average economic losses experienced in each of the six primary waterway
industries for the three network recovery strategies
Figure 4-8 and Figure 4-9 provide average behavior when 𝑑, the number of days
the waterway links are disrupted, is treated as a random variable. Figure 4-10 focuses on
four particular disruption lengths: 𝑑 = 5, 𝑑 = 10, 𝑑 = 15, and 𝑑 = 20. Behavior of
economic losses across all industries is depicted with the blue curve, corresponding to
the left axis. The trajectory of resilience over time is depicted with the green curve and
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the right axis ranging from 0 to 1. Note that resilience reaches 1 in a shorter time as the
recovery strategies are switched from W1 to W3 for one particular disruptive scenario,
and it takes longer for scenarios with larger impacts. Also, economic losses increase as
the disruptive event’s severity increases, but they decrease as recovery strategies
become faster. Total economic losses range roughly from $45 to $1,200 million for
disruptions lasting from 5 to 20 days under strategy W1, for example. The decrease in
total economic losses for faster recovery strategies can be observed by examining plots
of the same disrupted scenario as the y-axis of the total economic loss is the same across
different recovery strategies. Hence, observations in the plots below align with the
conclusions from Figure 4-8 and Figure 4-9. A faster recovery strategy means that at a
certain point in time the resilience of the disrupted system is larger than one of a slower
recovery strategy, and this results in more resilient industries being able to recover
faster which is portrayed by larger entries in the resilience matrix, according to Eq.
(4-6).
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(a) d = 5, W1 (b) d = 5, W2 (c) d = 5, W3
(d) d = 10, W1 (e) d = 10, W2 (f) d = 10, W3
(g) d = 15, W1 (h) d = 15, W2 (i) d = 15, W3
(j) d = 20, W1 (k) d = 20, W2 (l) d = 20, W3
Figure 4-10. Economic losses across the six primary waterway industries (blue curve,
left vertical axis) and network resilience (green curve, right vertical axis) over time for
the three recovery strategies and four disruptive scenarios. Note that the economic loss
axis is held constant for the same disruptive scenario
In order to compare the three recovery strategies W1, W2, and W3,
interdependent impacts for a particular disruption scenario (d = 10) are further
examined. Consider first the total economic loss, an aggregation of the losses incurred
across all sectors at each point in time from the disruption through recovery. This
function is cumulative, therefore naturally exhibits an increasing pattern, illustrated in
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Figure 4-11. Note that the total economic loss is larger under strategy W1, and W3
results in the lowest estimate for the total economic loss. Also, note that the different
recovery strategies are impacting the interdependency model through (i) the resilience
trajectory and (ii) the time to full recovery. A faster recovery of the disrupted system
leads to less total economic losses incurred as the economy can recover faster and more
effectively. However, the faster the strategy is, the more costly it is.
Figure 4-11. Total economic loss computed under three recovery activities
Strategies
Rather than a multi-industry economic loss perspective, focus could be given to
a particular industry. In particular, the industries whose commodities flow along the
disrupted waterway links are considered: food and beverage and tobacco products
(FBT), petroleum and coal products (PC), chemical products (CH), nonmetallic mineral
products (NMM), primary metal products (PM), and fabricated metal products (FM).
Figure 4-12 and Figure 4-13 are plots of the difference in the inoperability of the
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individual sectors when considering strategies W1 and W2, and strategies W2 and W3,
respectively.
In both cases, the sector of primary metal products is the most impacted by the
change of strategies while the rest of the sectors have comparable and a smaller change
in the estimated inoperability. Note that all observe the same pattern in the difference in
inoperability over the time to full network recovery. This pattern suggests that opting
for a faster recovery strategy is not always beneficial. After time 𝑡 = 15, the
magnitude of the tradeoff of switching to W2 from W1 starts to decrease, suggesting
that this might be a good time for decision makers to switch back to a cheaper strategy.
A similar conclusion can be drawn from the comparison between W3 and W2 with an
overall much smaller difference in the inoperability between the two strategies.
Figure 4-12. Impact on individual sector inoperability for adopting strategy W2 as
opposed to W1
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Figure 4-13. Impact on individual sector inoperability for adopting strategy W3 as
opposed to W2
Figure 4-14 shows the inoperability trajectory for the sector of primary metal
products under the three different recovery strategies. The results shown comply with
Figure 4-11, total economic loss under the three recovery strategies. Adopting strategy
W1 results in an estimation of a largest inoperability, while W3 results in a much
steeper decreasing trajectory for the inoperability of the Primary Metal Products sector.
With such an extensive analysis, decision makers have a range of metrics to
consider in order to choose the best recovery strategy that balances cost and risk
management. Using the metrics above, decision makers can examine the possible
strategies that minimize the cost (W1), or minimize the total economic losses (W3), or a
combination of the two based on how a strategy is significantly better than the other
over time.
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Figure 4-14. Sector inoperability over time for the Primary Metal Products industry for
the three recovery strategies
Concluding Remarks
Risk managers preparing for all such sources of disruptive events must plan for
the interconnected relationship of infrastructure networks with the industries that rely
upon them. Most work in infrastructure networks addresses esoteric graph theoretic
measures of topology (e.g., centrality, betweenness) that may provide little insight for
decision making (Hines et al., 2010). However, the ultimate usefulness of understanding
interdependent effects for a sustained decision making is not just a descriptor of
physical damage, but of economic interruption (the result of a lack of functionality)
(Tierney, 1997; Webb, Tierney, & Dahlhamer, 2000). That is, the benefit of physical
models of interdependence is lost unless they ultimately translate into (i) dollars of
losses incurred, and (ii) extent and duration of system inoperability.
This work presents a stochastic approach to compute three metrics of the
resilience of an infrastructure network following a disruption: (i) the loss of service
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cost, (ii) the total network restoration cost, and (iii) the cost of interdependent impacts.
These three metrics extend from prior work in stochastic network resilience (Ramirez-
Marquez & Rocco, 2012; Baroud et. al 2014d). The first two metrics are modeled using
simulation of probability distributions. The economic impacts of a disruptive event is a
well-studied topic (Rose, 2009; Hallegatte, 2008, 2014; Okuyama, 2004; Jiang &
Haimes, 2004), as a result, the third metric developed here represents a first step in
measuring the broader multi-industry impacts of resilience in infrastructure networks,
integrating a network resilience model and an economic interdependency model.
A case study involving the disruption of links on a waterway infrastructure
network, the Mississippi River Navigation System, illustrates these concepts, and
results demonstrate the importance of considering such measures in risk-informed
decision making problems. Incorporating resilience in the interdependency model is
helpful to accurately assess the patterns in the cost metrics over time as the system is
recovering. Strategies differ in their cost of implementation and interdependent impacts,
allowing a decision maker to understand tradeoffs among different objectives. In this
particular example, the Petroleum and Coal Products industry was most impacted on
average by the disruption, measured by a stochastic duration of commodity flow
stoppage, and the interdependent inoperability in the Primary Metal Products industry
was most affected by the change in recovery strategy.
