Review on Energy Resilience

Reliability Engineering and System Safety 170 (2018) 10–19

Contents lists available at ScienceDirect

Reliability Engineering and System Safety

journal homepage: www.elsevier.com/locate/ress

A Bayesian kernel approach to modeling resilience-based network

component importance

Hiba Baroud a , ∗ , Kash Barker b

a Department of Civil and Environmental Engineering, Vanderbilt University, 400 24th Ave S, Rm 274, Nashville, TN 37212, USA b School of Industrial and Systems Engineering, University of Oklahoma, 202 W. Boyd St., Rm. 124, Norman, OK 73019, USA

a r t i c l e i n f o

Keywords:

Resilience

Component importance

Bayesian kernel methods

Infrastructure systems

a b s t r a c t

The resilience of infrastructure networks is an increasingly important consideration in infrastructure planning

and risk management. One aspect of resilience-based planning is determining which components in the network

are most important to the resilience of the network. This work makes use of a resilience-based component im-

portance measure, the resilience worth, and proposes to model this measure under uncertainty using a Bayesian

kernel technique. Such a technique can be useful in modeling component importance as it enables the probability

distribution for the importance measure to be updated using data and prior information with a Bayesian kernel

model. The proposed approach is applied to study the importance of locks and dams along the Mississippi River

Navigation System. The highest predictive overall accuracy is achieved with a uniform prior distribution, and

using the posterior distribution and a multicriteria decision analysis technique, we identify the five locks and

dams with the largest impact on the system’s resilience.

© 2017 Elsevier Ltd. All rights reserved.

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. Introduction and motivation

Critical infrastructure networks, including roadway, railway, inland

aterway, and electric power networks, are vital to the regional and

ational economy as well as the community that relies on them. These

etworks are also prone to disruptive events that can disable their op-

ration. Such disruptions can be the result of natural disasters (e.g., in

012, Hurricane Sandy disabled the physical infrastructure networks in

he heavily populated NY/NJ area for several days [51] ), human-made

ttacks (e.g., in 2013, shooters armed with assault rifles did extensive

amage to 17 transformers in southern California [24] ), and failures

e.g., the American Society of Civil Engineers has assigned US public

nfrastructure a grade of D + [2] ). To reduce the effects of these events, ost research efforts have been devoted to developing traditional mea-

ures of protection [10,31] that can be expensive and degrade typical

erformance. However, given the inevitability of these events, recent

ttention has been placed on the ability “to withstand and rapidly re-

over from all hazards [52] , where the combination of “withstanding ”

nd “recovering ” from disruptions constitutes resilience . The Department

f Homeland Security [17] announced a set of grant programs targeting

ifferent areas prone to willful attacks or natural disasters, aiming to

rovide resources helpful in supporting the National Preparedness Goal

NPG] in ensuring “a secure and resilient Nation with the capabilities

equired across the whole community to prevent, protect against, miti-

∗ Corresponding author.

E-mail addresses: [email protected] (H. Baroud), [email protected] (K. Barker)

Я

ttps://doi.org/10.1016/j.ress.2017.09.022

eceived 7 December 2016; Received in revised form 9 August 2017; Accepted 30 September 2

vailable online 3 October 2017

951-8320/© 2017 Elsevier Ltd. All rights reserved.

ate, respond to, and recover from the threats and hazards that pose the

reatest risk ” [16] .

Resilience is broadly defined as the ability of a system to absorb

he shock of a disruptive event and bounce back from adverse effects.

odels and measures of resilience have increasingly been seen in the

iterature [29] . Historical references to resilience appeared in the ecol-

gy literature [27] , with other fields more recently adopting the terms,

ncluding psychology [9,47] , business [26] , economic impacts [42,43] ,

nd engineering [21,28,38] . A paradigm in the civil infrastructure field

s the “resilience triangle, ” which integrates robustness (initial impact)

nd rapidity (speed of recovery) for a disruptive event [12,15,48,54] .

n the network domain, Najjar and Gaudiot [34] and Rosenkrantz et al.

44] have proposed topological measures for resilience in networks.

This work adopts the paradigm for resilience based on system per-

ormance 𝜑 ( t ), as shown in Fig. 1 [25] . Three dimensions of resilience

re depicted in Fig. 1: (i) reliability, or the ability of the network to meet

erformance expectations prior to a disruption, (ii) vulnerability, or the

bility of disruptive event e j to impact the system performance in an

dverse manner, and (iii) recoverability, or the ability and speed of a

etwork to recover after e j . A time-dependent measure of resilience, or

he ratio of recovery over loss, accompanies this paradigm, provided in

q. (1) [4,8,35] .

𝜑

( 𝑡 𝑟 ||𝑒 𝑗 ) = 𝜑

( 𝑡 𝑟 ||𝑒 𝑗 ) − 𝜑 (𝑡 d ||𝑒 𝑗 )

𝜑 ( 𝑡 0 ) − 𝜑

( 𝑡 d ||𝑒 𝑗 ) (1)

.

017

H. Baroud, K. Barker Reliability Engineering and System Safety 170 (2018) 10–19

Fig. 1. Graphical depiction of network performance, 𝜑 ( t ), across several state transitions before and after disruptive event e j .

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A means to identify areas for investment in infrastructure networks is

o focus on those components found to be most critical to the operation

f the network. The field of reliability engineering identifies these most

ritical components through component importance measures, widely

sed to find the weakest components in a system and candidates for in-

estment [30] . Starting with the Birnbaum [11] importance measure,

hich measures how the change in reliability of component i influ-

nce a change in the reliability of the system as 𝐼 𝐵 𝑖 = 𝜕 𝑅 𝑠 ∕ 𝜕 𝑅 𝑖 , many

eliability-based component importance measures have been proposed.

hese include the risk reduction worth (RRW), an index that quantifies

he potential damage to a system caused by a particular component, and

he reliability achievement worth (RAW) of a component, the maximum

roportion increase in system reliability generated by that component

37] .

