security sensitivity committee deliverable...

RAIN–RiskAnalysisofInfrastructureNetworksinResponsetoExtremeWeather ProjectReference:608166 FP7-SEC-2013-1Impactofextremeweatheroncriticalinfrastructure ProjectDuration:1May2014–30April2017

ThisprojecthasreceivedfundingfromtheEuropeanUnion’sSeventhFrameworkProgrammeforresearch,technologicaldevelopmentanddemonstrationundergrantagreementno608166

SecuritySensitivityCommitteeDeliverableEvaluationDeliverableReference D5.2DeliverableName Reportonriskanalysisframeworkforcollateralimpactsof

cascadingeffectsContributingPartners TUDelftDateofSubmission April2017Theevaluationis:

• Thecontentisnotrelatedtogeneralprojectmanagement• Thecontentisnotrelatedtogeneraloutcomesasdisseminationandcommunication• Thecontentisnotrelatedtocriticalinfrastructurevulnerabilityorsensitivity

Diagrampath1-2-3.ThereforetheevaluationisPublic.DecisionofEvaluation Public Confidential Restricted EvaluatorName P.L.Prak,MSSMEvaluatorSignature DateofEvaluation 2017-05-02

RAIN – Risk Analysis of Infrastructure Networks in Response to Extreme Weather Project Reference: 608166 FP7-‐SEC-‐2013-‐1 Impact of extreme weather on critical infrastructure Project Duration: 1 May 2014 – 30 April 2017

Date: 30/04/2017 Dissemination level: (PU, PP, RE, CO): PU This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 608166

Deliverable 5.2 -‐ Report on risk analysis framework for collateral impacts of cascading effects

Authors Noel van Erp (TU Delft) Ronald Linger (TU Delft) Nima Khakzad (TU Delft)

Pieter van Gelder* (TU Delft)

*Corresponding author: [email protected]

D5.2- Report on risk analysis framework for impacts of cascading effects

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DOCUMENT HISTORY

Index Date Author(s) Main modifications

E01 04-30-2017 NVE, RL, NM, and PVG First Draft

05-11-2017 Reviewers: Vajda Andrea (FMI) and Timo Hellenberg (HI)

Review by FMI and HI

05-13-2017 NVE, RL, NM, and PVG Final Report

Document Name: Report on risk analysis framework for collateral impacts of cascading effects

Work Package: 5

Task: 5.2

Deliverable: 5.2

Deliverable scheduled date (36th Month) 30th April 2017

Responsible Partner: TU Delft


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Table of Contents

Executive Summary ................................................................................................................................ 5

1. Introduction ................................................................................................................................... 7

2. A Comparison Between Markov Chain and Dynamic Bayesian Network ....................................... 8

2.1. Markov Chain Analysis ........................................................................................................... 8

2.2. Dynamic Bayesian Network ................................................................................................. 11

2.3. Application of DBN to the Transportation Problem ............................................................. 12

2.4. Results .................................................................................................................................. 14

2.5. Interconnected systems ....................................................................................................... 17

3. An Alternative Methodology to Model Cascading Effects ........................................................... 21

3.1. A System and Its States ........................................................................................................ 21

3.2. The Physics of the Cascade Process ..................................................................................... 21

3.2.1. A Simple Example of a Probability Map ....................................................................... 22

3.2.2. Probability Maps that Capture Inhomogeneous Markov Processes ............................ 23

3.3. Modelling Cascades Through Time ...................................................................................... 24

3.3.1. Constructing Proxy State Probability Distributions ...................................................... 25

3.3.2. Applying the Product and the Sum Rules ..................................................................... 25

4. The Probability Sort Algorithm ..................................................................................................... 28

4.1. How to Graphically Represent Multivariate Pdfs ................................................................. 28

4.2. The Idea Behind the probability Sort Algorithm .................................................................. 31

4.3. A Probability Sort Analysis ................................................................................................... 33

4.4. The Information Entropy of a System .................................................................................. 40

5. Modelling of Homogeneous Markov Processes with the Probability Sort Algorithm .................. 42

5.1. The Physics of the Probability Map ...................................................................................... 42

5.2. Some Example Probability Maps .......................................................................................... 42

5.3. A Probability Sort Analysis of Cascading Effects ................................................................... 44

5.3.1. Time Evolving Marginal Damage State Probabilities .................................................... 45

5.3.2. Time Evolving ML Damage States ................................................................................ 47

6. Modelling of Inhomogeneous Markov Processes with the Probability Sort Algorithm ............... 50

6.1. A Topology and Landslide Physics ........................................................................................ 50

6.2. A First Cascading Effect Analysis .......................................................................................... 51

6.3. A Second Cascading Effect Analysis ..................................................................................... 54

6.4. Comparing Inhomogeneous and Homogenous Markov Assumptions ................................. 56


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6.5. A Third Cascading Effect Analysis ......................................................................................... 57

7. Concluding remarks ..................................................................................................................... 61

8. References ................................................................................................................................... 63

9. Appendix: The Probability Sort Algorithm ................................................................................... 64

9.1. Algorithmic Outline .............................................................................................................. 64

9.2. Pseudo-‐Code ........................................................................................................................ 64

9.3. Pen and Paper Algorithmic Run ........................................................................................... 73


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Executive Summary

The RAIN project is concerned about the behaviour of critical infrastructures, such as road, rail and electricity and telecommunication networks, when subjected to extreme weather events such as heavy rainfall, landslides, floods, etc. and combinations of these. These extreme weather events, as well as the consequent behaviour of the infrastructural elements, vary both spatially and temporally.

For example, the closer an infrastructural object is to the centre of some event, the greater will be its tendency to be in a damaged state. Also, if the damage state of one infrastructural object is dependent on the damage state of another (such as in cascading networks), then as the latter infrastructural object is damaged and time progresses the greater will be the probability of the former infrastructural object to be in a damaged state.

A risk-‐based decision-‐making framework for large-‐scale infrastructural networks under influence of extreme weather hazards has been developed in RAIN deliverables D5.1 and D.5.5. In the current deliverable, we will concentrate in particular on the modelling of cascading effects in large-‐scale infrastructural networks. Neglecting or underestimating (inter)dependencies between the failures or disruptions of the critical infrastructure components can cause designers, experts, managers and decision makers to underestimate the overall inter-‐infrastructural risks. It is therefore necessary to further develop approaches that consider the interconnected nature of critical infrastructure components and – systems, which will be the aim of this report.

Real life systems of infrastructural objects will be conceptualized as an event tree system, where all the specific damage states of the infrastructural system are leaves on the event tree. If we have N infrastructural objects and M damage states per object, then the event tree system will consist of

NM distinct states. Now, as the number of infrastructural objects N grows, the number of distinct

states , NM , of the corresponding event tree will grow exponentially. So, for non-‐trivial infrastructural systems the size of the corresponding state space will quickly become overwhelmingly large and, as a consequence, seemingly, make futile any attempt to come at some sort of (approximately) exact evaluation of the infrastructural system.

Furthermore, damage states of infrastructural objects are time dependent, due to deterioration, dissipation effects, but also due to human intervention after mitigation measures have been applied to the infrastructural system.

Finally, the damage states in one infrastructural system (for instance an electricity network) may affect the damage state in another infrastructural system (for instance a rail network). Interconnectedness between infrastructures causes the computational complexity to grow even more.

In this deliverable we have derived a general framework based on Bayesian probability theory and the Probability Sort algorithm in order to model time-‐dependent and cascading effects in infrastructural networks through space and time. This framework also allows for human intervention measures to mitigate risks or reduce failure probabilities of infrastructural components. The proposed probability sort approach allows one to come to some sort of exact evaluation of even


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exponentially large event trees of system states, as long as the information entropy in that event tree is low enough, as will be demonstrated in this deliverable.

Applications are presented for the Bayesian modelling of cascading effects in landslides and for the cascading effects in an electricity network. The outcomes are time-‐dependent probability maps of failure of the overall infrastructural systems, which serve as input for the decision-‐making by infrastructural managers.


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1. Introduction

In this report we propose to use Bayesian statistics and the Probability Sort algorithm in order to model cascading effects over time and space. The proposed probability sort approach allows one to come to some sort of exact evaluation of even exponentially large event trees of system states, as long as the information entropy in that event tree is low enough, as will be demonstrated in this report.

This report is structured as follows. In Chapter 2 there will be given a discussion of the (relatively) established Markov Chain (MC) and Dynamic Bayesian Networks (DBN) methodologies , as we will give for a simple three-‐bridge toy-‐problem a pen-‐and-‐paper MC analysis as well as a GeNIe DBN analysis. In Chapter 3 we then describe an alternative probability sort approach that was specifically developed for the RAIN project. In Chapter 4 we give the probability Sort algorithm which lies at the core of this alternative probability sort approach. In Chapter 5 the alternative probability sort approach is illustrated for a toy-‐problem under the assumption of homogeneous transition probabilities. In Chapter 6 then there are given two problems of infrastructural networks under cascading effects due to extreme weather events in which inhomogeneous transition probabilities are assumed (cascading landslides under extreme rainfall in Sec. 6.1 – 6.4 and cascading effects in an electricity network in Sec. 6.5).


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2. A Comparison Between Markov Chain and Dynamic Bayesian Network

In this chapter we critically compare the modelling techniques of Markov Chains and Dynamic Bayesian networks for modelling infrastructural networks, consisting of large numbers of components.

For the sake of illustration, consider a land-‐based road transportation system consisting of three bridges, B1-‐B3 (Figure 2.1), in which bridges B1 and B2 connect the northern part to an island, and bridge B3 connects further to the southern part.

Figure 2.1: A transportation system between H and W comprising three bridges B1, B2 and B3

The bridges are assumed to have constant failure rates; as such, the failure probability of each bridge as a function of time can be calculated using the exponential distribution as:

P(Bridge fails) = P(t) = 1-‐ exp (-‐λt) (2.1)

Since techniques such as fault tree or conventional Bayesian network are not able to account explicitly for temporal evolution of failure probabilities, application of Markov chain (Ebeling, 1997) and dynamic Bayesian network (Jensen & Nielsen, 2007) will be sought in this chapter while emphasizing their advantages and drawbacks.

2.1. Markov Chain Analysis

Markov chains are mathematical systems that ‘hop’ from one state to another state. Predictions for the future state are based only on its present state, but not on its states before. Hence, the system is conditional on the present state of the system, its future and past states are independent.

Considering bridges with binary states, i.e., Bridge ={work, fail}, the total number of states for the transportation system will be 23=8 as listed in Table 2.1.


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Table 2.1. State-‐space of the transportation system in Figure 2.1.

State B1 B2 B3 System

① √ √ √ √

② √ √ ╳ ╳

③ √ ╳ √ √

④ ╳ √ √ √

⑤ √ ╳ ╳ ╳

⑥ ╳ √ ╳ ╳

⑦ ╳ ╳ √ ╳

⑧ ╳ ╳ ╳ ╳

For each combination of the bridges’ states, the state of the system can be identified as being in the failure (one cannot get from Home, H, to Work, W) or operation mode. Accordingly, the total failure probability of the system can be identified as the complement of the total probability of operation as:

𝑃 𝑠𝑦𝑠𝑡𝑒𝑚 𝑓𝑎𝑖𝑙𝑠 = 1 − (𝑃!(𝑡) + 𝑃!(𝑡) + 𝑃!(𝑡)) (2.2)

where 𝑃!(𝑡) is the probability of the system being in the ith state at time t. To calculate the probabilities of the states reported in Table 2.1, the state-‐transition diagram in Figure 2.2 has been depicted. 𝜆!, 𝜆!, 𝑎𝑛𝑑 𝜆! refer, respectively, to the failure rates of the bridges B1, B2, and B3. It is worth noting that the Markov Chain developed in this way is a homogeneous Markov chain in that the failure rates (transition rates) are constant over time (a property which limits the application of homogeneous Markov chain to time-‐dependent failure rate processes).

1

2 3 4

765

8

λ3 λ3 λ2λ2λ1λ1

λ2λ2λ1 λ1 λ1 λ1

λ3λ3λ3λ3

λ2λ2

λ1 λ1 λ3λ3λ2λ2

1

2 3 4

765

8

λ3 λ2λ1

λ2λ1 λ1

λ3λ3

λ2

λ1 λ3λ2

Figure 2.2. Markov Chain for the transportation network shown in Figure 2.1.