Resilience in this chapter has been modeled using simulation techniques and
integrated into data-driven techniques, the interdependency model. In addition, the
decisions to be made using the analysis above would be based on the overall cost and
economic impact of the disruption. As a result, two questions arise. Is it possible to use
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data to model resilience metrics using data-driven tools and statistical methods as
opposed to simulation of probability distributions? Also, with respect to preparedness
and recovery decision making, how can risk managers identify critical components
based on their resilience as opposed to analyzing the overall resilience of the disrupted
system discussed in this chapter? The following chapter addresses both of these
questions to complement the resilience analysis framework of infrastructure systems.
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Chapter 5
Bayesian Kernel Methods for Resilience Importance Measures
Prediction
When planning for transportation networks such as inland waterways, it is
important to understand which components (e.g., locks, dams, waterway links) are most
influential on the performance of the entire network and are most influenced by other
components in the network. This is a well-studied topic in reliability engineering, with
component importance measures (CIMs) quantifying the influence of particular
components on the overall structural performance or reliability of the system (Leemis
2009; Kuo & Zhu, 2012). Other explorations of CIMs in a network context include
those by Murray-Tuite and Mahmassani (2004), who determine transportation link
importance based on the disruption of an optimal traffic assignment network, Jenelius,
Petersen, & Mattsson (2006), who provide several vulnerability-based importance
measures for transportation networks, and Nagurney and Qiang (2007) and Qiang and
Nagurney (2008), who develop a more general flow efficiency metric with which to
rank the importance of transportation network components. Natvig, Huseby, and
Reistadbakk (2011) suggest that importance measures are helpful in (i) determining
which components merit resources to improve overall system performance, and (ii)
preparing an efficient component repair checklist in the event of system failure. This
work addresses these two items in the context of waterway transportation resilience.
As the quantification of resilience has become a vital component of
infrastructure risk analysis, stochastic simulation and the Beta Bayesian kernel model
are used to estimate resilience metrics to analyze the recovery process of disrupted
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critical infrastructure systems. More specifically, stochastic resilience based component
importance measures are assessed using the component’s characteristics and disruption
data. Such estimates would help risk managers determine the overall best recovery
strategy of an infrastructure system in case of a disruption impacting multiple
components. The model is deployed in an application to an inland waterway
transportation network, the Mississippi River Navigation system, for which the recovery
activities sets are analyzed based on the components’ resilience importance measures.
These measures are estimated using either stochastic simulation techniques or statistical
tools such as Bayesian kernel models.
The contributions of this research lie in (i) deploying and validating resilience-
based importance measures (Barker, Ramirez-Marquez, & Rocco, 2013) to study the
important links of inland waterway networks and (ii) improving the prediction of
resilience-based importance measures using data-driven and statistical methods. The
importance measures, extended from stochastic measures of the time required for a
network to achieve full resilience after a disruption (Pant et al., 2014; Baroud et al.,
2014d), along with the cost of recovery activities, are aimed at determining the best
recovery set to restore the service in the disrupted links.
The first section of this chapter presents a stochastic analysis of the resilience
importance measures that highlights a decision making approach to prioritize recovery
for disrupted links of the river (Baroud, Barker, & Ramirez-Marquez, 2014b). The
second section provides a data-driven modeling approach to estimate and predict
resilience-based importance measures of locks and dams which constitute nodes in the
waterway network (Baroud & Barker, 2014). The case study of the Mississippi River
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Navigation System is used in each section to illustrate the two approaches in identifying
critical components of the waterway network.
Stochastic Analysis of Resilience Importance Measures
Literature Review
Common reliability-based CIMs include (Vasseur & Llory, 1999): (i) Birnbaum
importance, or 𝜕𝑅𝑆 /𝜕𝑅𝑖 where 𝑅𝑆 and 𝑅𝑖 are system and component 𝑖 reliability,
respectively, which describes the probability that component 𝑖 is critical to the
functioning of the system, (ii) reliability achievement worth (RAW), or the maximum
proportion increase in system reliability generated by a given component, (iii) risk
reduction worth (RRW), an index that quantifies the potential damage to a system
caused by a particular component, and (iv) Fussell-Vesely, or an index quantifying the
maximum decrement in system reliability caused by a particular component. Several
other discussions of importance measures include those by Ramirez-Marquez and Coit
(2005), Zio, Marella, and Podofillini (2007), and Rocco and Ramirez-Marquez (2012),
among others, and they generally calculate CIMs as a ratio of the measure of
component contribution to system reliability and a measure of system reliability itself.
When considering component importance in a resilience setting, one may want
to understand the effect that both the disruption magnitude and the recovery speed of
components will have on the time to full network service resilience, 𝑇𝜑(𝐱(𝑡0))(𝑒 𝑗 ),
introduced in the literature review of Chapter 4.
This first resilience-based CIM is illustrated in Eq. (5-1), where 𝑉𝑖 𝑗 refers to a
component’s vulnerability, or its ability to maintain performance after the disruptive
event 𝑒𝑗 , and 𝜑(𝑡) refers to the system’s performance measure (for more discussion on
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the performance measure, please refer to the literature review in Chapter 4). For
example, when 𝑥𝑖 (𝑡) describes traffic flow on the 𝑖th highway link, 𝑉𝑖 𝑗
= 0.3 suggests
a 30% reduction in flow at the onset of the event. Decreasing performance 𝜑(𝑡) occurs
until 𝑡𝑑 when the new disrupted state is reached, 𝑥𝑖 (𝑡d) = (1 − 𝑉𝑖 𝑗 )𝑥𝑖 (𝑡0). A complete
reduction in the functionality of the link occurs when 𝑉𝑖 𝑗
= 1, and 𝑉𝑖 𝑗
= 0 when the
event does not impact the functionality of link 𝑖. The numerator in Eq. (5-1) quantifies
the network service loss due to the disruption effect on link 𝑖, while the denominator
describes the maximum loss among all the links. This ratio is then multiplied by the
time required to restore the system service to its original state, providing the proportion
of restoration time attributed to each link.
CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 )
= 𝜑(𝐱(𝑡0)) − 𝜑 ((𝐱(𝑡0), 𝑥𝑖 (𝑡𝑑 |𝑉𝑖
𝑗 )))
max𝑖 {𝜑(𝐱(𝑡0)) − 𝜑 ((𝐱(𝑡0), 𝑥𝑖 (𝑡𝑑 |𝑉𝑖 𝑗 )))}
𝑇 𝜑(𝒙(𝑡0)|𝑉𝑖
𝑗 )
(5-1)
As 𝑉𝑖 𝑗 and 𝑇
𝜑(𝐱(𝑡0)|𝑉𝑖 𝑗
) are stochastic terms, a probability distribution can be generated
for CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) for 𝑡r ∈ [𝑡s, 𝑡f]. This CIM is comparable to the risk reduction worth
(RRW) importance measure (Ramirez-Marquez, Rocco, Gebre, Coit, & Tortorella,
2006)
The second resilience-based CIM addresses the perspective of reliability
achievement worth (RAW). That is, Eq. (5-2) defines the “resilience worth” of link ,
WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ), or an index that quantifies how the time to total network service
resilience is improved for event 𝑒𝑗 if link i is invulnerable (or 𝑉𝑖 𝑗
= 0).