Barker et al. [4] proposed two stochastic importance measures for

dentifying network components that contribute to the resilience of the

etwork. As these measures are stochastic, a distribution for the impor-

ance measure is produced for each network component. Naturally, this

istribution could be improved over time as more disruption data about

network are collected, including how the disruption impacts the per-

ormance of the network (vulnerability, in reference to Fig. 1 ) and how

uch time is required to restore the network to a desired performance

evel (recoverability).

Baroud et al. [7] proposed a simulation technique that deploys

tochastic importance measures to identify the impact of individual

omponents on the recovery of a disrupted infrastructure network. In

articular, this previous work was concerned with identifying the best

trategy to recover disrupted links of the river of an inland waterway

ased on different resilience-based importance measures of these links

s well as other factors such as cost of implementation of the strategy. In

his paper we focus on modeling and predicting resilience-based impor-

ance measures by exploring the possibility of updating the distribution

or the importance measure using data and prior information with a

ayesian kernel method. Locks and dams along an inland waterway can

iffer in many aspects, such as their physical characteristics (e.g., geo-

raphic location, age, capacity), their operations (e.g., number of lock-

ges, vessels, tons of commodity flowing), and their past performances

e.g., number of lock outages, average delay, percentage of vessels de-

ayed). As such, identifying the most important lock/dam based on these

ultiple criteria by simply examining data or relying solely on opera-

ors ’ expertise is challenging. Further, the ranking of these components

an be based on different attributes such as probability of failure or im-

act on system recovery time and effectiveness. The objective of this

ork is to rank the importance of locks/dams according to their im-

s

11

act on the overall resilience of the waterway system. The proposed ap-

roach combines decision maker expertise with historical data to predict

esilience-based importance metrics. Ultimately, when this approach is

mplemented in practice, decision makers would inform the prior distri-

ution based on their knowledge and preferences.

In contrast to the preliminary results of this work found in Baroud

nd Barker [6] , this paper focuses on the prediction accuracy and the

nterpretability of Bayesian methods to model resilience-based impor-

ance measures. Section 2 provides some background to the stochas-

ic resilience-based importance measures and Bayesian kernel methods.

ection 3 develops the Bayesian kernel approach to updating network

esilience, and Section 4 illustrates the use of the method in an inland

aterway network application whose locks and dams are disrupted. Fi-

ally, concluding remarks are found in Section 5 .

. Methodological background

This section offers a review of importance measures and of Bayesian

ernel methods, both of which are integrated in this work.

.1. Resilience-based importance measures

Component importance measures have been commonly used in the

eliability literature, examples of such measures include [50] : (i) Birn-

aum importance, or 𝜕 R S / 𝜕 R i where R S and R i are system and com-

onent i reliability, respectively, which describes the probability that

omponent i is critical to the functioning of the system, (ii) reliabil-

ty achievement worth (RAW), or the maximum proportion increase in

ystem reliability generated by a given component, (iii) risk reduction

orth (RRW), an index that quantifies the potential damage to a sys-

em caused by a particular component, and (iv) Fussell-Vesely, an index

uantifying the maximum decrement in system reliability caused by a

articular component. Several other discussions of importance measures

nclude those by Ramirez-Marquez and Coit [36] , Zio et al. [53] , and

occo and Ramirez-Marquez [39] , among others, and they generally

alculate these measures as a ratio of the measure of component contri-

ution to system reliability and a measure of system reliability itself.

Resilience-based importance measures calculate the contribution of

component to the resilience of the network as a function of its vulner-

bility (i.e., initial degradation in network performance) and recover-

bility (i.e., time required for the recovery of network performance).

number of such measures have been developed by analogy to the

eliability-based importance measures [4,18] .

In this section, we review one particular measure known as the “re-

ilience worth ”, W Я 𝜑,𝑖 ( 𝑡 |𝑒 𝑗 ) in Eq. (2) , which serves as an index quanti-

H. Baroud, K. Barker Reliability Engineering and System Safety 170 (2018) 10–19

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ying how the time to full network service restoration is improved when

component is assumed to be invulnerable. The resilience worth mea-

ure provides an analogous perspective to the reliability achievement

orth (RAW) from the reliability engineering field.

Я 𝜑,𝑖 ( 𝑡 |𝑒 𝑗 ) = 𝑇 𝜑

( 𝒙 ( 𝑡 0 ) |𝑉 𝑗 𝑖 ) − 𝑇 𝜑 (𝒙 ( 𝑡 0 ) |𝑉 𝑗 𝑖 =0 )

𝑇 𝜑

( 𝒙 ( 𝑡 0 ) |𝑉 𝑗 𝑖 )

(2)

Eq. (2) defines the resilience worth of component i as a function of

he magnitude of and time required to recover from disruptive event,

j . In particular, the resilience worth is computed during the recovery

rocess at time t ∈( t s , t f ) and the ratio is a function of the time to recov- ry, 𝑇

𝜑 ( 𝒙 ( 𝑡 0 ) |𝑉 𝑗 𝑖 ) , where 𝜑 ( 𝒙 ( 𝑡 0 ) |𝑉 𝑗 𝑖 ) describes the network performance onditioned on the vulnerability of the i th component as a function of

he disruptive event or its ability to maintain service after the disrup-

ion, 𝑉 𝑗

𝑖 ( 𝑒 𝑗 ) = 𝑉 𝑗

𝑖 where 𝑉

𝑗

𝑖 ∈ [ 0 , 1 ] . For example, 𝜑 ( t ) would represent

ommodity flows through an inland waterway network at time t and

( t 0 ) represents the as-planned performance state prior to the onset of

isruptive event e j . As such, 𝑇 𝜑 ( 𝒙 ( 𝑡 0 ) |𝑉 𝑗 𝑖 ) measures the total time spent

rom time t s when recovery activities are started to time t f when system

ervice is completely restored, Я 𝜑 ( 𝑡 |𝑒 𝑗 ) = 1 , and 𝑇 𝜑 ( 𝒙 ( 𝑡 0 ) |𝑉 𝑗 𝑖 =0 ) assumes the pecific case where the i th component is invulnerable. 𝑉

𝑗

𝑖 represents the

ercentage reduction in the component performance state at the onset

f the event. Decreasing performance 𝜑 ( t ) occurs until t d when the new

isrupted state is reached, Eq. (3) . A complete reduction in the function-

lity of the link occurs when 𝑉 𝑗

𝑖 = 1 , and 𝑉 𝑗

𝑖 = 0 when the event does

ot impact the functionality of link i .