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Using the state transition diagram in Figure 2.2, the state probabilities can be derived by solving a system of differential equations:

( ) ( ) ( )11 2 3 1

dP tP t

dtλ λ λ= − + + ;

( ) ( ) ( ) ( )23 1 1 2 2

dP tP t P t

dtλ λ λ= − +

( ) ( ) ( ) ( )32 1 1 3 3

dP tP t P t

dtλ λ λ= − +

( ) ( ) ( ) ( )41 1 2 3 4

dP tP t P t

dtλ λ λ= − +

( ) ( ) ( ) ( )52 2 3 3 1 5

dP tP t P t P t

dtλ λ λ= + −

( ) ( ) ( ) ( )61 2 3 4 2 6

dP tP t P t P t

dtλ λ λ= + −

( ) ( ) ( ) ( )71 3 2 4 3 7

dP tP t P t P t

dtλ λ λ= + −

( )7

81

1 ii

P P t=

= −∑

Solving the equations, we have:

( ) ( )1 2 31

tP t e λ λ λ− + +=

( ) ( ) ( )1 2 32 1t tP t e eλ λ λ− + −= −

( ) ( ) ( )1 3 23 1t tP t e eλ λ λ− + −= −

( ) ( ) ( )2 3 14 1t tP t e eλ λ λ− + −= −

M

Having the state probabilities, the failure of the transportation system can be estimated as:


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( ) ( ) ( ) ( ) ( )( )1 3 2 3 1 2 31 3 4system fails 1 1 t t tP P P P e e eλ λ λ λ λ λ λ− + − + − + += − + + = − + − (2.3)

2.2. Dynamic Bayesian Network

BN (Jensen & Nielsen, 2007) is a probabilistic method for reasoning under uncertainty in which random variables are represented by nodes while the conditional dependencies or cause-‐effect relationships among them are denoted by directed arcs (Figure 2.3(a)).

(a) (b)

Figure 2.3. Schematic of (a) conventional Bayesian network and (b) dynamic Bayesian network (Jensen and Nielsen, 2007)

The type and strength of the dependencies can be encoded in form of conditional probability tables assigned to the nodes. Using the chain rule and the concept of d-‐separation, the joint probability of a set of random variables 𝑈 = {𝑋!,𝑋!,… ,𝑋!} can be factorized as the product of marginal and local conditional probabilities:

𝑃 𝑈 = 𝑃(𝑋!|𝜋 𝑋! )!!!! (2.4)

where 𝜋 𝑋! is the parent set of the node 𝑋! . For instance, the joint probability distribution of the random variables 𝑋!,𝑋!,𝑋! and 𝑋! in the BN of Figure 2.3(a) can exclusively be expanded as 𝑃(𝑋!,𝑋!,𝑋!,𝑋!) = 𝑃 𝑋! 𝑃 𝑋! 𝑋! 𝑃 𝑋! 𝑋!,𝑋! 𝑃(𝑋!|𝑋!).

Dynamic Bayesian network (DBN) is an extension of ordinary BN that, compared to its ancestor, facilitates explicit modelling of temporal evolution of random variables over a discretized timeline (Figure 2.3(b)). Dividing the timeline to a number of time slices, DBN allows a node at ith time slice to be conditionally dependent not only on its parents at the same time slice but also on its parents and itself at previous time slices:

𝑃 𝑈!!∆! = 𝑃(𝑋!!!∆!|𝑋!! ,𝜋(!!!! 𝑋!!),𝜋(𝑋!!!∆!)) (2.5)

According to the DBN in Figure 2.3(b), the conditional probability of 𝑋!, for example, at the time slice 𝑡 + ∆𝑡 is 𝑃(𝑋!!!∆!|𝑋!!!∆! ,𝑋!! ,𝑋!!). For the sake of brevity, the DBN in Figure 2.3(b) can be depicted in an abstract from, as shown in Figure 2.4, where the numbers within the squares refer to the number of time slice taken into account in temporal dependencies.


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Figure 2.4. Abstract representation of DBN in Figure 2.3(b). The numbers attached to the temporal arcs indicate the number of previous time slices to be taken into account in temporal dependencies.

2.3. Application of DBN to the Transportation Problem

The road transportation system with 3 bridges B1, B2 and B3 shown in Figure 2.1 can be modelled as a reliability block diagram as depicted in Figure 2.5 (a).

B1

B2

B3H W

Subsystem 1

B1 B2

Subsystem S1 B3

System fails

(a) (b)

Figure 2.5. Transportation system as (a) a reliability block diagram and (b) a fault tree.

The BN presentation of the transportation system has also been given in Figure 2.6. To model the temporal evolution of the system failure, the BN can be replicated in sequential time steps as depicted in Figure 2.7.


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S1 B3

B1 B2

System

S1 B3

B1 B2

System

Figure 2.6. Modelling the transportation system as a conventional Bayesian network.

S1 B3

B1 B2

System

S1 B3

B1 B2

System

S1 B3

B1 B2

System

S1 B3

B1 B2

System

S1 B3

B1 B2

System

S1 B3

B1 B2

System

t=0 t=Δt t=2Δt

S1 B3

B1 B2

System

S1 B3

B1 B2

System

S1 B3

B1 B2

System

t=0 t=Δt t=2Δt

Figure 2.7. Modelling the transportation system as a dynamic Bayesian network.

In this regard, the conditional probabilities within each time slice can be modelled as simple AND/OR gates (as in the fault tree in Figure 2.5(b)) while the conditional probabilities between sequential time slices (e.g., from t=Δt to t=2Δt) can be determined as :

(I) in the first time step t=0

B1 work fail

B2 work fail

B3 work fail

1 0

1 0

1 0


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B1 Work Fail

B3 Work Fail

B2 work fail work fail

S1 work fail work fail

S1 work 1 1 1 0

System work 1 0 0 0

fail 0 0 0 1

fail 0 1 1 1

(II) in sequential time steps:

1tBΔ work fail

21tB Δ

Work 1 te λ− Δ 0

Fail 11 te λ− Δ− 1

2tBΔ work fail

22tB Δ



3tBΔ work fail

23tB Δ



2.4. Results

To make a comparison between Markov chain and DBN, the failure probability of the system is calculated for a 20-‐year period assuming λ1= 0.1/year, λ2= 0.15/year, and λ3= 0.2/year. As for Markov chain, the system’s failure probability can be calculated using Equation (2.3) while for the DBN analysis, the DBN displayed in Figure 2.7 is implemented and run in Bayesian network software GeNIe (Figure 2.8). the results of the Markovian analysis and the DBN analysis have been reported in Table 2.2 while the system’s failure probabilities calculated using the methodologies have been shown in Figure 2.9. As can be seen the results of the two methodologies are nearly the same.


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Figure 2.8. Modelling of the DBN in GeNIe


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Table 2. The comparison of results for a 20-‐year period

BN MC

t P(B1) P(B2) P(B3) P(System) P(system)

0 0 0 0 0 0

1 0.0952 0.1393 0.1813 0.1922 0.1921

2 0.1813 0.2592 0.3297 0.3612 0.3612

3 0.2593 0.3624 0.4513 0.5028 0.5027

4 0.3298 0.4512 0.5507 0.6176 0.6175

5 0.3936 0.5277 0.6322 0.7086 0.7085

6 0.4513 0.5935 0.6989 0.7795 0.7795

7 0.5036 0.6501 0.7535 0.8342 0.8341

8 0.5508 0.6988 0.7982 0.8759 0.8758

9 0.5936 0.7408 0.8348 0.9074 0.9074

10 0.6323 0.7769 0.8647 0.9312 0.9311

11 0.6673 0.8080 0.8892 0.9490 0.9489

12 0.6990 0.8347 0.9093 0.9622 0.9622

13 0.7276 0.8577 0.9258 0.9721 0.9721

14 0.7535 0.8776 0.9392 0.9794 0.9794

15 0.7770 0.8946 0.9502 0.9848 0.9848

16 0.7982 0.9093 0.9593 0.9888 0.9888

17 0.8174 0.9219 0.9666 0.9918 0.9918

18 0.8348 0.9328 0.9727 0.9940 0.9940

19 0.8505 0.9422 0.9776 0.9956 0.9956

20 0.8648 0.9502 0.9817 0.9967 0.9967


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Figure 2.9. Predictions for the system failure in 20 years (blue and red lines on top of each other)

2.5. Interconnected systems

Although the two methodologies have resulted in the same failure probabilities for the system of interest, they have advantages and drawbacks compared to each other. First, in both techniques, it is supposed that the failure rates do not change with time. The Markov chain model which is developed in this way is known as a homogeneous Markov chain (with constant transition rates); in a homogeneous DBN, the temporal probabilities – denoted by arcs between sequential time slices – remain constant from a time slice to the other (Figure 2.7). Such a DBN is also known as being based on Markovian property, where each node at time slice t only needs to be conditioned on nodes in the previous time slice t-‐1. This limits the application of both techniques when it comes to the modelling of time-‐dependent failure rate models, for example, where the failure of a bridge does not follow an exponential distribution but Weibull distribution. In the latter case, the failure rate of the bridge can increase or decrease over time.

To address this issue, non-‐homogeneous Markov models (Huang, 1997) have been developed with approximate algorithms. As for DBN, such a time dependency can be modelled by conditioning a node in the time slice t to nodes not only in the last time slice t-‐1 but also to the time slices before the last, i.e., t-‐2, t-‐3, so on. While the Markov chain analysis – whether homogeneous or non-‐homogeneous – can result in the notorious state-‐space explosion plight1, the development and simulation of a corresponding non-‐homogeneous DBN would easily become intractable and too time-‐consuming via available commercial software such as GeNIe.

In the context of constant failure rates, however, DBN outperforms MC. For make the discussion more concrete, consider the transportation system in Figure 2.10, where in addition to the bridges there is a rail way passing along B1 and B4.

1 The number of the states of the system grows exponentially with the number of systems components. For example, if the system consisted of five bridges instead of three, the number of states would increase from 8 to 32.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20

P(system

fails)

Time (year)

DBN MC


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Figure 2.10. Transportation system comprising three bridges B1, B2 and B3 for car passage and the bridges B1 and B4 for train passage.

The system’s fault tree and corresponding BN have been depicted in Figures 2.11(a) and (b), respectively. Having the failure rate of the bridge B4, the BN presented in Figure 2.11(b) can be replicated in sequential time intervals as a DBN to calculate the failure probability of the system over time. It is worth noting that modelling the transportation system in Markov chain would result in 16 states as reported in Table 2.3. Accordingly, the failure probability of the system could be calculated as:

P(System fails) = 1 – {P1(t) + P2(t)+ P3(t) + P5(t) + P6(t) + P7(t) + P9(t) + P10(t)}.

Modelling the transportation shown in Figure 2.10 as a DBN in GeNIe software, as depicted in Figure 2.12, the probability of the system failure over a 20-‐year period has been displayed in Figure 2.13. For sake of comparison, the failure probability of the transportation system shown in figure 2.1 (no rail way) has also been presented in figure 2.13. Obviously, the addition of a parallel transportation system, i.e., rail way, to the network has decreased the failure probability of the entire system.

B1 B2

Subsystem S1 B3

No car passes

B4 B1

No train passes

System fails

B1 B2

S1 B3

No car

B4

No train

System fails

(a) (b)

Figure 2.11. (a) Fault tree analysis and (b) Bayesian network analysis of the transportation system in Figure 2.10.


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Table 2.3. State-‐space of the transportation system in Figure 2.10.

States B1 B2 B3 B4 System

1 √ √ √ √ √

2 √ √ √ X √

3 √ √ X √ √

4 √ √ X X X

5 √ X √ √ √

6 √ X √ X √

7 √ X X √ √

8 √ X X X X

9 X √ √ √ √

10 X √ √ X √

11 X √ X √ X

12 X √ X X X

13 X X √ √ X

14 X X √ X X

15 X X X √ X

16 X X X X X


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Figure 2.12. Modelling the transportation system shown in Figure 2.10 in GeNIe as a dynamic Bayesian network.

Figure 2.13. Predictions for the system failure in 20 years with and without the railway.

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

P(system

fails)

Time (year)

Rail-‐Car Car


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3. An Alternative Methodology to Model Cascading Effects

In this chapter we proceed to give a general outline of a methodology by which to model cascading effects which offers an alternative to the previously discussed Markov Chain and Dynamic Bayesian Network methodologies. This alternative methodology is based upon the Probability Sort algorithm which has been developed for both this RAIN project as well as the EU FP7 InfraRisk project (van Erp, Linger, van Gelder, 2016). This probability sort approach has the advantage that it may be used to model cascading effects for systems that consist of a large number of elements, providing that the information entropy of that system is sufficiently low, as for low entropic systems the relevant probability components will be in a exponentially small sub-‐region of the state space.