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WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) =
𝑇 𝜑(𝒙(𝑡0)|𝑉𝑖
𝑗 )
− 𝑇 𝜑(𝒙(𝑡0)|𝑉𝑖
𝑗 =0)
𝑇 𝜑(𝒙(𝑡0)|𝑉𝑖
𝑗 )
(5-2)
These two resilience-based CIMs are illustrated using general networks in Barker et al.
(2013). In this work, these metrics are (i) integrated with a decision making approach,
(ii) used in a Beta Bayesian kernel model, and (iii) applied to the study of inland
waterway network resilience.
Methodology: Recovery Strategies Decision Process
A disruptive event could impact one or more links, and in the case of the
disruption of multiple links, a decision should be made regarding the order in which the
links should be repaired. That order is dependent on several factors such as the degree
of importance of the links, the time required to achieve full recovery, and the resources
available to perform recovery activities, among others. A heuristic approach is
presented here to determine the optimal recovery strategy as a function of three metrics
representing several factors impacting the overall recovery: (i) the time to network
restoration importance, CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ), (ii) the resilience worth, WЯ𝜑,𝑖 (𝑡r|𝑒
𝑗 ), and (iii)
the total cost of recovery.
Recall, 𝐶𝑖 (𝑒 𝑗 ) = 𝐶𝑖
𝑗 as the cost of repairing link 𝑖 disrupted by event 𝑒𝑗 . 𝐂(𝑒𝑗 )
is then the vector of costs for all the links disrupted by the same event.
For a disruptive event 𝑒𝑗 , the recoverability strategy, 𝐬𝑗 (𝑒𝑗 ) = (𝑠1 𝑗 , 𝑠2
𝑗 , … , 𝑠𝑚
𝑗 ),
is a vector of link recovery activities to restore the performance of the system following
disruptive event 𝑒𝑗 . Each element of the recovery activity vector 𝐬𝑗 (𝑒𝑗 ) is described by
(i) the order in which recovery is performed, and (ii) the cost required for recovery to
occur. This is represented with a duple, as shown in Eq. (5-3).
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𝐬(𝑒𝑗 ) = (𝐨(𝑒𝑗 ), 𝐂(𝑒𝑗 ))
= ((𝑜1 𝑗 , 𝐶1
𝑗 ), … , (𝑜𝑖
𝑗 , 𝐶𝑖
𝑗 ), … , (𝑜𝑚
𝑗 , 𝐶𝑚
𝑗 ))
(5-3)
To count link 𝑖 among those links that are disrupted by 𝑒𝑗 , �̃�𝑖 𝑗 is introduced in Eq.
(5-4).
�̃�𝑖 𝑗
= {1 if 𝑉𝑖 𝑗
> 0
0 otherwise (5-4)
As such, Eqs. (5-5) and (5-6) describe the (𝑜𝑖 𝑗 , 𝐶𝑖
𝑗 ) duple in more detail,
respectively. The order in which the recovery activity for link 𝑖 is accomplished is
represented by 𝑜𝑖 𝑗 and the cost required to complete this activity is 𝐶𝑖
𝑗 , and both are
respective to the disruptive event, 𝑒𝑗 . For example, if all disrupted links are repaired at
the same time, 𝑜𝑖 𝑗
= 1, ∀ 𝑖. Also, 𝐶𝑖 𝑗 is a random variable described by its probability
density function (pdf), 𝑓 (𝑐𝑖 𝑗 (𝑒𝑖
𝑗 )).
𝑜𝑖 𝑗 (𝑒𝑗 ) = {𝑜𝑖
𝑗 |𝑜𝑖
𝑗 = ℎ, ℎ ∈ 𝑍+, ∑ 𝑜𝑖
𝑗
𝑖
= ∑ �̃�𝑖 𝑗
𝑖
} (5-5)
𝐶𝑖 𝑗 (𝑒𝑗 ) = {𝐶𝑖
𝑗 |𝑃(𝐶s < 𝐶𝑖 (𝑒
𝑗 ) ≤ 𝐶𝑟 ) = ∫ 𝑓(𝑐𝑖 𝑗 )𝑑𝑐𝑖
𝑗 𝐶𝑟
𝐶𝑠
} (5-6)
If the recovery orders are known and the probability distributions for the
components recoveries are given, then a schedule for recovery can be devised. The set
𝐴ℎ = {𝑠𝑖 𝑗 |𝑜𝑖
𝑗 = ℎ, ∀𝑖} is the collection of all those components having the same order
of recovery planning. The recovery planning activity schedule is thus given in Eq. (5-7).
Each element in set 𝐴ℎ shows those activities which are planned in parallel (i.e., all
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occurring at order h), while the different sets show the series planning of the overall
recovery activities, 𝑊 𝑝(𝑒𝑗 ).
𝑊 𝑝(𝑒𝑗 ) = {𝐴1, 𝐴2, … , 𝐴𝑙 , 𝑙 ≤ ∑ �̃�𝑖 𝑗
𝑖
} where 𝑝 = 1, … , 𝑃𝐿 (5-7)
Special cases include scenarios where all the recovery activities are in series, 𝑙 = �̃�𝑖 𝑗 , or
when they are all in parallel, 𝑙 = 1. The number, 𝑃𝐿 , of possible recovery sets, 𝑊 𝑝, is
governed by the different combinations of recovery activities the sets contain. If the
number of element sets, 𝐴ℎ, is fixed to 𝐿, then the number of possible recovery sets can
be represented in Eq. (5-8).
𝑃𝐿 = ∑ … ∑ ∑ ∏ ( 𝑛 − 𝑛1 − ⋯ 𝑛𝑖
𝑛𝑖 )
𝐿
𝑖=1
𝑛
𝑛1=0
𝑛−𝑛1
𝑛2=0
𝑛−𝑛1−⋯−𝑛𝐿
𝑛𝑟=0
(5-8)
One way of solving this optimization problem is to consider all the possible, 𝑃𝐿 ,
combinations of recovery sets and choosing the best set with respect to either time to
full network resilience (Baroud et. al, 2014d) or with respect to the total cost of
recovery activities (Baroud et. al, 2014c). Another option would be to consider a bi-
objective optimization problem examining the tradeoffs between the time to full
network resilience and recovery cost. The approach considered here is different in a
way that it does not need to check on all the possible recovery set combinations. The
approach consists of first identifying recovery sets satisfying the priority ranking of the
disrupted links according to the resilience-based CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) and WЯ𝜑,𝑖 (𝑡r|𝑒
𝑗 ), of
each disrupted link. Once the recovery sets are identified, the best recovery set is chosen
as a function of the total cost required for each set to achieve full network resilience.
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Such an optimization problem is of a stochastic order in nature (Dentcheva &
Ruszczyński, 2003, 2004), hence a heuristic stochastic ranking approach based on the
Copeland Score (CS) method is used to rank the different recovery sets. The CS is a
technique used to rank objects characterized by a set of attributes (Al-Sharrah, 2010).