𝑖

( 𝑡 d ) = ( 1 − 𝑉 𝑗

𝑖

) 𝑥 𝒊 ( 𝑡 0 )

(3)

Parameter 𝑉 𝑗

𝑖 is a stochastic term, with Eq. (4) describing the likeli-

ood of 𝑉 𝑗

𝑖 lying in [ a,b ] ∈ [0,1].

( 𝑎 < 𝑉

𝑗

𝑖 ≤ 𝑏

) = ∫

𝑏

𝑎

𝑓

( 𝑣 𝑗

𝑖

) 𝑑𝑣

𝑗

𝑖 (4)

Given the information describing the disruptive event and its impact

n the individual components as well as the overall system, recover-

bility is the time required to recover the functionality of a link. As the

nitial effect of e j would impact recovery time, recovery time for the

th component is described as a function of 𝑉 𝑗

𝑖 , or 𝑈

𝑗

𝑖 ( 𝑉 𝑗 𝑖 ( 𝑒 𝑗 ) ) = 𝑈 𝑗

𝑖 ( 𝑉 𝑗 𝑖 ) .

imilar to the initial impact, recovery can also be stochastic. Eq. (5) pro-

ides the probability that component i recovers prior to time t ∈( t s , t f ). t is assumed that 𝑥 𝑖 ( 𝑡 ) = 𝑥 𝑖 ( 𝑡 d ) until the recovery time is met, suggesting step function to recovery.

( 𝑡 𝐬 < 𝑈

𝑗

𝑖

( 𝑉 𝑗

𝑖

) ≤ 𝑡

) = ∫

𝑡 𝑟

𝑡 𝑠

𝑓

( 𝑢 𝑗

𝑖

( 𝑉 𝑗

𝑖

)) 𝑑𝑣

𝑗

𝑖 (5)

As shown in Fig. 1 , vulnerability, 𝑉 𝑗

𝑖 , and recoverability, 𝑈

𝑗

𝑖 ( 𝑉 𝑗 𝑖 ) ,

ombine to describe network performance 𝜑 ( t ) for t ∈( t s , t f ) as a func- ion of component state variables 𝜑 ( 𝑡 ) = 𝜑 ( 𝐱( 𝑡 ) ) . If 𝑉 𝑗

𝑖 and 𝑈

𝑗

𝑖 ( 𝑉 𝑗 𝑖 ) are

oth stochastic terms, as presented above, a probability distribution for

he time to full network service resilience can be constructed. And since

he time to full network service resilience is stochastic, then W Я 𝜑,𝑖 ( 𝑡 |𝑒 𝑗 ) an be modeled using a probability distribution. Prior studies have con-

idered simulation methods that rely on assumptions of the severity of

he event, the time to full network service resilience, and the component

ulnerability to model the resilience worth. In this paper, we propose to

ombine probabilistic assumptions with data-driven methods to improve

he predictive accuracy and interpretability of modeling the resilience

orth importance measure. We propose to use the beta Bayesian kernel

ethod, which is reviewed in the following section.

.2. Bayesian kernel methods

Bayesian kernel methods integrate (i) the Bayesian property of im-

roving predictive accuracy as data are dynamically obtained, with (ii)

12

he kernel function which adds specificity to the model and can make

onlinear data more manageable. Kernel-based approaches to data clas-

ification have revolutionized data mining [46] . Kernel functions map

nput data that are potentially not easily classified with a linear classi-

er to a higher dimensional space, where algorithms (e.g., least squares

egression, support vector machines) enable classification or regres-

ion [14,45] . More recently, kernel functions have been integrated with

ayesian methods [3] .

Given that Bayesian methods make use of previous data to estimate

osterior probability distributions of the parameter of interest that fol-

ows a specific prior distribution, the integration of Bayesian and kernel

ethods allows for a classification algorithm that provides probabilis-

ic outcomes (i.e., probability of a data point belonging to a particular

lass) as opposed to deterministic outcomes (i.e., purely the classifica-

ion of a data point to a particular class). Several extensions of Bayesian

ernel models have assumed both Gaussian and non-Gaussian distribu-

ions for this classification probability to be estimated. In particular, for

he non-Gaussian case, models were developed with a beta conjugate

rior to model binary classification by estimating the probability of a

ata point belonging to one classification [32] , while another used a

oisson Bayesian kernel model to estimate the frequency of disruptive

vents [5,19] . In the beta Bayesian kernel model, the prior and posterior

istribution of the parameter of interest, 𝜃i , is a beta distribution with

arameters ( 𝛼, 𝛽) and ( 𝛼∗ , 𝛽∗ ), respectively. The relationship between

rior and posterior parameters in Eq. (6) is used to classify the obser-

ations of an unknown data point i represented by the vector x i . The

robability that data point i is positively labeled follows the beta dis-

ribution where y i represents the unknown classification of data point i

nd y is a vector of m known classifications (the training set).

∗ = 𝛼 + 𝑚 − 𝑚

∑ { 𝑗|𝑦 𝑗 =1} 𝑘

( 𝐱 𝑖 , 𝐱 𝑗

) ∗ = 𝛽 +

𝑚 + 𝑚

∑ { 𝑗|𝑦 𝑗 =−1} 𝑘

( 𝐱 𝑖 , 𝐱 𝑗

) (6)

The kernel function is k ( x i , x j ), 𝑚 + is the number of positive labels,

nd 𝑚 − is the number of negative labels in the training set of size m .

he ratios representing the proportions of each class ensure an unbiased

stimation of the posterior parameters in the presence of imbalanced

ata sets [32] . The model in Eq. (6) is considered to be a weighted

ayesian kernel model where 𝑚 − 𝑚 and

𝑚 + 𝑚 are weighting parameters. The

ernel function k ( x i , x j ) is determined by the model user. In this paper,

radial basis kernel function is used, Eq. (7) .