3.1. A System and Its States

If we have some system S which consists of n components and where the ith component of this system can be in one of im (damage) states, which may be encoded as

ii ms ,,1…= . (3.1)

Then the state of a system S may be encoded by the (system state) vector

[ ]nsss 21=s . (3.2)

It follows from (3.1) and (3.2) that the total number of states the system may be in equals

∏=

=n

iimN

1

. (3.3)

In case we have that the number of states for each component of the system admits the same number of states m , then it can be glanced from (3.3) that the number of states N of the system S will grow exponential in the number of elements n :

nmN = . (3.4)

3.2. The Physics of the Cascade Process

In order to model patterns of cascade propagations through the possible system states, we need to know the probability of a current system state ( )nows as a function of some previous system state ( )previouss . For the modelling of a first-‐order Markov process ( ) ( )( )1| −ttp ss , we need to determine the

system state probability distribution by way of the vector-‐function f of ( )1−ts , where ( )1−ts is the system state (3.2) at time-‐step 1−t .

It is in this function f , henceforth called the probability map, that the ‘physics’ of the cascading effect mechanism under consideration find their expression. The probability map f is a collection of n row-‐vectors, which may be of different length, depending on the possible states of the corresponding system elements, of probabilities which must sum to one:


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( )( ) ( )( )11 −− = tij

t f ssf , ⎩⎨⎧

=

=

.,,1,,,1

imjni

……

(3.5)

It follows that the probability ( ) ( )( )1| −ttp ss , for a given state vector ( )ts , may be written down as the

product

( ) ( )( ) ( )( )∏=

−− =n

i

tsi

tti

fp1

1,

1| sss , (3.6)

where it is to be understood that the possible states (i.e., labels) of the elements is of the state

vector ( )ts correspond with the column coordinates of the ith row-‐vector of the probability map f .

3.2.1. A Simple Example of a Probability Map

We now will give, as the physics for this example are quite simple, for pedagogical purposes, an outline on how to model the probability map f for a system of objects in which one or more failures may set off a cascade of failures. After that, cascading effects will be modelled for infrastructural networks under influence of extreme weather events (cascading landslides under extreme rainfall in Sec. 6.1 – 6.4 and cascading effects in an electricity network in Sec. 6.5).

Say, we have 2kn = objects arranged in a k-‐by-‐k grid with, say, a distance of 50 meters between

horizontally and vertically adjacent objects and a distance of 22 505071.70 += meters between diagonally adjacent objects. Let at each time step the state of each object be dichotomous, or, equivalently, 2=im (3.1), where the damage state vector ( )1-ts has the elements

( )

⎩⎨⎧

=−

failureafe

s ti ,2s,11 (3.7)

Then, if we assume that every failed object will generate a radiation influence of Q which falls off

as the inverse of the distance, generating a location dependent radiation r of

( )( ) ( )22

,ii

iyyxx

Qyxr−+−

= , (3.8)

where ( )ii yx , are the location coordinates of the ith object, and if we assume a superposition of

radiations, then we may determine the total radiation R at a given coordinate ( )yx, as the sum of

overpressures of all the failed objects, (3.7) and (3.8):

( ) ( ) ( ) ( )( )( ) ( )

∑∑=

−

= −+−−=−=

n

iii

ti

n

iii

yyxx

QsyxrsyxR1

22

1

1

1,1, . (3.9)

By way of a probit-‐function , we then may compute the probability of a non-‐failed object failing as a function of its location coordinates:


23

( )( ) ( )

,2,

erf121

2exp

21, 0

2,0

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ ++=⎟⎟

⎠

⎞⎜⎜⎝

⎛−= ∫

+

∞−

yxRduuyxP

yxR β

π

β

(3.10)

which for a total radiation of ( ) 0, =yxR will give a corresponding base-‐line damage probability of

( )[ ]2erf1 021 β+ , (3.10). It follows that the probability map f can be constructed as, (3.7) and (3.10):

( )( ) ( ) ( )[ ] ( )

[ ] ( )⎪⎩

⎪⎨⎧

=

=−=

−

−−

,2,10,1,,,1

1

11

ti

tiiiiit

ssyxPyxP

sf (3.11)

where ( )ii yx , are the location coordinates of the ith object, ( )ii yxP , is the (probit-‐)probability of a

failure for a non-‐failed object at location ( )ii yx , , and ( )1−tis is the state of the ith component of the

system.

Note that in (3.11) we have an example of an irreversible Markov process, as objects, after having failed, will remain in the failed state at all the consequent time steps t .

3.2.2. Probability Maps that Capture Inhomogeneous Markov Processes

The probability map (3.11) corresponds with a homogenous Markov processes in which the transition probabilities remain the same over time. Stated differently, in the above system of objects example possible ‘burn-‐out effects’ (i.e., the drop in radiation of failed objects due to the decrease in the amount of capacity that is contained in the object) are not taken into account. In order to model inhomogeneous Markov processes by way of the probability sort algorithm, we need to introduce the concept of counters.

For example, if we want to incorporate burn-‐out (3.7) through (3.11), then we may postulate radiation of the ith object iQ as some monotonic decreasing function g of the number of time steps

ik that this object is already ‘burning’:

( )ii kgQ = . (3.12)

Substituting (3.12) into (3.8), we obtain

( ) ( )( ) ( )22

|,ii

iii

yyxx

kgkyxr

−+−= , (3.13)

which gives a total radiation (3.9),

( ) ( )( ) ( )( ) ( )

∑=

−

−+−−=

n

iii

iti

yyxx

kgsyxR

122

1 1|, k , (3.14)

that is also a function of the counter vector k , where the elements ik are the number of time

steps that an object is in a failed state.


24

Substituting (3.14) into (3.10), we obtain the probit-‐function of the probability of a non-‐failed object failing as a function of its location coordinates and the counter vector k :

( ) ( ).

2|,

erf121|, 0

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ ++=

kk

yxRyxP

β (3.15)

It follows that the probability map f which is both dependent on the actual damage state vector ( )1−ts (i.e., the specific constellation of failed objects) as well as the counter vector ( )1−tk (i.e., the

specific time a given burning object is already burning) can be constructed as, (3.7) and (3.15):

( ) ( )( ) ( ) ( )[ ] ( )

[ ] ( )⎪⎩

⎪⎨⎧

=

=−=

−

−−−

,2,10,1,|,|,1

,1

111

ti

tiiiiitt

ssyxPyxP kk

ksf (3.16)

It is stated in (Khakzad, 2015) that inhomogeneous Markov processes are intractable, because of the conditional (transition) probability tables that fail exponentially in size as time effects are taken into account. It will be shown in this Chapter 6 that the proposed probability sort approach is particularly amenable to the modelling of inhomogeneous Markov processes.

3.3. Modelling Cascades Through Time

The conditional state probability distributions (3.6) may be combined, in principle, by the product rule of (Bayesian) probability theory:

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )1111 |||, −+−+ = ttttttt ppp sssssss . (3.17)

By way of the sum rule of (Bayesian) probability theory, the bivariate (3.17) may be summated, in principle, over the states at time step t :

( ) ( )( ) ( ) ( ) ( )( )( )∑ −+−+ =t

ttttt pps

sssss 1111 |,| . (3.18)

By repeated applications of (3.17) and (3.18), one may obtain the probability distribution of the system at time step Tt + , given the initial state of the system at time step 1−t .

It is stated that (3.17) and (3.18) may be used in principle as it follows from both (3.3) and (3.4) that the number of probabilities in the conditional probability distribution (3.6) will grow exponentially as the number of components n in the system S increases. In order to circumvent this exponential failure for systems that have a low information entropy it is proposed that the Probability Sort algorithm be used.

By way of this Probability Sort algorithm there may be found for each consecutive time step all the

states ( )tis for which we have that

( ) ( )( ) >0| ss tip cut-‐off. (3.19)

The way the Probability Sort algorithm accomplishes this is explained in the following paragraphs.


25

3.3.1. Constructing Proxy State Probability Distributions

At time step t the Probability Sort algorithm identifies the state that has the highest probability, or, equivalently, that corresponds with the Maximum Likelihood (ML), and then working its way down to

the next highest probability, and the next highest, and so on. This will give tN~ probability sorted

states ( )tis for which we have that

( ) ( )( ) >−1| ttip ss cut-‐off. (3.20)

Stated differently, the Probability Sort algorithm gives, for a given probability map f , a proxy state probability distribution p~ as a collection of state probabilities for a collection of state vectors (3.6):

( ) ( )( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )⎪⎪

⎩

⎪⎪

⎨

⎧

=

=

=

=

−

−

−

−

,,|

,,|,,|

|~

~1

~~

21

22

11

11

1

tN

ttN

tN

tttt

tttt

tti

tttpP

pPpP

p

sss

ssssss

ss

(3.21)

where ( ) ( )tj

ti PP ≥ for ji < , and where the probability coverage of the proxy p~ is equal or smaller

than 1:

probability coverage ( ) ( )( ) ( ) 1|~~

12

~

1

1 ≤== ∑∑==

−tt N

i

tN

i

tti Pp ss . (3.22)

Note that for a cut-‐off of zero the probability coverage (3.22) will be one, that is, if the corresponding state space probability distribution admits an exact evaluation.

3.3.2. Applying the Product and the Sum Rules

In order to come to (3.19), we start with some initial state of the system ( )0s , for which we compute the probability sorted proxy

( ) ( )( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )⎪⎪

⎩

⎪⎪

⎨

⎧

=

=

=

=

.,|

,,|,,|

|~

1~

01~

1~

12

012

12

11

011

11

01

11 tNNN

i

pP

pPpP

p

sss

ssssss

ss

(3.23)

We then determine for each of the probability sorted states ( )1is in (3.23) the corresponding proxy

distributions:

( ) ( )( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )⎪⎪

⎩

⎪⎪

⎨

⎧

=

=

=

=

,,|

,,|

,,|

|~

2~

12~

2|~

22

122

2|2

21

121

2|1

12

222 NiNiN

ii

ii

ij

pP

pP

pP

p

sss

sss

sss

ss

(3.24)


26

for 1~,,1 Ni …= , and where, in order to enforce (3.19),

( ) ( ) ( ) ( ) ( )( )>=⋅ 01212| |, sss ijiij pPP cut-‐off. (3.25)

Then by way of the product rule (3.17) we have, (Jaynes, 2003):

( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )( )

( )

( ) ( ) ( ) ( ) ( )( )( )

( )

( ) ( ) ( ) ( ) ( )( )( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎪⎩

⎪⎨⎧

=⋅

⎪⎩

⎪⎨⎧

=⋅

⎪⎩

⎪⎨⎧

=⋅

=

,,

,|,

,,

,|,

,,

,|,

|,~

2~

1012

~12

|~

22

1012

212

|2

21

1012

112

|1

012

2

22

N

iiNiiN

iiii

iiii

ij

pPP

pPP

pPP

p

ss

sss

ss

sss

ss

sss

sss

(3.26)

for 1~,,1 Ni …= . Then by way of the sum rule (3.18) , (Jaynes, 2003), we may obtain the marginal

state probability distribution

( ) ( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=⋅=

=⋅=

=⋅=

==

∑∑

∑∑

∑∑

∑

==

==

==

=

,,|,

,,|,

,,|,

|,~|~

2~

~

1

012~

~

1

12|~

2~

22

~

1

0122

~

1

12|2

22

21

~

1

0121

~

1

12|1

21

~

1

01202

2

1

2

1

22

11

11

1

N

N

iiN

N

iiiNN

N

ii

N

iii

N

ii

N

iii

N

iijj

pPPP

pPPP

pPPP

pp

ssss

ssss

ssss

sssss (3.27)

where for 2~,,1 Nj …= the ( )2

js for the initial state conditions ( )1is are collected and the

corresponding probabilities ( ) ( ) ( )( )122| | ijij pP ss= are summated over 1

~,,1 Ni …= , as only the vectors ( )2js are retained.

If repeat this process of applying the product and sum rules for another 2−t times, then we will obtain the proxy state probability distribution of interest:

( ) ( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⋅=

⋅=

⋅=

==

∑

∑

∑

∑

−

−

−

−

=

−

=

−

=

−

=

−

,,

,,

,,

|,~|~

~

~

1

1|~~

2

~

1

1|22

1

~

1

1|11

~

1

010

1

1

1

1

tN

N

i

ti

tiN

tN

tN

i

ti

ti

t

tN

i

ti

ti

t

N

i

ti

tj

tj

t

t

tt

t

t

t

PPP

PPP

PPP

pp

s

s

s

sssss

(3.28)


27

where, because of (3.25), we have that, for tNj ~,,1…= , all ( ) ( )( )0| ss tjp are greater than the cut-‐off

(3.19).