The technique assumes that the ranking of the objects could be defined without
considering the decision maker’s preferences and it is considered a nonparametric
approach. The CS is computed based on pairwise comparisons between objects in a set
and is defined as the difference between the number of times an object 𝑎 is better (with
respect to attribute 𝑞𝑘) than the other objects and the number of times that object 𝑎 is
worse (with respect to the same attribute 𝑞𝑘) to the other objects. 𝐶𝑘 (𝑎, 𝑏) provides a
value based on a comparison between object 𝑎 and object 𝑏 for attribute 𝑞𝑘, 𝑘 =
1, … , Ω, performed according to the rule in Eq. (5-9). Note that a minimum to the
objective is desired in this case. Before the first attribute, 𝑞1, 𝐶0(𝑎, 𝑏) is initialized at
zero, and Eq. (5-9) iterates through all Ω attributes.
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) (5-9)
The method by Al-Sharrah (2010) dictates that the CS of object 𝑎 is obtained by adding
𝐶𝑖 (𝑎, 𝑏) over all 𝑏, each representing the other objects, as shown in Eq. (5-10). The
object with the largest CS value is assumed to stochastically dominate all other objects
with respect to the set of attributes. Note that CS assumes that all attributes have the
same importance. If this assumption is not valid for a decision maker, then other
approaches could be considered such as the ordered weighted averaging (Yager, 1988).
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CS(𝑎) = ∑ 𝐶Ω(𝑎, 𝑏)
𝑏≠𝑎
(5-10)
Mathematically, the algorithm is performed according to the conditions outlined
in the steps in Eq. (5-11).
Step 1
𝑊1 = {𝐴1, 𝐴2, … , 𝐴𝑙 , 𝑙 ≤ ∑ �̃�𝑖 𝑗
𝑖 |𝐶𝑆CIЯ1 < 𝐶𝑆CIЯ2 < ⋯ < 𝐶𝑆CIЯ𝑙 }
Step 2
𝑊 2 = {𝐴1, 𝐴2, … , 𝐴𝑙 , 𝑙 ≤ ∑ �̃�𝑖 𝑗
𝑖 |𝐶𝑆WЯ1 < 𝐶𝑆WЯ2 < ⋯ < 𝐶𝑆WЯ𝑙 }
Step 3
𝑊 ∗ = {𝐴1, 𝐴2, … , 𝐴𝑙 , 𝑙 ≤ ∑ �̃�𝑖 𝑗
𝑖 | 𝑃 (𝑇𝐶(𝑒 𝑗 ) < 𝑐
{𝑊∗}
) ≥ 𝑃 (𝑇𝐶(𝑒𝑗 ) < 𝑐 {𝑊1,𝑊2}
) }
(5-11)
In Eq. (5-11), 𝐶𝑆CIЯℎ represents the Copeland Score for links that have an order ℎ of
recovery activity computed based on the CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) of those links. Similarly 𝐶𝑆WЯℎ is
the Copeland Score computed according to WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ). Step 1 selects one recovery
activity set satisfying the Copeland Score for CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ), meaning that the strategy
chosen in this case repairs network components that are most impactful according to the
importance of the component from a resilience perspective. In a second step, another
recovery set is selected based on the Copeland Score for WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ). Note that more
than one recovery set could be selected in each of steps 1 and 2. Step 3 chooses between
the recovery sets selected in the first step and second step, the selection in this step is
done with respect to the cost of implementing each recovery set.
Case Study: Stochastic Analysis of Locks and Dams Resilience Importance Measures
This section applies (i) the stochastic time-to-resilience metric, (ii) the
resilience-based component importance measures, and (iii) the resulting stochastic
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ranking approach to a case study focusing on the Mississippi River Navigation System.
More information on this waterway network and its important contribution to the U.S.
economy is highlighted in the case studies of Chapter 3 and Chapter 4. The case study
of this chapter analyzes the resilience of a known number of links that might go
completely or partially inoperable due to a disruptive event. Note that the links in this
illustration are randomly selected and are different than the links considered in Chapter
4 which were specifically selected to be near the port of Catoosa in order to analyze
regional interdependent impacts.
Each link is considered to be subject to a disruptive event. The impact on each
link is dictated by the vulnerability parameter 𝑉𝑖 𝑗 which is a random variable following
a uniform distribution, 𝑉𝑖 𝑗 ~ UNI(0,1). In addition to that, the time each link requires to
recover after the disruption also follows a uniform distribution with values going from 0
to 1 time unit, 𝑈𝑖 𝑗
(𝑉𝑖 𝑗 (𝑒𝑗 )) ~ UNI(0,1). The network performance measure 𝜑(𝑡) is the
total weight of commodities flowing throughout the Mississippi River Navigation
System in a certain period of time (or the sum of the commodity flow for each link over
all 3046 links comprising the network). Commodity flow data for each link in the
Mississippi River Navigation System is provided by the US Army Corps of Engineers,
comprised of the yearly tonnage of commodity flow, with the commodity flow of five
chosen links provided in Table 5-1. The IDs of the links are 210, 310, 500, 708, and 800
(the first three numbers are omitted in this case to simplify the notation). The daily
commodity flow is assumed to be the annual flow divided by 365. A disruption
resulting in a vulnerability of 0.25 for a certain link would decrease the commodity flow
for that link by 25%.
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Table 5-1: Annual commodity flow (in tons) across the five links chosen from the
Mississippi River Navigation System
Link ID Total annual commodity flow
200210 1,626,645
200310 20,402,014
200500 15,240,458
200708 62,011,198
Recall the linear behavior of 𝜑(𝑡f) between 𝑡e and 𝑡d in Figure 4-1 and Figure
4-2. Another functional form could be considered (e.g., exponentially decreasing),
though this case considers the linear form.
One means to identify the importance of each link is to investigate how the
vulnerability of each link affects the vulnerability of the entire system. According to the
plot in Figure 5-1, links 708 and 800 are the most impactful for the whole system in the
event they are disrupted. Link 210 is the least important in terms of adverse effects on
the system in a disruption.
Figure 5-1: Total network-wide vulnerability (in tons) as a function of individual
component vulnerability
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The first importance measure, CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ), is applied to the five links of the
waterway network. Figure 5-2 depicts the cumulative probability distribution
of CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) for each link constructed using 2000 simulations of possible disruptive
events. Note that links 310 and 800 are overall dominated by the rest of the links. Links
210, 500, and 708 have a CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) that is always less than roughly 0.1, while links
310 and 800 could reach much higher measures of resilience. This means that links 310
and 800 are the most impactful to the overall system’s resilience when a disruptive
event occurs. In this specific example, the cumulative distributions are easily
distinguishable, and an initial recovery set can be identified in this case to be 𝑊1 =
{𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐴5} such that 𝐴1 = {𝑠310 1 }, 𝐴2 = {𝑠800
1 }, 𝐴3 = {𝑠210 1 }, 𝐴4 = {𝑠708
1 }, 𝐴5 =
{𝑠500 1 }. This recovery set performs recovery activities with the sequence 310-800-210-
708-500.
Figure 5-2: Cumulative probability distribution for the resilience-based component
importance measure, 𝐂𝐈Я𝝋,𝒊(𝒕𝐫|𝒆 𝒋)
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Although a Copeland Score analysis is not necessarily needed in that case, the
Copeland scores are computed and shown for those links as an illustration and
validation of the method. According to the histogram in Figure 5-3, the same recovery
set is suggested based on the value of the Copeland score for each link, suggesting that
link 310 is the most impactful, followed by links 800, 210, 708, and 500. The choice of
order priorities in this case is solely based on the value of the Copeland Score, meaning
that the link with the highest score is first repaired. However, different decision makers
might interpret the scores differently: (i) as 310 and 800 stand out from the rest, they
could be repaired in parallel first, then the remaining three each repaired in series, or (ii)
310 and 800 could be repaired in parallel, then 210, 708, and 500 could be repaired in
parallel. These two recovery activities sets are defined below as 𝑊1′ and 𝑊1′′.