( 𝐱 𝑖 , 𝐱 𝑗

) = exp

( − 𝐱 𝑖 − 𝐱 𝑗 2

2 𝜎2

) (7)

With such a model, the probability distribution of the parameter of

nterest could be derived. And a point estimate of that parameter could

e the expected value of the posterior probability distribution or any

ther conditional expected value representing a more extreme case.

. Bayesian kernel approach to modeling resilience importance

Previous work computes the resilience worth by assuming that the

ime to full network resilience is stochastic and follows a particular

robability distribution. W Я 𝜑,𝑖 ( 𝑡 |𝑒 𝑗 ) is then computed by means of sim- lation. This paper incorporates data-driven tools to the modeling of

esilience worth, providing a similar approach to the non-Gaussian

ayesian kernel models discussed above and applies it to model the

esilience worth of the components of a network. The outcome of the

odel, W Я 𝜑,𝑖 ( t|𝑒 𝑗 ) , is a value between 0 and 1, where 0 represents a non- mpactful component and 1 represents a highly impactful component.

herefore, a suitable conjugate prior in this case is the beta distribution

or which the range of the random variable is [0, 1]. Eq. (8) is a repre-

entation of the beta prior probability distribution with parameters 𝛼 > 0

nd 𝛽 > 0, where W Я is the resilience worth described in Eq. (2) and

H. Baroud, K. Barker Reliability Engineering and System Safety 170 (2018) 10–19

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e

W

𝜃

( 𝛼, 𝛽) is the beta function. Bayesian kernel methods provide a more

ccurate estimate of the resilience worth as the posterior probability

istribution relies on prior information pertaining to the component’s

haracteristics and historical data of disruptions.

( W Я ) = W Я 𝛼−1 ( 1 − W Я ) 𝛽−1

B ( 𝛼, 𝛽) (8)

Prior parameters are often assumed to be given or otherwise chosen

o be noninformative. In this case, such parameters can be determined

rom expert elicitation to incorporate an opinion and prior knowledge on

he importance of network components. Data describing past disruptive

vents and component characteristics are then embedded into the kernel

atrix and integrated in the computation of the posterior parameters.

There are two ways to analyze the outcome of the resilience worth

rom the beta Bayesian kernel model. One possibility is to analyze com-

onents using a point estimate (e.g., the expected value of the posterior

istribution) and examine the resilience worth of a component based

n this estimate. The point estimate for the resilience worth importance

easure is found in Eq. (9) where the larger the estimate, the more

mpactful the component.

Я ∗ =

𝛼 + 𝑚 − 𝑚

∑ { 𝑗|𝑦 𝑗 =1} 𝑘 (𝐱 𝑖 , 𝐱 𝑗 )

𝛼 + 𝑚 − 𝑚

∑ { 𝑗|𝑦 𝑗 =1} 𝑘 (𝐱 𝑖 , 𝐱 𝑗 ) + 𝛽 + 𝑚 + 𝑚 ∑{ 𝑗|𝑦 𝑗 =−1} 𝑘 (𝐱 𝑖 , 𝐱 𝑗 ) (9)

Another possibility is to analyze components using the entire proba-

ility distribution in Eq. (8) instead of only the point estimate. Doing so

akes advantage of the entire distribution of resilience worth (e.g., not

nly central tendency but also the tails of the distribution). We later dis-

uss an approach for comparing the distributions of different locks and

ams with a decision analysis technique applied to stochastic ranking.

As the objective of this paper is to predict importance measures and

ssess the model interpretability, the main outcome used in this analy-

is is the importance of each component. If the decision maker is inter-

sted in identifying the most impactful attributes contributing to these

easures, further analysis within the kernel matrix would be required.

ach entry in the kernel represents a similarity measure between the

est data point and each data point in the training set. The similarity

easure corresponds to the summation of kernel function values across

ll the attributes. To learn more about the role of the attributes in the

alculation of importance measures, the decision maker could look into

he individual kernel function value for each attribute instead of the

verall summation across attributes to determine the covariate (or set

f covariates) that most significantly impacts the classification of the

omponent.

. Illustrative example: inland waterway network resilience

The framework discussed above is applied to analyze the resilience

orth of locks on the Mississippi River Navigation System, which is

odeled as a network of nodes representing ports and locks/dams, and

inks representing sections of the river. The system performance upon

hich the importance measures are calculated is assumed to be the total

mount of commodity being shipped through the entire navigation sys-

em in a given year. If a disruption results in a river section or lock/dam

losure, its impact would be measured in terms of a decrease in total

ommodity flow regardless of the directionality. The importance of a

aterway component is then calculated based on the time required for

he system to recover from such a disruption as a function of the vulnera-

ility of that component. Prior studies have considered resilience-based

mportance measures to analyze the impact of links (i.e., sections of the

iver) on the recoverability of the waterway after a disruption [6] . The

isk and impact of lock outages play a different role than closures of river

ections. As a result, the two analyses can result in different investment

trategies and corresponding disruption losses. This article, however, is

ot concerned with identifying risk management and recovery strate-

ies, but rather the focus is given to effectiveness of using data-driven

13

ayesian techniques to identify important locks based on (i) the pre-

ictive accuracy of the Bayesian approach, and (ii) the interpretability

f resilience-based importance measures. As such, we assume that deci-

ion makers have some prior knowledge on the importance measures of

ach lock/dam instead of calculating or simulating their values based

n system performance, and we update this knowledge with data using

he Bayesian kernel model.

The Mississippi River Navigation System has 29 locks acting as key

onnectors between different ports nationwide. The data, retrieved from

he database collected by the US Army Corps of Engineers [49] , contain

etailed information on each lock’s characteristics including the river

ile, the total number of vessels passing by the lock, the total tonnage,

he frequency and average delay for the vessels and tows experiencing

elay time due to the lock’s closure, and the yearly frequency of closure

or each lock. A sample of the data is presented in Table 1 . No prior

ata are available for the resilience worth, but we assume that such

ata can be elicited from risk managers or government officials. Given

he characteristics of each lock and dam, an individual can be asked to

lassify each lock and dam as either impactful or non-impactful.