28

4. The Probability Sort Algorithm

4.1. How to Graphically Represent Multivariate Pdfs

We now discuss how to represent highly multivariate probability distribution functions on a two dimensional plane (Skilling, 2004), as this will the groundwork for the upcoming discussion of the Probability Sort analysis.

Say we wish to numerically evaluate the integral of the bivariate normal distribution ( )Σ,µMN

where

⎟⎟⎠

⎞⎜⎜⎝

⎛=00

µ , and ⎟⎟⎠

⎞⎜⎜⎝

⎛=Σ

17.07.01

, (4.1a)

or, equivalently,

( ) ( ) ( )⎥⎦⎤

⎢⎣

⎡++−

−= 22

2

4.121exp

27.01

, yxyxyxpπ

, (4.1b)

where 5,5 ≤≤− yx .

Figure 4.1: Function p

Then the total volume under the curve ( )yxp , in Figure 4.1 is given by the integral

( ) ( ) 9993.04.1

21exp

27.015

5

5

5

222

=⎥⎦

⎤⎢⎣

⎡++−

−∫ ∫− −

dydxyxyxπ

. (4.2)

We may evaluate the integral (4.2) through brute force. We partition the yx, -‐plane in little squares

with area kj dydx , 20,,1…=j , 20,,1…=k , then define the centre of these areas as ( )kj yx ~,~ , and

compute the strips of volume jkV as

( ) kjkjjk dydxyxpV ~,~= . (4.3)


29

In Figure 4.2 we give all the volume elements jkV together:

Figure 4.2: Volume elements of function p

The total volume under the curve ( )yxp , may be approximated as

9994.020

1

20

1

==∑∑= =j k

jkVvolume . (4.4)

Now, we may map these 3-‐dimensional volume elements jkV to corresponding 2-‐dimensional area

elements iA . This is easily done by introducing the following notation

kji dydxdw = , ( )[ ] ( )kji yxpyxp ~,~~,~ = , (4.5)

where index i is a function of the indices j and k :

( ) kji +−≡ 201 (4.6)

and 400,,1…=i . Using (4.5), we may rewrite (4.3) as

( )[ ] iii dwyxpA ~,~= . (4.7)

In Figure 4.3 we give all the 400 are elements iA together:

Figure 4.3: Area elements of function p


30

Since (4.7) is equivalent to (4.3), we have that the mapping of the 3-‐dimensional volume elements

jkV to their corresponding 2-‐dimensional area elements iA has not led to any loss of information;

that is,

volumeVAareaj k

jki

i === ∑∑∑= ==

20

1

20

1

400

1

. (4.9)

We now may, trivially, rearrange the elements iA in Figure 3 in descending order, so we obtain

Figure 4.4.

Figure 4.4: Ordered area elements of function f

Note that the horizontal axis of Figure 4.4 is non-‐dimensional. This is because we are looking at an collection of rectangular area elements ordered in one of many possible configurations.

Now all these rectangular elements have a base of 25.0== dydxdw , being that there are 400 area

elements we might view Figure 4.4 as a representation of some monotonic descending function ( )wg , where 1000 ≤≤ w .

Figure 4.5: Plot function g


31

What we have accomplished is that we have mapped 3-‐dimensional volume elements, (Figure 4.2), of the bivariate probability distribution p , (Figure 4.1), to 2-‐dimensional area elements, (Figure 4.3),

after which we have rearranged these area elements in descending order, (Figure 4.4), so as to get a monotonic descending ‘function’ g , (Figure 4.5).

We now may integrate the univariate function g and, again, get the volume (4.2) we are looking for.

Moreover, in going from Figure 4.1 to Figure 4.5, all the pertinent probability density information is retained, as every point on Figure 4.5’s w-‐axis corresponds with a ( )yx, -‐coordinate. Note that any k-‐

variate function probability distribution p may be thus reduced to its corresponding monotonic

descending univariate representation g , where it is understood that every point on the univariate

w-‐axis corresponds with some 1×N coordinate x , (Skilling, 2004).

4.2. The Idea Behind the probability Sort Algorithm

We now discuss the basic idea behind the Probability Sort algorithm for the case where we have only two damage states 2=M ; that is, the states damaged and not-‐damaged. The pseudo-‐code for the general case of arbitrary M is given in Appendix A.

For the case where 2=M , the number of possible damage states will be N2 . The Probability Sort algorithm goes from the most likely state 1s , to the next likely state 2s , to the next likely damage

state 3s , and so on, such that

( ) ( )ji pp ss ≥ , for ji < . (4.10)

The selection of the is is done such that there are no rejections in the damage state proposals.

Moreover, the selection itself only takes ( )NO time.

For the specific case 2=M , the state vectors is , for ni 2,,2,1 …= , may be constructed as vector

consisting of 0 and 1’s. The probabilities of an element ks in is being either 0 or 1 is

( )ksp , for .2,1=k (4.11)

So, the probability which is associated with a given is may be computed as

( ) ( )∏=

=N

kki spp

1

s . (4.12)

Now, let

( ) ( )[ ]1,0maxmax === kkk spspp (4.13)

be the maximum possible damage state probability for component k , and let

{ }1,0max ∈ks (4.14)


32

be the damage state which corresponds with this maximum probability. Then the damage state vector with maximum probability,

∏=

=N

kkpP

1

maxmax , (4.15)

is given as

{ }maxmax2

max1

max ,,, nsss …=s . (4.16)

Now, it stands to reason that this ‘Most Likelihood’ damage state vector maxs should be the first and foremost of all the possible damage state scenarios of which the consequences should be evaluated; that is,

max1 ss = , (4.17)

where, by construction,

( ) ( ) maxmax1 Ppp == ss . (4.18)

If we follow this line of reasoning, then the second best damage state proposal would be that damage state vector which has the second highest probability.

Now, the minimum possible probability for a damage state for component k is given as (4.13):

( ) ( )[ ]1,0min1 maxmin ===−= kkkk spsppp , (4.19)

with corresponding damage state (4.14):

{ }1,01 maxmin ∈−= kk ss . (4.20)

Let

{ }minmin2

min1

min ,,, Nppp …=p (4.21)

be the vector with the minimum probabilities for the n components. Then the damage state minqs

which corresponds with the maximum of the minimum vector (4.21)

( )minmin max p=qp (4.22)

is the only possible candidate for as state switch (4.20):

{ }maxmax1

minmax1

max2

max12 ,,,,,,, Nqqq ssssss …… +−=s (4.23)

So, the probability which corresponds with this second best proposal is (4.13), (4.15) and (4.22):

( ) maxmax

min

2 Ppp

pq

q=s . (4.24)


33

as the qth state probability has switched from its maximum probability state to its minimum.

Now, the third most probable damage state vector necessarily will also be of the form where only one damage state, say us , is being switched, as we reset qs to its original damage state value in

(4.16):

{ }maxmax1

maxmax1

max1

minmax1

max2

max13 ,,,,,,,,,,, Nqqquuu sssssssss ……… +−+−=s . (4.25)

But as we come to the fourth most probable damage state vector, then we find that we bifurcate into the possibility of either both us and qs being switched,

{ }maxmax1

minmax1

max1

minmax1

max2

max14 ,,,,,,,,,,, Nuuuqqqa sssssssss ……… +−+−=s , (4.26)

or us being reset to its original damage state value in (4.16), as we switch some other element, say

ws :

{ }maxmax1

minmax1

maxmaxmax2

max14 ,,,,,,,,,,, Nwwwuqb ssssssss ………… +−=s . (4.27)

In Appendix A the Probability Sort switching algorithm is given which produces scenario proposals ordered by their probabilities.

4.3. A Probability Sort Analysis

Say, we have 121=N objects arranged in a 11-‐by-‐11 grid with a distance of 50 meters between

horizontally and vertically adjacent objects and a distance of 22 505071.70 += meters between diagonally adjacent objects. We then let the centre object with coordinates, say, ( )300,300 be in

flames, which generates a radiation of, say, 500 which falls of as the inverse of the distance, generating a location dependent overpressure R of

( )( ) ( )22 300300

500,−+−

=yx

yxR , (4.28)

and a corresponding (probit) probability of being damaged (i.e. 2=M ) of

( )( ) ( ) ,

2,7534.41

21

2exp

21,

2,7534.4

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡ +−+=⎟⎟

⎠

⎞⎜⎜⎝

⎛−= ∫

+−

∞−

yxRerfduuyxPyxO

π (4.29)

which for a radiation of ( ) 0, 00 =yxR will give a corresponding base-‐line damage probability of

( ) 600 10, −=yxP . Then we may obtain the following damage probability map for our 121=N

objects.


34

Table 4.1: Damage probability map

0.0004 0.0007 0.0012 0.0019 0.0026 0.0029 0.0026 0.0019 0.0012 0.0007 0.0004 0.0007 0.0014 0.0029 0.0059 0.0100 0.0121 0.0100 0.0059 0.0029 0.0014 0.0007 0.0012 0.0029 0.0083 0.0239 0.0558 0.0778 0.0558 0.0239 0.0083 0.0029 0.0012 0.0019 0.0059 0.0239 0.1116 0.3892 0.5974 0.3892 0.1116 0.0239 0.0059 0.0019 0.0026 0.0100 0.0558 0.3892 0.9898 1.0000 0.9898 0.3892 0.0558 0.0100 0.0026 0.0029 0.0121 0.0778 0.5974 1.0000 1 1.0000 0.5974 0.0778 0.0121 0.0029 0.0026 0.0100 0.0558 0.3892 0.9898 1.0000 0.9898 0.3892 0.0558 0.0100 0.0026 0.0019 0.0059 0.0239 0.1116 0.3892 0.5974 0.3892 0.1116 0.0239 0.0059 0.0019 0.0012 0.0029 0.0083 0.0239 0.0558 0.0778 0.0558 0.0239 0.0083 0.0029 0.0012 0.0007 0.0014 0.0029 0.0059 0.0100 0.0121 0.0100 0.0059 0.0029 0.0014 0.0007 0.0004 0.0007 0.0012 0.0019 0.0026 0.0029 0.0026 0.0019 0.0012 0.0007 0.0004

The state space which corresponds with this damage probability map is

361201121 1033.122 ×==− , (4.30)

as the number of states is 2=M , the number of objects is 121=N , and as the centre element is known to be in a damaged (i.e. failed) state. We now proceed to do a Probability Sort analysis for the raw damage probability map in Table 4.1.

The Probability Sort algorithm gives a list with the most probable damage states, starting with most probable damage state. The maximum probability of any of these damage state vectors is

max1 00041.0 PP == . (4.31)

From all the 361033.1 × possible damage states, (4.30), we take the 6102× most likely damage states, sorted from high to low probabilities. This probability sort has a probability coverage of

( ) 6997.06102

1

=∑×

=iip s , (4.32)

which for a univariate Normal distribution

( ) ( ) ⎥⎦⎤

⎢⎣

⎡−−= 2

221exp

21,| µ

σσπσµ xxp

would roughly correspond with the total ‘state space’ enclosed in the 1-‐sigma interval ( )σµσµ +− ,, as shown Figure 4.6


35

Figure 4.6: Normal distribution together with 1-‐sigma probability coverage region

since

( ) 6827.0,| =∫−

−

dxxpσµ

σµ

σµ . (4.33)

The probability sort of the damage states allows us to graphically represent the 120-‐variate damage state probability distribution on a two dimensional plane (see also Figure 4.5), Figure 4.7.

Figure 4.7: Probability sorted damage states

Note that in Figure 4.7 the probabilities as a function of the probability sort order i number fall off so quickly that its graph is basically a composition of a vertical and a horizontal line which, respectively, hug the y-‐ and x-‐axis. In order to obtain to get a better sense of the probability contours we may take

-6 -4 -2 2 4 6

0.1

0.2

0.3

0.4


36

the base 10 log of the probabilities, where (per MATLAB’s scientific notation) =× ba 10 ae+b, Figure 4.8.

Figure 4.8: Base 10 log-‐probability sorted damage states (raw probability map)

Because of the monotonic descending character of the probability sorted damage state proposals is ,

we have that for general c and C

( ) cp i <s , as Ci ≥ . (4.34)

So from Figure 4.8 we may get a sense for the probability contours, or, equivalently, the associated c and C in (4.34):

Table 4.2: Probability contours

( ) cp i <s Ci ≥ 410−=c 244=C 510−=c 5874=C 610−=c 92691=C 710−=c 855808=C

We also may plot the probability coverage (4.32) as a function of the probability sort order i , Figure 4.9.


37

Figure 4.9: Probability coverage as function of damage state order i

Moreover, this figure gives us a sense of the probability coverage contours:

( ) PpN

ii =∑

=1

s . (4.35)

In Table 4.3 we give the probability coverage P which is associated with the first N~ probability sorted damage state vectors.