(i) 𝑊1′ = {𝐴1, 𝐴2, 𝐴3, 𝐴4} such that 𝐴1 = {𝑠310, 1 , 𝑠800
1 }, 𝐴2 = {𝑠210 1 }, 𝐴3 =
{𝑠708 1 }, 𝐴4 = {𝑠500
1 }, 𝐴4 = {∅}, this recovery set performs recovery activities
with the sequence 310-210-708-500 in series with 800 in parallel.
(ii) 𝑊1′′ = {𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐴5} such that 𝐴1 = {𝑠310, 1 , 𝑠800
1 }, 𝐴2 =
{𝑠210 1 , 𝑠708
1 , 𝑠500 1 }, 𝐴3 = {∅}, 𝐴4 = {∅}, 𝐴5 = {∅}, this recovery set repairs
links 310 and 800 in parallel at first then repairs links 210, 708, and 500 in
parallel.
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Figure 5-3: Copeland Score for each link computed based on the resilience-based
component importance measure, 𝐂𝐈Я𝝋,𝒊(𝒕𝐫|𝒆 𝒋)
The second importance measure, WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ), quantifies how time to full
network resilience improves when a particular link is not affected by the disruptive
event. Given that the time to full network resilience is stochastic, it is possible to
construct the corresponding cumulative probability distribution of the resilience worth
of a link using simulation techniques. Figure 5-4 illustrates the cumulative probability
distribution for WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) for all five links, expressing the probability that the
resilience worth of a link is smaller than a target value. The distributions are constructed
using the same simulation technique in Figure 5-3. For example, consider the target
value WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) = 0.5 on the horizontal axis. Note that links 708 and 800 have a
smaller probability of having a resilience worth less than 0.5, while the rest of the links
have a higher probability for this target value suggesting that links 708 and 800 are
more impactful in terms of time to full recovery. That is, the time to full network
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resilience is much shorter when either link 708 or 800 is not disrupted. Also note that
the probability distributions are not easily ranked as in the CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) case. For
example, comparing the distributions of WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) for links 210 and 310 depends on
the target value considered. For resilience worth values smaller than 0.5, link 310 is
more impactful, while link 210 is more impactful in cases where the resilience worth is
larger than 0.5. The Copeland Score method is deemed useful in distinguishing those
differences. The plot for the cumulative probability distribution helps in identifying the
first three links to be repaired which are links 800, 708, and 500. The ranking of the last
two links will be determined by the corresponding Copeland Score of each link.
Figure 5-4: Cumulative probability distribution for the component resilience-worth,
𝐖Я𝝋,𝒊(𝒕𝐫|𝒆 𝒋)
Figure 5-5 is a histogram of the values of the Copeland scores for each link, as
expected from the analysis of the cumulative probability distributions, the links with the
highest scores are links 800, 708, and 500 in descending order. The comparison of links
210 and 310 is much clearer with the Copeland Score, where link 310 is deemed more
impactful with a higher score than link 210. Another candidate for the optimal recovery
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set based on WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) is 𝑊 2 = {𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐴5} such that 𝐴1 = {𝑠800
1 }, 𝐴2 =
{𝑠708 1 }, 𝐴3 = {𝑠500
1 }, 𝐴4 = {𝑠310 1 }, 𝐴5 = {𝑠210
1 }, performing series recovery activities
with the sequence 800-708-500-310-210. Similarly to the analysis of the CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ),
different decision makers might have different perspectives on how to interpret such
results and might choose different recovery sets. For the purpose of this research, the
value of the score is used to rank the links, and having links repaired in parallel would
only be the case where the links have the same value of the Copeland Score.
Figure 5-5: Copeland Score for each link computed based on the component resilience-
worth, 𝐖Я𝝋,𝒊(𝒕𝐫|𝒆 𝒋)
Consider only recovery sets 𝑊1 corresponding to 310-800-210-708-500 from
the CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) and 𝑊 2 corresponding to 800-708-500-310-210 from the
WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ). These recovery strategies are compared with the third step of Eq. (5-11)
using the cost of implementation. This example is for illustration purposes, and the cost
for repairing each link is drawn from a discrete uniform distribution with parameters
randomly assigned to each link that range from 1 to 10 time units, larger values were
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randomly assigned to lengthier links without having a specific relationship between the
parameters of the distribution and the length of the link. Ideally, such a distribution and
such parameters would either be determined using expert solicitation or in the case of
the availability of historical data, they would be determined using statistical analysis
and distribution fitting of the data. For instance, in this particular case, risk managers
might want to fit a triangular distribution determining the minimum, maximum and
most likely value for the cost of repairing a particular link. The parameter might highly
depend on the available resources and workers who perform repair activities, it is then
linked to the geographic location of the disrupted links, among other factors. By
examining Figure 5-6, it is impossible to choose the best recovery set as the cumulative
probability distributions overlap in some instances while in other cases one recovery set
outperforms the other. The best recovery set is then determined using the Copeland
Score method with a slight variation from Eq. (5-9). Since a smaller value of cost is
preferred, a larger Copeland Score value represents the preferred option in Eq. (5-12).
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) (5-12)
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Figure 5-6: Cumulative probability distribution of the total cost (in thousands of dollars)
of each recovery set
With a Copeland score equal to 4, recovery set 𝑊1 is chosen to be the optimal recovery
set in terms of cost. That is, the recovery set according to CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) results in a
lower cost to implement than WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ). Thus, an appropriate balance between link
importance and cost is met.
Decision makers might choose to include more options into the last analysis by
taking more possibilities for the recovery sets. For example, if more than five links were
considered, a mix of the results from the two importance rankings could help guide
series and parallel repair to appropriately balance cost considerations. The analysis in
this case study has been guided by a simulation of possible values of the resilience-
based importance measures that rely on random generations of the time, vulnerability,
and cost of the recovery process. A more accurate way of quantifying resilience
importance measures is to use statistical methods to draw inference from data.
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Data-Driven Prediction of Resilience-Based Importance Measures
There has been a particular emphasis on data-driven methods recently as more
decision makers are incorporating statistical analysis of historical data into the decision
making process to provide accurate quantification of risk and resilience metrics.
Chapter 2 of this dissertation developed a new class of machine learning tools to predict
critical infrastructure risk by estimating the frequency of disruptions using historical
data, experts’ knowledge, and characteristics of the system. The work here is concerned
with using data-driven tools to improve the way resilience importance measures are
computed. Resilience has often been evaluated qualitatively. In this dissertation,
methods were developed to quantify resilience. More specifically, this section is
reinforcing the quantification mechanism with data-driven methods and statistical
modeling.