We focus our analysis of the waterway system on the two main com-

onents of the model: (i) the prior distribution and its impact on model

esults, and (ii) the interpretation of the posterior distribution of the

esilience worth to infer ranking of the locks and dams.

.1. Prior distribution impact analysis

In this section, we explore the predictive ability, the sensitivity, and

he interpretability of the prior distribution.

.1.1. Predictive accuracy

The ultimate goal of quantifying and analyzing the resilience of in-

rastructure systems is to develop risk and recovery management strate-

ies, and an ordered ranking of important system components can assist

hese strategies. However, some critical characteristics of these statisti-

al methods for quantifying and predicting resilience metrics is the ac-

uracy of such models, their interpretability, and their flexibility. In the

ollowing analysis, we address the ability of Bayesian kernel methods

o address these characteristics. This class of models offers a great deal

f benefits by integrating the Bayesian property with kernel functions.

he Bayesian property is used to account for uncertainty, to improve

redictive accuracy as new information becomes available, and to in-

orporate decision maker expertise and knowledge. The kernel function

akes into account the influence of multiple factors on the resilience of

n infrastructure system through the integration of covariates into the

ayesian model.

We first consider the predictive accuracy of the beta Bayesian ker-

el model by assessing the ability of the model to classify correctly each

ock as either impactful or not impactful. Table 2 provides a summary of

redictive accuracy metrics which are (i) the ability to correctly classify

lock as impactful (true positive rate, TP ), (ii) the ability to correctly

lassify a lock as not impactful (true negative rate, TN ), and (iii) the

ccuracy score, 𝐴𝐶𝐶 = √ 𝑇 𝑃 × 𝑇 𝑁 . These accuracy metrics were com-

uted for the Bayesian kernel model under multiple assumptions for the

rior distribution.

The predictive accuracy metrics are used to assess the ability of the

odel to correctly classify new data points given their attribute infor-

ation. Since the outcome of the model is a probability distribution of

he parameter of interest, we will use a point estimate that is the aver-

ge of the posterior distribution, shown in Eq. (10) . If the estimate of

esilience worth is greater than or equal to a threshold (e.g., 0.5), the

ock is classified as impactful, indicating a resilience worth of 1. If the

stimate is below 0.5, the lock would be classified as non-impactful with

Я = 0 .

̄ 𝑖 =

𝛼∗

𝛼∗ + 𝛽∗ (10)

H. Baroud, K. Barker Reliability Engineering and System Safety 170 (2018) 10–19

Table 1

Example data describing lock and dam characteristics.

Lock & Dam Closure frequency River Mile Vessels Tonnage Lockages . . .

L&D 3 0 797 9397 6747 4406 . . .

L&D 13 6 523 2810 14,545 3155 . . .

L&D 2 0 815 4478 6735 2893 . . .

L&D 20 23 343 2508 20,828 3582 . . .

L&D 22 40 301 2280 22,476 3486 . . .

L&D 8 6 679 4333 10,277 2620 . . .

… … … … … … . . .

Table 2

Predictive accuracy of the beta Bayesian kernel model (BK).

BK GLM Bayesian GLM

TP TN ACC TP TN ACC TP TN ACC

Uniform 0.80 0.96 0.87 0.76 0.74 0.72 0.88 0.88 0.87

Jeffrey’s Prior 0.80 0.96 0.87

Empirical ( 𝛼 = 1 ) 0.85 0.75 0.76 Empirical ( 𝛽 = 1 ) 0.81 0.75 0.72

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The different priors considered for the Bayesian kernel model are

he following, (i) a uniform distribution where both prior parameters are

qual to 1, (ii) Jeffrey’s prior [13] where both prior parameters are equal

o 0.5, and (iii) two variations of an empirical prior. Empirical ( 𝛼 = 1) ssumes that 𝛼 = 1 and calculates 𝛽 using the method of moments by ssuming that the mean of the prior distribution is equal to the propor-

ion of positively classified data points in the training set, and Empirical

𝛽 = 1) assumes that 𝛽 = 1 and calculates 𝛼 using the same method of oments. The objective of considering these priors is to assess the im-

act of eliciting a prior distribution on the predictive accuracy of the

odel. This is often a crucial and impactful factor in Bayesian model-

ng techniques, however, it is also challenging and expensive to perform

hese elicitations, and as such, it would be helpful to form an idea on

he impact of prior information on future prediction accuracy.

A cross-validation technique is used to assess the predictive accuracy

f the models: 50% of the data is used to train the model, an additional

0% is used to tune the parameter in the kernel function, and the model

s tested on the remaining 30% of the data. Table 2 provides a sum-

ary of the TP, TN, and ACC metrics for the beta Bayesian kernel (BK)

echnique, as well as two other classical statistical techniques, the Gen-

ralized Linear Models (GLM) [33] and the Bayesian GLM [22] . More

pecifically, for GLM, we fit the data with a logistic regression model

hich assumes that the outcome variable follows a binomial distribu-

ion and computes the logit of the probability of success which is our

arameter of interest, 𝜃i , as a linear function of the attributes, shown in

q. (11) .

ogit ( 𝜃𝑖 ) = 𝛽0 + 𝛽1 𝑥 1 + …𝛽𝑑 𝑥 𝑑 (11)

For Bayesian GLM, we consider a Bayesian version of logistic regres-

ion that models the coefficients as random variable and updates their

rior distribution accordingly, the default case for which has the prior

istribution for all the covariates following a Cauchy distribution with

enter 0 and scale 2.5 [23] . As a result, the model prediction is in the

orm of the probability of a lock being impactful as opposed to a deter-

inistic outcome such as the case in the logistic regression.