Table 4.3: Probability coverage contours

( ) PpN

ii =∑

=

~

1

s

1.0=P 1290~=N

2.0=P 6333~=N

3.0=P 23623~=N

4.0=P 68597~=N

5.0=P 206435~=N

6.0=P 611092~=N

7.0=P 610002.~×≈N

… … 0.1→P 361033.1~

×→N


38

It may be seen from Table 4.3 that the active probability components of the damage state probability distribution ( )ip s which corresponds with the probability map in Table 4.1 are located in

an exponential small region of the total state space of 361033.1 ×=N ; that is, by a margin of a one followed by 30 zeros.

The first seven most likely damage states are given below. The most probable damage state is where all probabilities in Table 4.1 greater than 0.5 are set to 1 and all probabilities smaller are set to 0, Table 4.4.

Table 4.4: Most Likelihood (ML) damage state (P = 0.00041)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The four next most probable states (because of the probability symmetry present in Table 1), are the ones where the probability which is closest to 0.5 is switched from either 0 to 1 or, as is actually the case, from 1 to 0 (boldface underlined), Tables 4.5 through 4.8.

Table 4.5: Second through fifth best ML damage state (P = 0.00028)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


39


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


40

The four then next most probable states (again, because of the probability symmetry present in Table 4.1), are the ones where the probability which are then closest to 0.5 is switched from either, as is actually the case, 0 to 1 or from 1 to 0. In Table 4.9 we give underlined and boldface the switched state, whereas the underlining without a boldface signifies the symmetrical switching locations.

Table 4.9: Sixth through ninth best ML damage state (P = 0.00026)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4.4. The Information Entropy of a System

Real life systems of infrastructural objects which are under influence of extreme weather events can be conceptualized as an event tree system, where all the specific damage states of the infrastructural system are leaves on the event tree. If we have n infrastructural objects and M

damage states per object, then the event tree system will consist of nMN = distinct states (3.4).

Now, as the number of infrastructural objects n grows, the number of distinct states N of the corresponding event tree will grow exponentially. So, for non-‐trivial infrastructural systems the size of the corresponding state space will quickly become overwhelmingly large and, as a consequence, seemingly, make futile any attempt to come at some sort of evaluation of the system. The Probability Sort algorithm, however, allows one to come to an approximate evaluation of even exponentially large event tree systems, as long as that systems entropy is low enough.

For example, an entropy of zero is achieved when it is known with certainty for each of the n components in which state they will be in. In this case there will be only one possible state, which gives a Shannon information entropy H of (Shannon, 1946)

01log0log01loglog1

=−=⋅−−=−= ∑∑≠= ki

N

iii ppH , (4.36)

as we have 1 damage state with a probability of one and ( )1−N damage states with a probability of

zero. In contrast, a maximum information entropy is achieved when for each of the n components the probability of being in one of the damage states is M1 . For the maximum information entropy

case all damage states are equally probable, all of those damage states having a probability of


41

( ) nMNp 11state damage == ,

which gives a Shannon information entropy of

MnMMNN

HnM

inn

N

i

log1log11log111

=−=−= ∑∑==

. (4.37)

Now, the closer the information entropy of the system under consideration is to zero, (4.36), the more comprehensive will be the probability coverage of the Probability Sort analysis. Also, the larger

the number of distinct states nM , the closer one will need the system to be to (4.36), in order to obtain the same probability coverage with the same amount of evaluations.

The take home point here is that when assessing the computational feasibility of an exact evaluation of some event tree system, one need not only take into account the number of components n and the number of damage states M , but also the information entropy H which is present in the system.

There were the information entropy H is too great the Probability Sort algorithm will tend to a Most Likelihood (ML) algorithm, where a pre-‐specified number of most probable damage states is identified and produced, together with corresponding probability coverage. So instead of just the one modus, or, equivalently, the most probable damage state, as is customary produced in ML algorithms, the Probability Sort algorithm gives the probability ridge surrounding the solitary ML probability peak.


42

5. Modelling of Homogeneous Markov Processes with the Probability Sort Algorithm

In this chapter we will discuss the modelling of homogeneous Markov process by way of the probability sort algorithm, using an infrastructural system of objects example in which an initial failure cascades through the spatial field at consecutive time steps. In the modelling of homogenous Markov processes the transition probabilities remain the same. Stated differently, in the here discussed infrastructural system of objects example, we will not take into account possible time-‐dependent burn-‐out effects; i.e., the drop in radiation of burning objects due to the decrease in capacity per object over time, or human intervention measures to reduce failure probabilities of infrastructural components. The modelling of inhomogeneous time-‐dependent Markov processes by way of the probability sort algorithm, will be discussed in the following chapter.

5.1. The Physics of the Probability Map

For our system of objects it is assumed that the failure of a given object generates a radiation of, say 200 , which falls of, say, as the inverse of the distance. Moreover, it is assumed that the total radiation for multiple failures is a superposition of the radiation of the separate failures.

For example, if we have n failed objects, having coordinates ( )ii yx , , for ni ,,1…= . Then the total

radiation R which is experienced by an intact object having coordinates ( )yx, is given as

( )( ) ( )

∑= −+−

=n

iii yyxx

yxR1

22

200, . (5.1)

The corresponding (probit) probability of being damaged is given as

( ) ( ) ,2

,7534.4121, ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡ +−+=

yxRerfyxP (5.2)

which for a radiation of ( ) 0, 00 =yxR will give a corresponding base-‐line damage probability of

( ) 600 10, −=yxP .

5.2. Some Example Probability Maps

With the probability map (3.2), the probability sort algorithm can be invoked, as explained in (3.23) through (3.28), in order to model the cascade of failure through the system of objects as time progresses.

Say we have 25=N objects arranged in a 5-‐by-‐5 grid with, say, a distance of 50 meters between horizontally and vertically adjacent objects and a distance of

22 505071.70 +=


43

meters between diagonally adjacent objects. If we let the objects with coordinates ( )150,150 and

( )100,150 fail, then we obtain the state matrix in Table 5.1.

Table 5.1: State matrix 1

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0

The corresponding probability map may be constructed by way of (7.1) and (7.2), Table 5.2.

Table 5.2: Failure probability map corresponding with state matrix 2

0.0129 0.0446 0.0778 0.0446 0.0129 0.0605 0.4459 0.8937 0.4459 0.0605 0.1674 0.9810 1 0.9810 0.1674 0.1674 0.9810 1 0.9810 0.1674 0.0605 0.4459 0.8937 0.4459 0.0605

Alternatively, if we let the objects with coordinates ( )150,150 , ( )200,150 , ( )150,100 fail, then we

obtain the state matrix in Table 5.3.


0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0


Table 5.4: Failure probability map corresponding with state matrix 1

0.5943 0.9688 0.9988 0.8994 0.3296 0.9688 1.0000 1 0.9999 0.6180 0.9988 1 1 1.0000 0.6438 0.8994 0.9999 1.000 0.9508 0.3877 0.3296 0.6180 0.6438 0.3877 0.1313

It may be glanced from Tables 11 and 13 that the superposition of radiation in (5.1) in all likelihood will lead to a cascade of failures.


44

5.3. A Probability Sort Analysis of Cascading Effects

Say we have 25=N objects arranged in a 5-‐by-‐5 grid with, say, a distance of 50 meters between horizontally and vertically adjacent objects. The primary initiating event, at time step 0=t , is the event where the centre object with coordinates ( )150,150 has failed, Table 5.5.

Table 5.5: State matrix corresponding with the primary event at t = 1

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0


Table 5.6: Failure probability map for the primary event at t = 1

0.0004 0.0015 0.0029 0.0015 0.0004 0.0015 0.0271 0.2256 0.0271 0.0015 0.0029 0.2256 1 0.2256 0.0029 0.0015 0.0271 0.2256 0.0271 0.0015 0.0004 0.0015 0.0029 0.0015 0.0004

The number of damage states is 2=M , the number of objects is 25=N , and the number of elements in a damaged (i.e. failed) state is 1=K . So the total state space which corresponds with the failure probability map in Table 5.6 is

724125 1068.122 ×=== −−KNM . (5.3)

It follows that following the primary event in Table 5.5, we will have 242 possible event scenarios at each time step. Among these higher order event scenarios are the scenarios in Tables 5.1, 5.3, and 5.5, with corresponding probability maps Tables 5.2, 5.4, and 5.6.

As we have an irreversible process (i.e., failed objects cannot ‘unfail’), the total number of scenario

routes at a given time step may be modelled by way of a 242 -‐by-‐ 242 Markovian transition matrix, having

142424 1081.222 ×=× (5.4)

elements. Now, a state matrix with K failures will map to possible K−252 end points. So, our

hypothetical 242 -‐by-‐ 242 Markovian transition matrix has

( )

1124

1

2424 1082.22!24!

!242 ×=−

+∑=

−

i

i

ii (5.5)


45

non-‐zero probability elements. In other words, at a given time step 0>t there are 111082.2 ×

admissible routes in which we go from one of the 71068.1 × possible starting scenarios to some

admissible subset of the total scenario space, with subsets ranging from 71068.1 × scenarios to 1 scenario.

This overwhelming number of admissible routes (5.5) notwithstanding, it is found that the Probability Sort algorithm will give very decent probability coverages over the time steps for given probability cut-‐offs for the primary event in Table 5.5, with a probability map ‘physics’ of (5.1) and (5.2), Table 5.6. In Table 5.7 these probability coverages are given together with the number of active probability components at each time step.

Table 5.7: Probability coverages and number of active probability components for model (5.1) and (5.2)

Time Step Cut-‐off = 10-‐6 Cut-‐off = 10-‐7 coverage # components coverage # components

1 0.9995 1094 0.9999 2459 2 0.9177 33100 0.9754 111430 3 0.8745 16104 0.9608 61476 4 0.8529 7069 0.9527 32864 5 0.8426 2417 0.9484 15976 6 0.8382 651 0.9463 7045 7 0.8365 126 0.9452 2373

The probability cut-‐offs in Table 5.7 are enforced such that the probability for a given scenario, (4.3), at a given time step is not smaller than that cut-‐off.

It may be glanced from the time progression of the number of active probability components in Table 5.7 that the primary event in Table 5.5, together with (5.1) and (5.2), will lead us from an initial low-‐entropic state, to an intermediate higher-‐entropic state, back to a final low entropic state. This may be explained as follows. Initially, we only expect the objects which are horizontally and vertically adjacent to the failed object to reach a failed state, Table 5.6. Because of the superposition of radiation we expect (see Tables 5.2 and 5.4) the objects to cascade as time progresses to a total conflagration state. But we are uncertain as to the route that will take us from the initial low entropic state to this final low entropic state. This uncertainty translates to an intermediate higher entropic state where the probabilities are more spread out over the total state space and, consequently, more active probability components are in play.

5.3.1. Time Evolving Marginal Damage State Probabilities

If, for the cut-‐off of 10-‐7, we weigh the damage state ‘matrices’ is by the normalized probabilities

∑ =

=N

i i

ii

P

PP ~

1

~ , (5.6)


46

where iP is the probability of is and N~ is the total number of active probability components, or,

equivalently, probability sort scenario proposals, then we obtain the following expected marginal probabilities, say, ( )θE , where

( ) ∑=

=N

iiiPE

~

1

~ sθ , (5.7)

which corresponds with marginal probability of being in a failed state, Tables 5.8-‐5.14, as we have chosen our event labels (i.e., Tables 5.1, 5.3, and 5.5) such that being in a damage state corresponds with a Bernoulli event.