This work uses the Beta Bayesian kernel model to assess the resilience
importance of critical infrastructure networks. More specifically, the model is used to
predict the resilience worth, WЯ𝐹,𝑖 (𝑡r|𝑒 𝑗 ), that was deployed in a stochastic analysis in
the preceding section. Most of the methodological background has been reviewed in
earlier sections and chapters. The literature review of this section will present the Beta
Bayesian kernel method followed by a discussion of how the model is used to assess the
resilience worth of critical infrastructure systems. The model is deployed in a case study
of inland waterways. More specifically, the application illustrates how the model can
help analyze the resilience of locks and dams along the Mississippi River Navigation
System.
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Literature Review
Recall that kernel functions are used to map input data, for which no pattern can
be recognized to model their behavior, to a higher dimensional space, where patterns
are more readily detected. Such functions enable algorithms designed to detect
relationships among data in the higher dimensional space. On the other hand Bayesian
methods make use of previous data to estimate posterior probability distributions of the
parameter of interest that follows a specific prior distribution. As a result, the
integration of Bayesian and kernel methods allows for a classification algorithm which
provides probabilistic outcomes (i.e., probability of a data point belonging to a
particular class) as opposed to deterministic outcomes (i.e., the mere classification of a
data point to a particular class). A detailed overview of Bayesian methods, kernel
functions, and Bayesian kernel models is presented in Chapter 2.
The basic form of Bayesian kernel methods assumes a Gaussian distribution.
Several extensions have been applied to Bayesian kernel models which assume both
Gaussian and non-Gaussian distributions for this classification probability to be
estimated. In particular, for the non-Gaussian case, models were developed with a Beta
conjugate prior to model binary classification by estimating the probability of a data
point belonging to one class (MacKenzie et al. 2014b), while another used a Poisson
Bayesian kernel model based on the Gamma conjugate prior to estimate the frequency
of disruptive events (Baroud et al., 2013a; Floyd et al., 2014). The Poisson Bayesian
kernel model has been developed, tested, and applied to a case study in Chapter 2 of this
dissertation.
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In this chapter, a Beta Bayesian kernel model is presented and deployed to
estimate the resilience of components of a critical infrastructure system. The Beta
Bayesian kernel model developed by MacKenzie et. al (2014b) is reviewed here. The
other methodological background used in this section has been reviewed earlier in the
chapter such as the concept of resilience worth, WЯ𝐹,𝑖 (𝑡r|𝑒 𝑗 ) and the Copeland score
method to rank probability distributions.
Since the Beta Bayesian kernel model is developed based on a conjugate prior,
both the prior and the posterior distributions of the parameter of interest, 𝜃𝑖 , are a Beta
distribution with parameters (𝛼, 𝛽) and (𝛼∗, 𝛽∗), respectively. Eq. (5-13) shows the
relationship between prior and posterior parameters.
𝛼∗ = 𝛼 + 𝑚− 𝑚
∑ 𝑘(𝑥𝑖 , 𝑥𝑗 )
{𝑗|𝑦𝑗=1}
𝛽∗ = 𝛽 + 𝑚+ 𝑚
∑ 𝑘(𝑥𝑖 , 𝑥𝑗 )
{𝑗|𝑦𝑗=−1}
(5-13)
The kernel function is 𝑘(𝑥𝑖 , 𝑥𝑗 ) and 𝑚+ is the number of positive labels while
𝑚− is the number of negative labels in the training set of size 𝑚. The ratios representing
the proportions of each class insure an unbiased estimation of the posterior parameters
in the presence of imbalanced data sets. Finally, a radial basis kernel function is used,
Eq. (2-20), to compute the kernel matrix.
Methodology: Resilience Worth Data-Driven Prediction
The work presented so far in this research computes the resilience worth by
assuming that the time to full network resilience is stochastic and follows a particular
probability distribution. WЯ𝐹,𝑖 (𝑡r|𝑒 𝑗 ) is then computed by means of simulation. This
135
section provides a similar approach to the non-Gaussian Bayesian kernel models
discussed above and applies it to model the resilience worth of the components of a
network. The outcome of the model, WЯ𝐹,𝑖 (𝑡r|𝑒 𝑗 ), is a number between 0 and 1, where
0 represents a non-impactful component and 1 represents a highly impactful component.
Therefore, a suitable conjugate prior in this case is the Beta distribution for which the
range of the random variable is [0,1]. Eq. (5-14) is a representation of the Beta
probability distribution with parameters α > 0 and β > 0, where WЯ is the resilience
worth described in Eq. (5-2) and Β(α, β) is the beta function. Bayesian kernel methods
provide a more accurate estimate of the resilience worth as the posterior probability
distribution relies on information pertaining to experts’ knowledge, the component’s
characteristics, and historical data of disruptions.
𝑃(WЯ) = WЯ𝛼−1(1 − 𝑊Я)𝛽−1
Β(α, β) (5-14)
There are two ways to analyze the outcome of the resilience worth from the Beta
Bayesian kernel model. One possibility is to analyze components using a point estimate
(e.g., the expected value or condition expected value of the posterior distribution) and
examine the resilience worth of a component based on this estimate. The larger the
estimate is, the more impactful the component is. Another possibility is to analyze
components using the entire probability distribution instead of only the point estimate.
This has a great benefit in assessing the resilience worth of components with similar
characteristics as information about the entire probability distribution is being used in
the analysis. The former way was used in Chapter 2 to analyze the outcome of the
Poisson Bayesian kernel model and estimate the frequency of disruption. The predicted
frequency in that case was equal to the mean of the posterior distribution, although
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other possibilities could have been considered such as the conditional mean in the case
of a risk averse decision maker. Conditional expectations were used in Chapter 3 to
inform preparedness decision making given the interdependent economic impacts of a
critical infrastructure disruption.
The analysis in this chapter will look into the entire probability distribution
instead of considering one point estimate (e.g. expected value or conditional expected
value). In order to compare probability distributions of the resilience worth the
Copeland Score (CS) method is used, which is, in general, a multiattribute ranking
technique (Al-Sharrah, 2010). As applied here, the CS method will be used to compare
different components according to the distribution of their WЯ importance measure,
where the attributes studied here are different percentiles of the distribution. Since the
most critical components (largest WЯ) needs to be identified, a maximum Ck(a,b) is
desired according to Eq. (5-9). For more discussion on how the Copeland Score is
computed, please refer to the literature review of the previous section in this chapter.
In case of a disruptive event impacting several components in the system,
determining the components’ resilience worth helps decision makers in identifying the
best strategy to recover the disrupted critical infrastructure by ordering the components’
repairs according to their resilience worth.
Case Study: Bayesian Kernel Modeling of Locks and Dams Resilience Importance
Measures
The framework discussed above is applied to analyze the resilience worth of
locks and dams on the Mississippi River Navigation System network. The river has 29
locks acting as key connectors between different ports nationwide. The data, retrieved
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from the database collected by the U.S. Army Corps of Engineers (2011) contains
detailed information on each lock’s characteristics including the river mile, the total
number of vessels passing by the lock, the total tonnage, the frequency and average
delay for the vessels and tows experiencing delay time due to the lock’s closure, and the
yearly frequency of closure for each lock. A sample of the data is presented in Table
2-4. No prior data are available for the resilience worth, but it is assumed that such data
can be elicited from risk managers or government officials. Given the characteristics of
each lock and dam, an individual can be asked to classify each lock and dam as either
impactful or non-impactful.