The Uniform and Jeffrey’s prior provide the same results for the BK

s the form of these distributions is quite similar and in this case the

light difference in the prior parameters did not impact the outcome of

he model. The best overall accuracy of the model is provided by the BK

ethod with either a Uniform or a Jeffrey’s prior at an accuracy rate of

7%, the same as the Bayesian GLM. The Bayesian GLM also resulted

n the best overall true positive rate at 88%, though the model did not

ffectively capture the points in the negative class. The BK resulted in

igher rates of true negatives, with the best rate being 96% for the BK

nder both the Uniform and Jeffrey’s priors. The BK models with the

14

mpirical priors performed similarly to the logistic regression, with the

xception of a better true positive rate for the BK models. We analyze

he prior distributions further in the next section.

The selection of a model to predict future resilience worth for the

ocks and dams of the Mississippi River Navigation System rely heavily

n the risk attitude of the decision maker. For example, if the decision

aker is risk averse, they may not able to tolerate a low rate of true

ositives, as an underestimation of the importance of the lock and dam

ight lead to an underinvestment in preparedness or recovery strategies

hat might result in potentially large losses that a risk averse decision

aker would not tolerate. In such a case, a Bayesian logistic regression

s the model selected. If the decision maker is risk neutral, they would

e interested in a predictive model with a good overall accuracy without

ny particular preference toward the true positive or true negative rate,

s such, they would be indifferent between the BK and the Bayesian

ogistic regression. Finally, if the decision maker is risk seeking with a

trong preference to minimize the budget allocated for rehabilitation

r recovery, making them willing to tolerate an underestimation of the

esilience worth, then they would want to use a model that accurately

redicts the locks and dams that are not so impactful in order to avoid

nnecessary costs. In that case, the decision maker would select a BK

odel that predicts low resilience worth components with an accuracy

f 96%.

.1.2. Sensitivity analysis on the prior distributions

The predictive accuracy analysis has so far considered point esti-

ates. In this section, we look at the impact of priors on the shape of

he posterior distribution. Using the beta Bayesian kernel model and a

niform beta distribution for the prior, we compute the posterior distri-

ution parameters 𝛼∗ and 𝛽∗ across all the locks in the data for 200 iter-

tions of randomly selected training and testing sets. At each iteration,

e obtain the posterior probability distribution of the resilience worth

or each lock and dam and use its expected value as a point estimate.

s a result, we have 200 realizations of the resilience worth estimate

or each lock and dam, and we plot the distribution of these estimates

n a histogram in Fig. 2 . Note that the distribution is dispersed around

range of values going from approximately 0.25 to 0.4. Variability is

ainly due to the data set being small. Also, the median of the distri-

ution reflects the actual number of positive classification originally in

he data. With such information, risk managers can identify the degree

o which the lock and dam is impactful with the probabilistic outcome

ather than a simple classification of 0 or 1. This helps in a more accurate

llocation of recovery resources.

We can examine the impact of different priors on the posterior prob-

bility distribution of the resilience worth to identify whether or not

H. Baroud, K. Barker Reliability Engineering and System Safety 170 (2018) 10–19

Fig. 2. Probability distribution of the posterior expected value of the resilience worth.

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Fig. 3. Resilience worth posterior frequency distribution using the BK model with (a) a uniform

𝜷 = 1 , (e) a prior skewed to the right, and (f) a prior skewed to the left.

15

he model is robust to potential noise that can result from the elicita-

ion process. Mentioned previously, it would be ideal to integrate the

pinion and knowledge of experts in the field with historical data to

btain the best predictive results. However, eliciting such qualitative

nformation in the form of probability distributions is often challeng-

ng. To account for uncertainty, we look at the impact of the structure

f the prior on the posterior distribution. Fig. 3 depicts the histogram of

he posterior probability distribution for the resilience for a number of

riors including (a) uniform, (b) Jeffrey’s prior, (c) and (d) the empirical

riors discussed previously, and (e) and (f) a couple of skewed priors.

he uniform and Jeffery ’s prior result in less variability in the poste-

ior distribution, whereas the empirical prior results in more breadth

nd thicker tails which means there is more variability in the prediction

hen we use empirical priors relative to uniformly structured priors.

The skewed priors are considered to be extreme cases where the elic-

tation results in misleading information that contradicts the historical

prior, (b) Jeffrey’s prior, (c) an empirical prior where 𝜶 = 1 , (d) an empirical prior where

H. Baroud, K. Barker Reliability Engineering and System Safety 170 (2018) 10–19

Fig. 4. Resilience worth posterior cumulative distribution with varying predictive accuracies, (a) ACC = 0.65, (b) ACC = 0.68, and (c) ACC = 0.80.

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ata. The data are skewed to the right with 30% of the locks and dams

onsidered to be impactful on the resilience of the inland waterway,

hich is why because of the weights in the computation of the posterior

arameters, the BK model is able to account for this imbalance. How-

ver, a prior that is strongly defined to be extremely skewed to either

he left or the right will result in a similar shape for the posterior re-

ardless of the shape of the historical data, as shown in plots (e) and (f)

n Fig. 3 .

.1.3. Accuracy and interpretability using different priors

Before implementing the model into a decision making framework,

e must also check the trade-off between its accuracy and its inter-

retability. In this section, we examine the accuracy of the model using

he entire posterior distribution of W Я . We plot the cumulative posterior distribution of all 29 locks and

ams considered in this study with three different prior distributions

hat consider different levels of variability in the prior and, as such,

esult in different predictive accuracy levels. The three prior distribution

onsidered to produce the posteriors in Fig. 4 are (a) 𝛼 = 𝛽 = 0 . 5 , (b) = 𝛽 = 3 , and (c) 𝛼 = 𝛽 = 10 . Examining the plots in Fig. 4 , we notice that it is easier to distinguish

he different resilience worth probability distributions for the different

ocks and dams in (a) than it is in (b) and (c), where in (c), all distri-

utions seem to overlap. The reason behind this effect is the amount of

ariance assumed in the prior distribution that gets transferred to the es-

imation of the posterior distribution. The less variability there is in the

osterior, the higher the accuracy is: plot (a) has an accuracy of 65%,

hile accuracy improves to 80% in plot (c). However, the interpretabil-

ty diminishes considerably, making it more difficult for decision mak-

rs to visualize the importance of different locks and dams on the inland

aterway system. If a high predictive accuracy is desired, a stochastic

anking approach would enable the ranking of probability distributions

hat may not otherwise be distinguishable. We present a multicriteria

ecision analysis technique in the following section to address this chal-

enge.