Table 5.8: Estimated probability map for the primary event at t = 1 (compare with analytical Table 5.6)

0.0004 0.0015 0.0029 0.0015 0.0004 0.0015 0.0271 0.2256 0.0271 0.0015 0.0029 0.2256 1.000 0.2256 0.0029 0.0015 0.0271 0.2256 0.0271 0.0015 0.0004 0.0015 0.0029 0.0015 0.0004

Table 5.9: Estimated probability map for the primary event at t = 2

0.1426 0.2902 0.3703 0.2902 0.1426 0.2902 0.5637 0.7353 0.5637 0.2902 0.3703 0.7353 1.000 0.7353 0.3703 0.2902 0.5637 0.7353 0.5637 0.2902 0.1426 0.2902 0.3703 0.2902 0.1426


0.7318 0.7776 0.8024 0.7776 0.7318 0.7776 0.8624 0.9161 0.8624 0.7776 0.8024 0.9161 1.000 0.9161 0.8024 0.7776 0.8624 0.9161 0.8624 0.7776 0.7318 0.7776 0.8024 0.7776 0.7318


0.9178 0.9315 0.9390 0.9315 0.9178 0.9315 0.9571 0.9736 0.9571 0.9315 0.9390 0.9736 1.000 0.9736 0.9390 0.9315 0.9571 0.9736 0.9571 0.9315 0.9178 0.9315 0.9390 0.9315 0.9178


47


0.9754 0.9794 0.9815 0.9794 0.9754 0.9794 0.9868 0.9918 0.9868 0.9794 0.9815 0.9918 1.000 0.9918 0.9815 0.9794 0.9868 0.9918 0.9868 0.9794 0.9754 0.9794 0.9815 0.9794 0.9754


0.9929 0.9939 0.9945 0.9939 0.9929 0.9939 0.9960 0.9974 0.9960 0.9939 0.9945 0.9974 1.000 0.9974 0.9945 0.9939 0.9960 0.9974 0.9960 0.9939 0.9929 0.9939 0.9945 0.9939 0.9929


0.9981 0.9983 0.9984 0.9983 0.9981 0.9983 0.9988 0.9992 0.9988 0.9983 0.9984 0.9992 1.000 0.9992 0.9984 0.9983 0.9988 0.9992 0.9988 0.9983 0.9981 0.9983 0.9984 0.9983 0.9981

It may be glanced from Tables 5.8 through 5.14, that the marginal probabilities of being in a failed state will increase in magnitude as time progresses. Also note that the estimated marginal probabilities of being in a failed stated at time step 1, Table 5.8, are the same as the analytical probability map in Table 5.6, which was obtained by way of the primary event in Table 5.5 and the probability map model (5.1) and (5.2).

5.3.2. Time Evolving ML Damage States

We now will focus on the change in probabilities of four representative fixed damage state scenarios, Tables 5.15 through 5.18.


0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

In Table 5.15 we have the total containment scenario, where no additional objects fail.


48


0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0


0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

In Tables 5.16 and 5.17 we have limited spill-‐off scenarios, where, respectively, one and two additional objects have failed.


1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

In Table 5.18 we have the total destruction scenario, where all the objects have failed. We now take a look at the progression of the probabilities of these damage states as time progresses, where we put the Most Likelihood (ML) probabilities in boldface, Table 5.18.

Table 5.19: Probabilities of the state matrices in Tables 5.15-‐5.18

Time Step P(State matrix 1) P(State matrix 2) P(State matrix 3) P(State matrix 4) 1 0.3140 0.0915 0.0267 9.17 5610−× 2 0.0986 0.0287 0.0084 0.0282 3 0.0310 0.0090 0.0026 0.6703 4 0.0097 0.0028 0.0008 0.8650 5 0.0031 0.0009 0.0003 0.9226 6 0.0010 0.0003 8.14 510−× 0.9390

7 0.0003 8.78 510−× 2.56 510−× 0.9433

At both time steps 1 and 2 the total containment scenario is the ML scenario. From time step 3 onwards, the total destruction scenario becomes the ML scenario. At time step 1 there is still a considerable likelihood that there is either full containment or limited spill-‐off:


49

( ) ( ) 8402.00267.024

0915.014

3140.0 =⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+ . (5.8)

At time step 2 this likelihood has dropped off dramatically:

( ) ( ) 2638.00084.024

0287.014

0986.0 =⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+ . (5.9)

At time step 3 the likelihood of either full containment or limited spill-‐off has dwindled to a mere

( ) ( ) 0826.00026.024

0090.014

0310.0 =⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+ , (5.10)

while the probability of the total destruction scenario is a hefty 0.6703, and as time progresses this probability approaches certainty. Especially so, if we take into account that total probability coverage has not been achieved; compare the right hand probability coverages in Table 5.7 with the of State Matrix 4 probabilities in Table 5.19.


50

6. Modelling of Inhomogeneous Markov Processes with the Probability Sort Algorithm

In this chapter we will discuss the modelling of inhomogeneous Markov process by way of the probability sort algorithm, using a landslide example in which an initial landslide (triggered by an extreme weather event) cascades downstream at consecutive time steps. In the modelling of this inhomogeneous Markov process, the transition probabilities change as a function of time. So, when a landslide occurs somewhere, then there will be an initial increase in the probability of a knock-‐on landslide occurring in the areas below this area. But is assumed that this danger will quickly dissipate over time, if the knock-‐on effect fails to materialize, as it is assumed that landslides do not keep on gushing. The time dependency being caused by physical processes and/or by human intervention measures in the system.

6.1. A Topology and Landslide Physics

It is assumed that the landslide area consists of several connected sub-‐areas. Landslides are assumed to cascade in the direction of the arrows, from upstream to downstream, Figure 6.1.

Figure 6.1: Topology of landslide area

It is assumed that there is a general landslide probability of 01.0=P , due to exposure to intensive rainfall. Also, vertical connections represent an initial probability of a knock-‐on landslide of 9.0=P , as the counter for the initiating landslide area is initially set to 1=k . Diagonal connections represent an initial probability of a knock-‐on landslide of 7.0=P , as the counter for the initiating landslide area is initially set to 1=k . Sideways connections represent an initial probability of a knock-‐on landslide of 7.0=P , as the counter for the initiating landslide area is initially set to 1=k . In order to compute the probability of a landslide event, the probability of no event is first computed of which


51

the complement is then computed, and as the initiating landslides remain in their landslide statuses, the corresponding counters ik in the corresponding inhomogeneous probability map ( ) ( )( )11 , −− tt ksf ,

see Section 3.2.2, are incremented over each time step. The corresponding probabilities P will drop off as a power ik of the ‘”half-‐life” parameters of 1.0 in Figure 6.2.

Figure 6.2: Landslide physics of the probability map

6.2. A First Cascading Effect Analysis

In Figure 6.3, we give an initiating landslide event. In Figures 6.4 through 6.8 we give for subsequent time steps the available cascade pattern, and in Tables 6.1 through 6.5 we give the corresponding marginal probabilities of a landslide occurring for a cut-‐off criterion of 10-‐7, (3.19).


52

Figure 6.3: Initiating landslide event

Figure 6.4: Time step t = 1


1.000 0.0100 0.0100 0.0100 0.9001 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100



1.000 0.0196 0.0196 0.0196 0.9102 0.0284 0.0331 0.0283 0.8134 0.6407 0.0284 0.0283 0.0284 0.0350 0.0331 0.0351 0.0284 0.0283 0.0351 0.0284


53



1.000 0.0284 0.0284 0.0284 0.9122 0.0455 0.0585 0.0451 0.8323 0.6802 0.0567 0.0526 0.7403 0.7902 0.0691 0.0712 0.0529 0.0586 0.0763 0.0585



1.000 0.0362 0.0360 0.0356 0.9135 0.0607 0.0804 0.0589 0.8368 0.6970 0.0852 0.0734 0.7666 0.8224 0.4434 0.1141 0.6807 0.7239 0.1306 0.0965



1.000 0.0430 0.0426 0.0419 0.9150 0.0743 0.0998 0.0713 0.8403 0.7105 0.1111 0.0922 0.7741 0.8336 0.4928 0.1535 0.7128 0.7609 0.4617 0.1412

In Table 6.6 we give the probability coverages over the time steps for cut-‐off criteria of 10-‐7 and 10-‐8

for the primary event in Figure 6.3, with a probability map ‘physics’ as shown in Figure 6.2, where the probability coverages are given together with the number of active probability components at each time step.


54

Table 6.6: Probability coverages and number of active probability components for Figures 6.2 and 6.3


1 0.9999 1160 1.0000 1976 2 0.9984 16590 0.9996 48175 3 0.9920 62208 0.9973 246954 4 0.9780 158551 0.9913 727034 5 0.9591 256441 0.9820 1309611

6.3. A Second Cascading Effect Analysis

In Figure 6.9, we give two simultaneously occurring initiating landslide events. In Figures 6.10 through 6.14 we give for subsequent time steps the available cascade pattern, and in Tables 6.7 through 6.11 we give the corresponding marginal probabilities of a landslide occurring for a cut-‐off criterion of 10-‐7, (3.19).

Figure 6.9: Initiating landslide events

In the Figures 6.10 through 6.14, we let the overlap between the cascade patterns between the two respective initiating events be coloured purple. If we compare the marginal probabilities in these regions of overlap in the Tables 6.7 through 6.11 with the corresponding regions in the Tables 6.1 through 6.5, then it may be seen that landslides may “conspire” together in their “attacks”, as the probabilities of a landslide occurrence for these regions are significantly larger under the scenario in Figure 6.9 than under the scenario in Figure 6.1.


55



1.000 0.0100 0.0100 0.0100 0.9000 0.0100 0.5000 1.000 0.0100 0.0100 0.0100 0.9000 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100



1.000 0.0195 0.0196 0.0196 0.9101 0.0283 0.5336 1.000 0.8133 0.7647 0.4605 0.9102 0.0282 0.0349 0.0330 0.8146 0.0283 0.0282 0.0349 0.0282



1.000 0.0279 0.0280 0.0280 0.9124 0.0448 0.5437 1.000 0.8326 0.7985 0.4990 0.9125 0.7407 0.8488 0.4409 0.8866 0.0521 0.0577 0.5930 0.7418



1.000 0.0350 0.0351 0.0353 0.9139 0.0591 0.5510 1.000 0.8374 0.8087 0.5132 0.9147 0.7672 0.8757 0.6855 0.8962 0.6816 0.7754 0.7749 0.8149


56



1.000 0.0422 0.0422 0.0427 0.9156 0.0732 0.5586 1.000 0.8410 0.8165 0.5249 0.9166 0.7748 0.8834 0.7133 0.9007 0.7135 0.8079 0.8642 0.8266

In Table 6.12 we give the probability coverages for cut-‐off criteria of 10-‐7 and 10-‐8 over the time steps for the primary event in Figure 6.3, where the probability coverages are given together with the number of active probability components at each time step.

Table 6.12: Probability coverages and number of active probability components for Figures 6.2 and 6.3


1 0.9999 1878 1.0000 3698 2 0.9980 24816 0.9995 74844 3 0.9898 97527 0.9967 359967 4 0.9747 199711 0.9905 837620 5 0.9602 274479 0.9834 1264532

6.4. Comparing Inhomogeneous and Homogenous Markov Assumptions

If we assume homogeneous probabilities that do not change over the time steps, Figure 6.15, then it is found that there will be a severe overestimation of the landslide probabilities, Tables 6.13 and 6.14, relative to the Tables 6.5 and 6.7.

Figure 6.15: Homogenous, time-‐independent landslide physics of the probability map


57

Table 6.13: Estimated probability map at t = 5, physics as in Figure 6.15

1.000 0.0423 0.0421 0.0405 1.0000 0.0753 0.1127 0.0708 0.9997 0.9917 0.1215 0.0934 0.9934 0.9967 0.7229 0.1669 0.9317 0.9500 0.4794 0.1504

Table 6.14: Estimated probability map at t = 5, physics as in Figure 6.15

1.000 0.0408 0.0418 0.0411 1.0000 0.0729 0.9713 1.000 0.9998 0.9994 0.9390 1.0000 0.9941 0.9994 0.9584 1.0000 0.9332 0.9702 0.9926 0.9963

Note that the Tables 6.13 and 6.14 correspond with probability cut-‐offs of 10-‐7 and, respective, probability coverages of 0.9521 and 0.9506.

6.5. A Third Cascading Effect Analysis

We now will do a cascading effect analysis on the Malborghetto electricity network (provided by RAIN partner AIA and reported in WP4), in case of voltage instabilities in the (grey) units X3 and X4, Figure 6.16. It is assumed that there is a general probability of voltage instability of

( ) 510−=yinstabilitp , (6.1)

a probability of voltage instability conditional on there being instability in a neighbouring blue node of

( ) 73.0| =blueyinstabilitp , (6.2)

and a probability of voltage instability conditional on there being instability in a neighbouring orange node of

( ) 89.0| =orangeyinstabilitp , (6.3)

where it is assumed that these voltage instabilities peak and are only relevant for the time period in which they occur; i.e., there is assumed to be an extreme dissipation effect:

610−=halflifeP . (6.4)


58

Figure 6.16: Malborghetto electricity network

By way of the modelling assumptions (6.1) through (6.4) and the topology in Figure 6.16, the probability of X6 failing at the first time step when X3 and X4 have failed at time step zero, is given as

( )( ) ( ) ( )[ ]21175351t 73.0101011011X5X4,X3,,X2,X1|X6 −−−−= −−−−=p , (6.5)

whereas at the second time step this probability, as the voltage instabilities in both X3 and X4 are assumed to have dissipated, is given as

( )( ) ( ) ( )[ ]

( ) .1011

73.0101011011X5X4,X3,,X2,X1|X6

55

21275352t

−

−−−−=

−−≈

−−−−=p (6.6)

In Table 6.15 we give the estimated marginal probabilities of shut down due to a cascade of the voltage instabilities for a cut-‐off criterion of 10-‐7.