Using the Beta Bayesian kernel model and a uniform Beta distribution for the
prior, the posterior distribution parameters 𝛼∗ and 𝛽∗ are computed and the distribution
of the expected value is presented in Figure 5-7. Note that the distribution is dispersed
around a range of values going from approximately 0.25 to 0.4. Variability is mainly
due to the data set being small. Also, the median of the distribution reflects the actual
number of positive classification originally in the data. With such information, risk
managers can identify the degree to which the lock and dam is impactful with the
probabilistic outcome rather than a simple classification of 0 or 1. This helps in a more
accurate allocation of recovery resources.
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Figure 5-7: Distribution of the posterior expected value of the resilience worth
The five most impactful locks/dams are considered judging by the expected
value of resilience worth. Based on the cumulative probability distributions of the
resilience worth of these five most impactful locks and dams in Figure 5-8, it is difficult
to distinguish their actual ranking of importance, which is the case for locks and dams
with similar characteristics. Such cases arise when disruptions occur in a particular
region and result in the closure of a number of similar locks and dams. Since it is
possible to construct the posterior probability distribution with the Beta Bayesian kernel
model, the locks and dams can be ranked according to their Copeland score with
approximated percentiles of the resilience worth as attributes (the top five appear in
Figure 5-9). Table 5-2 shows the ranking of the locks and dams based on (i) the
posterior expected value and (ii) the posterior Copeland Score. Note that each method
results in a different ranking, the reason for which is that the Copeland Score represents
the entire distribution (lower and upper tails) while the expected value is only a point
estimate of the average resilience worth.
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Figure 5-8: Posterior cumulative probability distribution of the five most impactful
locks and dams of the navigation system
Figure 5-9: Copeland score of the five most impactful locks and dams of the navigation
system
Table 5-2: Lock and dam repair order based on the resilience worth values
𝑾Я ranking Posterior 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
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Recall the ranking provided by the Poisson Bayesian kernel method in Table
2-6, it was based on locks and dams that have the highest frequency of closures and
could have also been used to inform recovery decision making. The ranking in Table
5-2 is based on the contribution these locks and dams have on the recovery of the
overall waterway network in the case of a disruptive event. Note that some locks and
dams are common between the two rankings such as L&D 27. There is not one single
correct answer to which component is the most critical or needs to be prioritized in
resource allocation. Instead, decision makers should account for multiple factors when
determining the importance of a link or a node in an infrastructure system. This
dissertation has provided several means to approach this problem by considering
different criteria to produce a ranking of the most critical components in the inland
waterway network.
Concluding Remarks
The ability of a network to “bounce back” from seemingly inevitable disruptive
events is a vital consideration. Hence, the ability to quantify the vulnerability and
recoverability of inland waterways, as it pertains to the inclusive measure of network
resilience, is addressed in this chapter.
A means to quantify vulnerability, or the initial impact experienced in a network
following a disruptive event, and recoverability, or the ability of a network to recover
functionality in a timely manner is presented. As such the work in this chapter
contributes a new way to implement two approaches to measure the importance of
network components from the perspective of component contribution to network
resilience as a function of stochastic vulnerability and recoverability terms.
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The first resilience-based component importance measure, CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) in Eq.
(5-1), quantifies the potential adverse impact on system resilience at time 𝑡𝑟 when
disruption 𝑒𝑗 affects link 𝑖. Analogous to the risk reduction worth CIM common in the
reliability engineering literature, it measures the proportional contribution of link 𝑖 to
the time required to achieve full network service resilience. The second resilience-based
component importance measure, WЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ) in Eq. (5-2), quantifies the potential
positive impact on network resilience when vulnerability-strengthening measures are
put into place such that link 𝑖 cannot be disrupted.
The two importance measures discussed in this chapter can serve as a guide to
prioritize resilience improvement activities. Using the two measures, the links are
ranked with respect to their importance to the overall system resilience, the rank is then
used to construct candidates for the optimal recovery strategy that would be later
determined based on the cost of implementation of the strategy. Due to the stochastic
nature of the elements comprising CIЯ𝜑,𝑖 (𝑡r|𝑒 𝑗 ), WЯ𝜑,𝑖 (𝑡r|𝑒
𝑗 ), and the cost, ordering
the components according to such measures requires a stochastic ranking technique and
the ranking of links and recovery sets is done using the Copeland Score method.
The case study illustrates a particular case of systems, which is the waterway
transportation system. Such system is not redundant, and hence, component importance
analysis has a large impact on the decision making under uncertainty. As illustrated by
the results of the analysis of disruptive events along the Mississippi River Navigation
System, some links are deemed more important than other and might require special
consideration from risk managers. While the case study in this chapter considers the
disruption of five links, the methodology can be used to solve more complex problems
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due to the inexpensive computation effort of the Copeland Score and the simplification
strategy of the feasible region in the optimization.
Another methodology is presented in this chapter to quantify resilience
importance metrics using data-driven and statistical methods. A Beta Bayesian kernel
model is applied to analyze the resilience of critical infrastructure networks by
estimating the resilience worth of each component in the network using prior
information as well as the component’s characteristics and historical data. The
methodology is applied to an inland waterway transportation network, and the resilience
worth of locks and dams is estimated to rank components depending on how impactful
they are to the rest of the network. Results show that while the expected value can be
used as an estimator, a more accurate metric is the Copeland Score which considers the
entire posterior distribution and accounts for more uncertainty in all the possible
disruption scenarios. Such analysis assists risk managers and decision makers in
allocating resources and determining the ranking order of the repair activities in case of
an event resulting in multiple disrupted components.
Based on the ranking of the locks and dams and the analysis of the best recovery
set, improvement activities can be suggested to address the prioritized components or
recovery sets. Such activities can be in the form of vulnerability reduction policies (i.e.,
protecting or hardening components) or in the form of accelerating the speed of
recovery activities.
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Chapter 6
Conclusions
The contributions of this research are focused on three main areas: (i) risk
analysis, (ii) interdependent impacts, and (iii) resilience modeling. The tools developed
and deployed in this research are used to predict risk, analyze its consequences, and
suggest recovery strategies.
This chapter includes an overview of the key findings in this dissertation and a
discussion of the future direction of this research.
Insights and Lessons
Statistical Modeling for Risk Analysis
The theoretical contribution of this research is made to the statistics and machine
learning literature through the Poisson Bayesian kernel model. It is the first Bayesian
kernel method that can be applied to count data. The model has been tested with sample
count data and found to outperform traditional count data modeling approaches such as
the Poisson and the Negative Binomial GLM in the prediction accuracy. The majority
of the data sets (five out of seven sets) in the empirical study had better predictive
accuracy for the proposed model across the four different metrics considered. The
Poisson Bayesian kernel model appears to be a good model for prediction purposes
when the data set is small with a small number of predictors, which is common among
risk analysis problems.