16

.2. Analysis of the Bayesian kernel posteriors using stochastic ranking

echniques

The average resilience worth for every lock is listed in Table 3 and

anked by order of importance from the largest to the smallest. A num-

er of the locks and dams have multiple chambers (mostly a main and

n auxiliary lock chamber). As such, we calculate the resilience worth

or each chamber in any lock. More than half of the locks and dams

re considered to be critical components that will contribute to the re-

overy of the waterway navigation system in the event of a large scale

isruption.

Included in this table are a subset of the characteristics of each lock

nd dam, noting that the strength of the BK model is its ability to ac-

ount for the non-linear relationship between the attributes and the re-

ponse variables in a semi-parametric way without establishing a direct

ink between the variables. Results in Table 3 suggest that the five locks

nd dams with the highest resilience worth are 18, 24, 17, 25, and 12.

e then look at these five most impactful locks/dams judging by the

xpected value, and we notice from their cumulative probability distri-

utions in Fig. 6 that it is difficult to distinguish their actual ranking

f importance, which can be the case for locks and dams with similar

haracteristics or geographical locations. Such cases arise when disrup-

ions occur in a particular region and result in the closure of a number

f similar locks and dams. Since we are able to construct the posterior

robability distribution, we propose to use a multicriteria decision anal-

sis technique to perform stochastic ranking of the posterior distribution

f the resilience worth of these locks and dams. Consider the comparison

f alternatives a and b , which are compared with measure X , a random

ariable. Fig. 5 illustrates how we can compare the percentiles of the cu-

ulative distribution function (CDF) of X a and X b using a multicriteria

ecision analysis technique to perform stochastic ranking of these two

lternatives [40] . In this application, we consider the Copeland score

CS) method which is, in general, a multicriteria ranking technique [1] ,

nd we use it here as a stochastic ranking tool.

H. Baroud, K. Barker Reliability Engineering and System Safety 170 (2018) 10–19

Table 3

Ranking of all the locks based on the resilience worth point estimate.

Rank Lock & Dam 𝑊 Я Closures River Mile Average delayed tows

1 L&D 18 0.658 35 273 1.84

2 L&D 24 0.571 4 738 1.85

3 L&D 17 0.555 105 186 1.95

4 L&D 25 0.537 23 343 2.59

5 L&D 12 0.535 40 301 1.23

6 L&D 15-1 0.528 4 201 1.9

7 L&D 14-1 0.513 6 679 1.74

8 L&D 13 0.513 20 186 1.15

9 L&D 21 0.511 16 325 1.95

10 L&D 22 0.507 15 583 2.43

11 L&D 16 0.507 0 615 1.33

12 L&D 10 0.502 2 714 0.96

13 L&D 20 0.5 47 411 2.27

14 L&D 3 0.5 0 797 0.71

15 L&D 27-1 0.5 6 523 2.06

16 L&D 27-4 0.5 0 815 1.71

17 L&D 19 0.5 5 753 3.42

18 L&D 5A 0.5 84 241 0.66

19 L&D 15-4 0.5 21 437 1.46

20 L&D 9 0.499 36 364 0.99

21 L&D 7 0.498 1 702 1.01

22 L&D 11 0.495 1 493 0.91

23 L&D 6 0.479 29 493 0.99

24 L&D 14-4 0.47 0 854 0

25 L&D 4 0.467 33 457 0.73

26 L&D 5 0.454 1 729 0.77

27 St Anthony Falls —Upper 0.438 4 483 0.02

28 L&D 1 0.433 3 647 0.25

29 Mel Price L&D 4 0.431 107 556 0.13

30 L&D 8 0.402 0 848 1.18

31 L&D 2 0.379 93 483 0.86

32 St Anthony Falls —Lower 0.312 0 853 0.2

Fig. 5. Comparison of distributions of performance for alternatives a and b as a multicriteria decision problem [40] .

Fig. 6. Cumulative posterior probability distributions of the five most impactful locks and

dams of the navigation system.

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17

The Copeland score method is considered to be a nonparametric

echnique in that each criterion is given an equal weight. Other multi-

riteria decision analysis techniques that can accommodate importance

eights applied to different criteria include the Preference Ranking Or-

anization Method for Enrichment Evaluation (PROMETHEE), Ordered

eighted Averaging, and the Technique for Order of Preference by Simi-

arity to Ideal Solution (TOPSIS), among others. Rocco et al. [41,41] and

loyd et al. [20] discuss PROMETHEE, Ordered Weighted Averaging,

nd TOPSIS, respectively, for stochastic ranking where (i) the alterna-

ives to be ranked exhibit uncertainty, (ii) this uncertainty is manifested

n probability distributions (histograms), and (iii) rather than comparing

he central tendency of these distributions, the entirety of the distribu-

ion is considered across multiple percentiles that are treated as criteria

n the multicriteria technique.

The CS is computed based on pairwise comparisons between alter-

atives in a set and is defined as the difference between the number of

imes an alternative a is better (with respect to criterion q k ) than the

H. Baroud, K. Barker Reliability Engineering and System Safety 170 (2018) 10–19

Fig. 7. Copeland score of the five most impactful locks and dams of the navigation system.

Table 4

Lock and dam repair order.