Note that for element X15 we have a jump in the probability of a shut down as we go from time step 5 to time step 6. And it can be seen in Figure 6.16 that element X15 can be reached by the indirect voltage instability route X14-‐X19-‐X18, even if element X15 initially escapes in time step 4 a direct voltage instability knock-‐on by element X14. Likewise, element X20 may be approached by the indirect route X14-‐X15-‐X18-‐X19, rather than the direct route X14-‐X19, which explains the slight jump in shut down probability at time step 7 for this element.


59

Table 6.15: Marginal probabilities of shutdown due to voltage instability

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

X1 0.0000 0.8251 0.8252 0.8252 0.8252 0.8252 0.8252

X2 0.0000 0.8251 0.8252 0.8252 0.8252 0.8252 0.8252

X3 1.000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

X4 1.000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

X5 0.0000 0.8251 0.8252 0.8252 0.8252 0.8252 0.8252

X6 0.9271 0.9271 0.9271 0.9271 0.9271 0.9271 0.9271 X7 0.0000 0.8252 0.8252 0.8252 0.8251 0.8251 0.8251

X8 0.0000 0.0001 0.7344 0.7345 0.7344 0.7344 0.7344

X9 0.0000 0.0001 0.0002 0.5362 0.5362 0.5361 0.5361

X10 0.0000 0.0001 0.0001 0.0002 0.3915 0.3914 0.3914

X11 0.0000 0.0001 0.0001 0.0001 0.0002 0.2858 0.2858

X12 0.0000 0.0001 0.0001 0.0001 0.0002 0.2858 0.2858

X13 0.0000 0.0001 0.0001 0.0002 0.3915 0.3914 0.3914

X14 0.0000 0.0001 0.7345 0.7345 0.7344 0.7344 0.7344 X15 0.0000 0.0001 0.0002 0.6538 0.6537 0.7003 0.7003

X16 0.0000 0.0001 0.0001 0.0002 0.4773 0.4772 0.5113

X17 0.0000 0.0001 0.0001 0.0002 0.0002 0.4972 0.4972

X18 0.0000 0.0001 0.0002 0.0003 0.6810 0.6809 0.6809

X19 0.0000 0.0001 0.0002 0.6538 0.6537 0.6919 0.6919

X20 0.0000 0.0001 0.0002 0.0003 0.5819 0.5818 0.6159

X21 0.0000 0.0001 0.0001 0.0002 0.0002 0.5179 0.5685

X22 0.0000 0.0001 0.0002 0.0003 0.0003 0.5179 0.5594 X23 0.0000 0.0001 0.0002 0.0003 0.0003 0.0002 0.4610

X24 0.0000 0.0001 0.0002 0.0002 0.0002 0.0002 0.0002

X25 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 X26 0.0000 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001

X27 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 X28 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

X29 0.0000 0.0001 0.0002 0.0003 0.0002 0.0002 0.0002

X30 0.0000 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001

X31 0.0000 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001

X32 0.0000 0.0001 0.0002 0.0002 0.0002 0.0001 0.0002

X33 0.0000 0.0001 0.0002 0.0002 0.0002 0.0001 0.0001

X34 0.0000 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001


60

In Table 6.16 we give the probability coverages for cut-‐off criteria of 10-‐7 and 10-‐8 over the time steps for the primary event in Figure 6.16, where the probability coverages are given together with the number of active probability components at each time step.

Table 6.16: Probability coverages and number of active probability components for Figure 6.16


1 1.0000 64 1.0000 64 2 1.0000 1061 1.0000 1919 3 0.9998 2577 1.0000 7500 4 0.9995 4858 0.9999 20186 5 0.9987 7691 0.9995 43001 6 0.9980 13929 0.9989 70372 7 0.9975 19855 0.9985 89521


61

7. Concluding remarks

Work package 5 of the RAIN project has developed a risk-‐based decision framework for large-‐scale infrastructural networks under influence of extreme weather hazards which is able to take into account the (collateral) impacts of cascading effects. In deliverables 5.1 and 5.5 it had been shown that WP5’s framework can serve as a template to integrate and harmonize all content-‐owning WPs within the project (as well as other large multi-‐disciplinary EU projects), as WPs will be forced to produce (conditional) probability distributions that connect in a meaningful way, based on the Bayesian paradigm. For example, severe rainfall impacts the structural integrity of pylons. So, structural engineer specifies to rainfall expert which rainfall levels are relevant for pylons. Rainfall expert then produces a tailor made rainfall probability distribution for the structural engineer, which the structural engineer can connect with his pylon structural integrity probability distribution in a meaningful manner. Stated differently, the (conditional) probability distributions are the inference modules that capture the expertise of the content-‐owning WPs. The same is true for mitigating measures and crisis response mechanisms, such as ICPR arrangements (Hellenberg et al, 2017). Crisis response experts specify the effectivity of response measures, in terms of a reduction in negative consequences. The capturing of such expertise is subsequently fed into the Bayesian framework. Due to their high impact low probability nature, cascading/domino effect hazards have started to be recognized as a priority issue in technical standards and legislation concerned with the control of major accident hazards, such as in the Council Directive 96/82/EC (EU Seveso-‐II Directive) and the European Parliament and Council Directive 2012/18/EU. In D5.2 a Bayesian methodology by which system state probabilities may be estimated for systems that are subjected to cascading/domino effect hazards, has been developed. This methodology makes use of a newly developed Probability Sort algorithm in order to estimate Markov Chains for what otherwise would have been intractable (in)homogeneous transition matrices. Neglecting or underestimating (inter)dependencies between the failures or disruptions of the critical infrastructure components can cause designers, experts, managers and decision makers to underestimate the overall inter-‐infrastructural risks. It is therefore necessary to further develop approaches that consider the interconnected nature of critical infrastructure components and – systems, which was the aim of this report. The generic mathematical framework developed in D5.2 is based on Bayesian probability theory and the Probability Sort algorithm in order to model time-‐dependent, inhomogeneous and cascading effects in infrastructural networks through space and time. This framework also allows for human intervention measures to mitigate risks or reduce failure probabilities of infrastructural components. The proposed Probability Sort approach allows one to come to some sort of exact evaluation of even exponentially large event trees of system states, as long as the information entropy in that event tree is low enough.


62

Applications are presented for the Bayesian modelling of cascading effects of landslides and for the cascading effects in an electricity network. The outcomes are time-‐dependent probability maps of failure of the overall infrastructural systems, which serve as input for the decision-‐making by infrastructural managers.


63

8. References

Ebeling, C.E. (1997). An Introduction to Reliability and Maintainability Engineering. 2nd ed. New Delhi: McGrawHill. Hellenberg, T. (2017). RAIN Deliverable 7.4. Techniques for Mitigation of, Adaptation to, and Coping with the potential impacts of extreme weather on critical infrastructures, with reference to the EU integrated political crisis response -‐ IPCR – arrangements. Huang, C.C. (1977). Non-‐homogeneous Markov chains and their applications. Iowa State University, PhD thesis. Jaynes, E.T. (2003). “Probability Theory; the Logic of Science”, Cambridge University Press. Jensen, F.V, Nielsen, T.D. (2007). Bayesian networks and decision graphs. 2nd ed. New York: Springer. Khakzad, N. (2015). Application of dynamic Bayesian network to risk analysis of domino effects in chemical infrastructures. Reliability Engineering & System Safety 138: 263-‐272. Shannon, C.E. (1948). A Mathematical Theory of Communication, Bell Sys. Tech. J., 27, 379-‐423. Skilling, J. (2004). Nested Sampling, In Bayesian Inference and Maximum Entropy Methods in Science and Engineering, (eds. Erickson G., Rychert J.T., and Smith C.R.) AIP Conference Proceedings, American Institute of Physics, New-‐York. 395-‐405. Van Erp, H.R.N., Linger, R.O. , van Gelder, P.H.A.J.M. (2016). Stress Test Framework for Systems, Nov. 2016, https://www.infrarisk-‐fp7.eu, Deliverable 6.2.


64

9. Appendix: The Probability Sort Algorithm

9.1. Algorithmic Outline

Step 1: Permutate the stateMatrix and the probMatrix into their desired base-line states. Step 2: Define the function undoPermutate(.) which undoes these base-line permutations in the final probability sorted damage state vectors. Step 3: Set up the output list probabilitySort and the intermediate Proposals list. Step 4: enter a While-loop, until the desired cut-off log-probability value crit, has been obtained, or until all possible damage state vectors have been passed through, whichever comes first. Step 5: take that damage state vector entry from the Proposals list which has the

maximum probability, make the components of that entry available within the While-loop, clean up the Proposals list, update the probabilitySort list by way of these components, and update the total probability coverage variable sumProb.

Step 6: replenish the Proposals list which with a maximum of three new proposals. These new proposals guarantee that all the remaining leaves of the event tree of the damage state space may still be explored, and that the next best probability is always in the updated Proposals list.

Step 7: Print the sumProb probability coverage value and terminate the algorithm. The probabilitySort list consisting of the probability sorted damage state vectors and their corresponding probabilities is now available for the user.

9.2. Pseudo-‐Code

INPUT stateMatrix: State matrix/list of the N components under consideration. probMatrix: State probability matrix/list of the N components under consideration. crit: log-probability cut-off criterium OUTPUT


65

probabilitySort: list consisting of damage state proposals ( )sx with corresponding probabilities ( )sP , ordered in descending order by way of the probabilities ( )sP ;

sumProb: the total probability density covered by the probabilities of the damage state vectors in the list probabilitySort ALGORITHM Step 1.a If we have M possible damage states for each of the N possible infrastructural elements, then we may define the N-by-M matrix

stateMatrix =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

M

MM

21

2121

(1)

the corresponding probability matrix which has its rows the damage state pdf of the corresponding infrastructural element may be given as the N-by-M matrix ,

probMatrix =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

NMNN

M

M

θθθ

θθθ

θθθ

21

22221

11211

(2)

We then do a Sort over the rows of probMatrix so that per rows the probabilities are in descending order, from large to small: permutatedProbMatrix =

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

NMNNNMNNNMNN

MMM

MMM

θθθθθθθθθ

θθθθθθθθθ

θθθθθθθθθ

,,,min,,,nextMax,,,max

,,,min,,,nextMax,,,max,,,min,,,nextMax,,,max

212121

222212222122221

112111121111211

………………………

.

(3) where we track the permutations of each of the rows that take us from probMatrix to permutatedProbMatrix. These permutations are then applied to the corresponding rows in stateMatrix. A possible realization of the resulting permutatedStateMatrix may be, say


66

permutatedStateMatrix =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

21

1212

M

MMM

. (4)

The permutatedStateMatrix allows us to keep track of which probabilities in the rows of permutatedProbMatrix point to which damage state. The first column of the permutatedStateMatrix then gives the damage state vector that has the highest probability of occurring, with a probability of

P = 1; For[ i =1, i ≤ N,

P = P × permutatedProbMatrix(i, 1); i++] N.B.: Instead of the specific case of a N-by-M matrix, we alternatively, and more generally, may have a list of length N , say, probList, with in each row of that list a discrete probability distribution over iM damage states, where Ni ≤≤1 , which are given in the corresponding rows of the list, say, stateList. For this more general case we may compute a permutatedProbList and a permutatedStateList. The first column of the permutatedStateList then also will give the damage state vector that has the highest probability of occurring, with a probability of P. Step 1.b then do a row Sort over the entire permutatedProbMatrix such that in its second column the probabilities are arranged in descending order from large to small. This results in, say, for short, the doublePermutatedProbMatrix, where second column of doublePermutatedProbMatrix =

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( ) ( ) ( )[ ] ⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

NMNNMM

NMNNMM

NMNNMM

θθθθθθθθθ

θθθθθθθθθ

θθθθθθθθθ

,,,nextMax,,,,,nextMax,,,,nextMaxmin

,,,nextMax,,,,,nextMax,,,,nextMaxnextMax

,,,nextMax,,,,,nextMax,,,,nextMaxmax

212222111211

212222111211

212222111211

…………

…………

…………

(5)


67

Step 2 Keeping track of the permutations that take us from the permutatedProbMatrix to the doublePermutatedProbMatrix, we may construct the corresponding doublePermutatedStateMatrix. For example, if we have the index vector