The Poisson Bayesian kernel model has been applied to predict the frequency of
disruptions in critical infrastructure systems, which is considered to be a methodological
contribution to the risk analysis field in which data-driven tools have received little
144
attention due to the limited availability of data. A case study to an inland waterway
network, the Mississippi River Navigation System, illustrates the advantages of using
the Poisson Bayesian kernel model as opposed to GLM. The model proposed in this
research provides more accurate predictions for the number of closures of locks and
dams along the waterway. In addition, the Poisson Bayesian kernel is a flexible tool
where decision makers are able to input their knowledge and expertise into the prior
distribution to guide the inference process. Also, the decision makers’ risk preference
can be taken into account in the analysis. The case study assumed a risk neutral decision
maker and considered the expected value of the posterior distribution as a point estimate
of the frequency of closures. However, risk averse decision makers will be interested in
either a different point in the upper tail of the posterior distribution expressed through a
conditional expected value, or the variability in the prediction.
Overall, the Poisson Bayesian kernel model is a tool that integrates several
sources of information (such as the decision maker’s expertise, prior information,
historical data, and predictors) to provide a more comprehensive quantification and
analysis of risk.
Economic Impacts of Disruptions
The first part of this dissertation answered questions related to the likelihood of
a disruptive event: What can go wrong? What are the chances of something going
wrong? The second part is concerned with the impact of a disruptive event: What are
the consequences if the undesirable event occurs? (Kaplan & Garrick, 1981).
Analyzing the impacts of a disruption in critical infrastructure systems goes
beyond assessing the losses pertaining to disrupted system. As discussed in the
145
introduction and case studies throughout this dissertation, critical infrastructure systems
are highly interdependent. As a result, accounting for the impact of a disruption is
concerned with a number of factors at different levels. Chapter 3 and Chapter 4
developed tools that integrate the inoperability interdependency model with decision
analysis and resilience modeling techniques aimed at (i) quantifying the interdependent
economic impacts of a disruption, (ii) assessing the efficacy of risk management
strategies, and (iii) analyzing the impact of infrastructure resilience on the economic
losses.
The deployment of these methods in the case study of the Mississippi River
Navigation System revealed insights in inland waterway preparedness strategies and
interdependent impacts modeling. Results from the Stochastic Multiobjective
Inoperability Decision Tree suggest that while an increase in the investment is
providing better protection to the port in terms of the average and conditional
expectation of the total economic losses distribution as well as the likelihood of a
disruptive event occurring, the decision maker should be aware of the value this
increased investment is adding. Sometimes investing an additional dollar in port
security might not greatly improve the port security, depending on the risk preference of
the decision maker. In addition, the interdependent analysis provided commodity
specific resilience insights. In the particular case of the Mississippi River Navigation
System, the Petroleum and Coal Products industry was most impacted on average by
disruptions, measured by a stochastic duration of commodity flow stoppage, and the
interdependent inoperability in the Primary Metal Products industry was most affected
by the change in recovery strategy.
146
The tools developed in Chapter 3 and Chapter 4 allow for a general analysis of
the interdependent impacts and risk management strategies assessment of the overall
inland waterway system, while also considering the impact of industry-specific
resilience of the disrupted and interdependent systems. Such methods can inform
decision making at a higher systems level as well as provide specific insights about the
different components of the system and any other interdependent systems for a more
targeted resource allocation effort.
Resilience Modeling
The third part of this dissertation is concerned with resilience modeling.
Resilience is identified by four dimensions: reliability, vulnerability, survivability, and
recoverability. It has been modeled from different perspectives such as analyzing the
time to full network recovery, assessing the system’s performance measure, and
modeling the cost of recovery activities, among others.
This research deploys resilience-based importance measures developed by
Barker et al. (2013) to assess the resilience importance of inland waterways. More
specifically, the stochastic analysis gives insights on the system’s resilience contributed
by certain links of the network. The links that are deemed more important than others
would require special consideration from risk managers.
In order to analyze the resilience-based importance of locks and dams which are
considered to be nodes in the waterway network, a data-driven approach is used to rank
the components based on their resilience worth. The method utilizes prior belief in the
form of experts’ knowledge integrated with characteristics of the network component to
produce a probability distribution expressing the magnitude of the resilience worth.
147
Risk managers can utilize the outcome of this research to inform their decisions
by either relying on a particular ranking of components or recovery sets, or by
integrating the different results for a more comprehensive analysis of the overall system
resilience.
Future Research
A number of research extensions can be investigated to expand on the ideas
developed in this dissertation. This section provides a discussion on the future direction
of the research in each of the three parts of the dissertation.
Bayesian Kernel Methods
Modeling extensions can be incorporated into the Poisson Bayesian kernel
method to make it more flexible and generalized.
Recall the discussion on the prior parameters estimation in the case study of
Chapter 2. Results show that the selection of the priors has an impact on the posterior
parameters estimation and prediction accuracy. One way to overcome the
misspecification of the priors is to assign a hyperprior distribution to account for the
uncertainty in estimating the prior parameters. This class of methods is called
Hierarchical Bayesian models which have been used in different applications but were
never integrated with Bayesian kernel models.
The analysis performed for the Poisson Bayesian kernel model did not account
for the impact of covariates on the performance metrics. To maintain consistency all
covariates were included in the models for the empirical analysis. However, for the case
study, reduced models were considered to compare the Poisson Bayesian kernel model
to the best version of the GLM. A next step would be to explore model selection criteria
148
and techniques for Bayesian kernel methods to investigate the possibility of improving
the model’s performance as a function of the predictors.
Interdependency Modeling
The uncertainty modeling of the stochastic multiobjective decision tree
developed in Chapter 3 considered two sources of uncertainty: (i) the uncertainty
associated with the likelihood of a disruption occurring, and (ii) the uncertainty
associated with the severity of a disruption and its impact on the demand perturbations.
The two uncertainties considered are not related to the interconnectedness of the
economy represented by the 𝐀⋆ matrix. Therefore, although the interdependent impact
between industries involves uncertainties, this is not analyzed in the study. An
integration of the suggested approach with models that consider not only the uncertainty
of the inoperability but also the uncertainty of the 𝐀⋆ matrix (Barker & Rocco, 2011;
Oliva, Panzieri, & Setola, 2011) constitutes the subject of future research.
Another extension to this research would be to consider multi-period decision
trees and incorporate resource allocation in addition to preparedness decisions. Also,
using a multi-regional dynamic interdependency model is useful in indicating the
regional expected total economic losses.
The case study in Chapter 4 analyzed three possible recovery strategies. A more
exhaustive set of strategies could be considered and compared with a stochastic
ordering technique (Copeland score or optimization). Future work would include the
resilience-based analysis of a more extensive set of disruptive scenarios and recovery
strategies, as well as the exploration of network examples where disruptive events are
not as localized as inland waterways.
149
Resilience Modeling
Note that the approach considered in the analysis of resilience importance
measures in Chapter 5 does not take into account the cascading effect of a disruption in
the system, this might be a concern in applications that are not related to waterway
network systems, such as in power grids or other transportation networks. In that case,
the methodology described here could be extended to a dynamic version whereby the
measures are updated with newly disrupted links due to the cascading effect at each
point in time.
While the resilience importance measures are key factors in determining the
recovery strategy, it is equally important to account for the overall cost and time of
implementing the strategy. Future research is involved in determining the optimal
recovery strategy by taking into account the tradeoff of the Bayesian kernel estimates of
the component importance and the time and cost of recovery.
150
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