𝑊 Я ranking Posterior expected value Posterior Copeland score

1 L&D 18 L&D 17

2 L&D 24 L&D 25

3 L&D 17 L&D 24

4 L&D 25 L&D 18

5 L&D 12 L&D 12

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ther alternatives and the number of times that alternative a is worse

with respect to the same criterion q k ) to the other alternatives. C k ( a,b )

rovides a value based on a comparison between alternative a and alter-

ative b for attribute q k , k = 1,…, Ω, performed according to the rule in q. (12) . As applied here, the CS method will be used to compare differ-

nt components according to the CDF of their W Я importance measure, here the criteria represent different percentiles of the distribution.

ince we would like to identify the most critical components (largest

Я ), a maximum C k ( a,b ) is desired. C 0 ( a, b ) is initialized at zero, and q. (13) iterates through all Ω criteria (percentiles).

𝑘 ( 𝑎, 𝑏 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝐶 𝑘 −1 ( 𝑎, 𝑏 ) + 1 𝑞 𝑘 ( 𝑎 ) > 𝑞 𝑘 ( 𝑏 ) 𝐶 𝑘 −1 ( 𝑎, 𝑏 ) − 1 𝑞 𝑘 ( 𝑎 ) < 𝑞 𝑘 ( 𝑏 ) 𝐶 𝑘 −1 ( 𝑎, 𝑏 ) 𝑞 𝑘 ( 𝑎 ) = 𝑞 𝑘 ( 𝑏 )

(12)

Eq. (13) shows that the CS of alternative a is obtained by adding C i ( a,

) over all b , each representing the other alternatives [1] . The compo-

ent with the largest CS value is assumed to stochastically dominate all

ther components with respect to the set of criteria [4] .

S ( 𝑎 ) = ∑ 𝑏 ≠𝑎

𝐶 Ω( 𝑎, 𝑏 ) (13)

Using this stochastic ranking technique, the locks and dams can be

anked according to their Copeland score with approximated percentiles

f the CDF for resilience worth as criteria (the top five of which appear

n Fig. 7 ). Table 4 shows the ranking of the locks and dams based on (i)

he posterior expected value and (ii) the posterior Copeland score. Note

hat each method results in a different ranking, the reason for which

s that the Copeland score represents the entire distribution (lower and

pper tails) while the expected value is only a point estimate of the

verage resilience worth.

In case of a disruptive event impacting several components in the sys-

em, determining the component resilience worth helps decision makers

n identifying the best strategy to recover the disrupted critical infras-

ructure by ordering the component repairs according to their resilience

orth.

Note that the Copeland score method is a nonparametric decision

nalysis technique that weights all percentiles equally. A different de-

ision analysis technique (e.g., PROMETHEE, TOPSIS) could allow for

ifferent weights on the percentiles (e.g., upper 10% to more effectively

ccount for risk).

18

. Concluding remarks

This paper applies a beta Bayesian kernel model to analyze the re-

ilience of critical infrastructure networks by estimating the resilience

orth of each component in the network using prior information as well

s historical data on the component’s characteristics. The methodology

s applied to an inland waterway transportation network, the Mississippi

iver Navigation System, and the resilience worth of locks and dams is

stimated to rank components depending on how impactful they are to

he rest of the network. Resilience worth is a resilience-based component

mportance measure derived from the concept of reliability achievement

orth in the reliability engineering field.

The performance of the model is first analyzed, whereby the pre-

ictive accuracy of the Bayesian kernel model was compared to tradi-

ional statistical methods (GLM and Bayesian GLM) under different as-

umptions of the prior distribution. The metrics considered to evaluate

he predictive accuracy were the rate of true positives, the rate of true

egatives, and the overall accuracy. The best overall models were the

ayesian kernel model with either a Uniform or a Jeffrey’s prior and

he Bayesian logistic regression. Depending on the decision maker’s risk

reference, a model can be selected based on either the highest rate

f true positives (for a risk averse decision maker who cannot tolerate

n underestimation of the resilience worth) or the highest rate of true

egatives (for a risk taking decision maker whose main objective is to

inimize the amount of resources spent and wants to accurately identify

he locks and dams that are not impactful to the resilience of the sys-

em to avoid unnecessary investment costs). The posterior distribution is

ensitive to extremely skewed prior distributions that dictate the struc-

ure and form of the posterior regardless of the historical data, though

he model is robust when the parameters have different values under

he same distribution form, such as the case with the Uniform, Jeffrey’s

riors, and empirical priors.

Also, the accuracy of the model might compensate for its inter-

retability particularly when considering the posterior distribution of

he resilience worth for all the components of the inland waterway sys-

em. Visualizing the different ranking of importance of the locks and

ams becomes more challenging as we aim toward a higher level of ac-

uracy. However, this should not pose a problem, as stochastic ranking

echniques, such as the Copeland score method, can be used to distin-

uish and rank overlapping probability distributions. Results show that

hile the expected value can be used as an estimator, a more compre-

ensive metric is the Copeland score which considers the entire posterior

istribution and accounts for more uncertainty and all possible disrup-

ion scenarios.

Such an analysis can assist risk managers and decision makers in

llocating resources and determining the ranking order of the repair ac-

ivities in case of an event resulting in multiple disrupted components.

main assumption in this study is that the directionality of commod-

ty flow does not impact these importance measures. The direction in

hich commodity is flowing can result in a wide range of impacts on

he disrupted component as well as on the overall system recovery. For

nstance, if a lock was closed for hours or days, the impact on the rest

f the network as well as its recovery will depend on the traffic level in

ach direction, potentially resulting in a different importance measure

or the component.

While the resilience worth is a key factor in determining the recovery

trategy, it is equally important to account for the overall cost and time

f recovery of the strategy. Future research is involved in determining

he optimal recovery strategy by taking into account the tradeoff of the

ayesian kernel estimates of the component importance, the time, and

he cost of recovery.

cknowledgment

This work was supported by the National Science Foundation

hrough awards 1635717 and 1635813 .

H. Baroud, K. Barker Reliability Engineering and System Safety 170 (2018) 10–19

R

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eferences

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gram. J Chem Inf Models 2010;50(5):785–91 .

[2] American Society of Civil Engineers. 2013. Report Card for America’s Infrastructure

2013.

[3] Barber D . Bayesian reasoning and machine learning. Cambridge, UK: Cambridge

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[4] Barker K , Ramirez-Marquez JE , Rocco CM . Resilience-based network component im-

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