[ ]10987654321 (6) of the original row ordering in permutatedProbMatrix, then the corresponding row ordering in both the doublePermutatedProbMatrix and doublePermutatedStateMatrix may be, say, [ ]97310682514 (7) Now let undoPermutate be that function that rearranges the index vector (7) back the original index vector, or, equivalently, (3) and (4) undoPermutate[ doublePermutatedProbMatrix ] = permutatedProbMatrix (8) undoPermutate[ doublePermutatedStateMatrix ] = permutatedStateMatrix. Step 3.a Set the vector stateVector as the first column of the doublePermutatedStateMatrix: stateVector = doublePermutatedStateMatrix(1, :); (9) or, equivalently, depending on the programming language used, stateVector = doublePermutatedStateMatrix(1, All); Likewise, set

P = 1; For[ i =1, i ≤ N,

P = P × doublePermutatedProbMatrix(i,1); (10) i++] Store both the probability ( )1P = P (11) and the unsorted stateVector, see (8),

( )1x = undoPermutate[ stateVector ] (12)


68

in a list ( ) ( ){ }11 , xP and insert that list entry into the list probabilitySort

probabilitySort = ( ) ( ){ }{ }11 , xP . (13) Step 3.b Now the stateVector in (9) gives the damage state vector that has the highest probability of occurring, while being arranged such that that the switching of the first damage state to the damage state of the second entry in the first row of the doublePermutatedStateMatrix will have the next highest probability; that is, stateVector(1) = doublePermutatedStateMatrix(1, 2); (14) has a corresponding next best probability of (10) P = [ P/ doublePermutatedProbMatrix(1,1) ] × doublePermutatedProbMatrix(1,2);

(15) In order to reflect the switch operation (14) we initialize the switchVector as the base-line vector switchVector = zeros(N, 1) ; (16a) which gives

switchVector = [ ]000 … ; (16b)

after which we switch the first entry of this vector from 0 to 1, so as to reflect the switch operation in (14): switchVector(1) = 1 ; (17a) which gives switchVector = [ ]001 … ; . (17b) Also, we set the active switch location as

activeSwitch = 1. (18) Store the adjusted probability (15), the adjusted state vector (14), the switch vector (17b), and the active switch location (18) in a list


69

{ P , stateVector, switchVector, activeSwitch } (19) and insert that list entry into the list Proposals Proposals = { { P , stateVector, switchVector, activeSwitch } }. (20) N.B.: Instead of performing multiplications and divisions on the probabilities in (10) and (15), we also may perform summations and subtractions from the corresponding log-probabilities; this will guard against the potential underflow of the product of N probabilities for large N . Step 4 We now have come to the core of the Probability Sort algorithm. This core consists of a While-loop which runs until the desired cut-off crit of probability sorted damage state vectors has been obtained or until the Proposals list is empty, signifying that the total state space has been explored. count = 1; While[ (Length[Proposals] > 0) OR (log(prob) < crit) Repeat Steps 5 and 6; (21)

count++ ] Step 5.a In each iteration of this While-loop the current Proposals list is updated by taking the list entry which has the greatest path probability P ; that is, take that list { P, stateVector, switchVector, activeSwitch }. (22) in Proposals where P is maximal. Step 5.b We then make available the entities in the list (18) for the algorithmic steps that will follow, by setting

workP = P ; workStateVector = stateVector ; workSwitchVector = switchVector ; (23) workActiveSwitch = activeSwitch ;


70

Step 5.c After which we remove the list entry (18) from the Proposals list. Step 5.d We then set ( )1count+P = P (24) and the unsorted stateVector, see (8),

( )1count+x = undoPermutate[ stateVector ] (25)

in a list ( ) ( ){ }1count1count , ++ xP and insert that list entry at the back of the list probabilitySort, so we obtain the updated list:

probabilitySort = ( ) ( ){ } ( ) ( ){ } ( ) ( ){ }{ }1count1count2211 ,,,,,, ++ xxx PPP … . (26) Step 5.e Finally, we update the total probability coverage variable:

sumProb = sumProb + ( )1count+P ; (26) Step 6 Now, the candidate with the greatest path probability P , that is, (18), is allowed to generate offspring before it gets moved to the probSort list. Each candidate can get a maximum of three ‘children’. As these children take the place of their progenitor in the Proposals list, they guarantee that

a) all the remaining leaves of the event tree of the damage state space may still be explored, and

b) that the next best probability is always in the updated Proposals list,

Offspring may be produced as follows: Step 6.a


71

Flip active switch one layer deeper, to a more improbable state, if permissible given maximum layer depth, and set that switch as the active switch and update the corresponding probability P and stateVector; that is, % first determine the number of possible damage states % for the infrastructural element under consideration:

q = workActiveSwitch ; M = length(doublePermutatedProbMatrix(q, :) ;

% elements in the switchVector take on values % from 0 (base-‐line damage state with the highest probability) % to M – 1 (damage state the lowest probability) % So we have below that 0 ≤ r ≤ M – 1.

r = workSwitchVector(q) ; If[ r < M – 1,

% Set

P = workP ; stateVector = workStateVector ;

switchVector = workSwitchVector activeSwitch = workActiveSwitch ;

%Then update P = [ P/ doublePermutatedProbMatrix(q, r) ] × doublePermutatedProbMatrix(q, r + 1); stateVector(q) = doublePermutatedStateMatrix(q, r + 1) ; switchVector(q) = r + 1;

%Store the list offSpring1 = { P, stateVector, switchVector, workActiveSwitch } ;

% anywhere in the Proposals list, Proposals = Insert [Proposals, offSpring1] ;

%and close the If-statement. ];

Step 6.b If active switch is a first layer switch, then de-activate switch and position switch one step forward if permissible given (a) row length or (b) a zero spot being available at that position, and activate that switch for that forward position.

q = workActiveSwitch ; If[ (workSwitchVector(q) == 1) AND (q + 1 ≤ N ) AND (workSwitchVector(q + 1) == 0),


72

% Set P = workP ; stateVector = workStateVector ;


%Then update

P = [ P/ doublePermutatedProbMatrix(q, 2) ] × doublePermutatedProbMatrix(q, 1); P = [P / doublePermutatedProbMatrix(q + 1, 1) ] × doublePermutatedProbMatrix(q + 1, 2);

stateVector(q) = doublePermutatedStateMatrix(q, 1) ; stateVector(q + 1) = doublePermutatedStateMatrix(q + 1, 2) ;

switchVector(q) = 0; switchVector(q + 1) = 1;

activeSwitch = q + 1;

%Store the list offSpring2 = { P, stateVector, switchVector, activeSwitch } ;



Step 6.c If the first entry in the switchVector is an available zero spot, then set that zero spot to a first level active switch, and update the corresponding probability P and stateVector; that is,

If[ switchVector(1) == 0,

% Set P = workP ; stateVector = workStateVector ;


%Then update

P = [ P/ doublePermutatedProbMatrix(1,1) ] × doublePermutatedProbMatrix(1,2); stateVector(1) = doublePermutatedStateMatrix(1, 2) ; switchVector(1) = 1;

activeSwitch = 1;

%Store the list offSpring3 = { P, stateVector, switchVector, activeSwitch } ;


73



Step 7: Print the probability coverage value sumProb and STOP. The list probabilitySort is now available to the user. N.B.: The probability criterion crit may prohibit the elements of the switch vector from the (N – m)th element onwards to leave their optimal base-line states of 0, as the probabilities of switch state 1 from the (N – m)th element onwards puts the probability of the probability P of the adjusted base-line state vector

P = 1; For[ i =1, i ≤ N,

P = P × permutatedProbMatrix(i, 1); i++] below the admissible threshold; that is, Pm < … < P2 < P1 < crit where P1 = [ P/ doublePermutatedProbMatrix(N – m + 1,1) ] × doublePermutatedProbMatrix(N – m + 1, 2); P2 = [ P/ doublePermutatedProbMatrix(N – m + 2,1) ] × doublePermutatedProbMatrix(N – m + 1, 2);

…

Pm = [ P/ doublePermutatedProbMatrix(N,1) ] × doublePermutatedProbMatrix(N, 2); So, under a probability criterion crit we may set the length of the switch vector in (16) to be of the reduced length (N – m), rather than the full length N.

9.3. Pen and Paper Algorithmic Run

We now give a pen and paper algorithmic run of the proposed Probability Sort algorithm for state vectors having probabilities greater or equal to crit = 610− , for the most simple non-trivial case where we have 3=N elements each having 3=M possible damage states. This in order to give the reader/programmer a concrete sense of the here proposed algorithm.


74

Let

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

10000001

1000000999

1000000999000

100001

1000099

100009900

1001

1009

10090

probMatrix (1)

and let the state matrix be such that its column is equivalent to the switch vector and its subsequent switching layers:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

210210210

xstateMatri (2)

Then ML proposal [ ]( )1000 has a probability of (1) and (2)

( )12

61

1010890109

1000000999000

100009900

10090 ×

=××=P

So, we set the ProbabilitySort list as

ProbabilitySort = [ ]( )⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ × 1

12

6

000,10

10890109

The next best state vector in terms of being the next most probable state vector then is given as:

[ ]( ) [ ]( )12

521

1010890109,00000 ×

→ 1 ,

as (1) and (2)

( )12

52

1010890109

1000000999000

100009900

1009 ×

=××=P ,

And where the bold face underlined switch state points to the active switch position. So, we set the Proposals list as


75

Proposals = [ ]( )

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ × 2

12

5

00,10

10890109 1

In the first iteration of the While-loop we then insert the most probable of the proposals in the probability sort list

ProbabilitySort = [ ]( ) [ ]( )

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ × 2

12

51

12

6

001,10

10890109,000,10

10890109

After we clean up the Proposals list as Proposals = { } We then go through the three offspring checkpoints:

[ ]( )

[ ]( )

[ ]( )

( )[ ]( ) n.a.,001,110

10890109,00

101098901,00

00 12

44

12

53

2

reject

×

×

→ 1

2

1

So as we store the proposals [ ]( )3002 and [ ]( )400 1 into the Proposals list with the corresponding probabilities

Proposals = [ ]( ) [ ]( )⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ × 4

12

43

12

5

00,10

10890109,00,10

1098901 12

In the next While-iteration we then have the most probable state vector [ ]( )3002 and add it together with its probability to the probability sort list, as we remove that entry from the proposals list:

ProbabilitySort =

[ ]( )

[ ]( )

[ ]( )⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

312

5

212

5

112

6

002,10

1098901

,001,10

10890109

,000,10

10890109


76

Proposals = [ ]( )

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ × 4

12

4

00,10

10890109 1

We have in this While-iteration

[ ]( )[ ]( )

[ ]( )[ ]( )reject

reject

reject

001,20000

00 3 13

2 →

So, we have at the end of the While-iteration proposals list is not replenished.

Proposals = [ ]( )

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ × 4

12

4

00,10

10890109 1

In the next While-iteration we then have the most probable state vector [ ]( )400 1 and add it together with its probability to the probability sort list, as we remove that entry from the proposals list:

ProbabilitySort =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

412

4

312

5

212

5

112

6

010,10

10890109

,002,10

1098901

,001,10

10890109

,000,10

10890109

Proposals = { } We have in this While-iteration the parent [ ]( )400 1 , which begets the offspring

[ ]( )

[ ]( )

[ ]( )

[ ]( )12

37

12

36

12

45

4

1010890109,01

1010890109,00

10108991,00

00

×

×

×

→

1

1

2

1


77

So, we have at the end of the While-iteration the replenished proposals list

Proposals =

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

712

3

612

3

512

4

01,10

10890109

,00,10

10890109

,00,10108991

1

1

2

In the next While-iteration we then have as the most probable state vector a choice between [ ]( )600 1 and [ ]( )7011 . We may choose either of these proposals and add it together with its probability to the probability sort list, as we remove that entry from the proposals list, say

ProbabilitySort =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

612

3

412

4

312

5

212

5

112

6

100,10

10890109

,010,10

10890109

,002,10

1098901

,001,10

10890109

,000,10

10890109

Proposals =

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

712

3

512

4

01,10

10890109

,00,10108991

1

2

We have in this While-iteration the parent [ ]( )600 1 , which begets the offspring


78

[ ]( )

[ ]( )

[ ]( )

[ ]( ) 12

29

12

38

6

1010890109,10

n.a.,0001010891,00

00×

×

→

1

1

2

1 reject


Proposals =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

912

2

812

3

712

3

512

4

10,10

10890109

00,1010891

01,10

10890109

,00,10108991

1

2

1

2

In the next While-iteration we then have as the most probable state vector [ ]( )7011 . We add it together with its probability to the probability sort list, as we remove that entry from the proposals list:

ProbabilitySort =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

712

3

612

3

412

4

312

5

212

5

112

6

011,10

10890109

,100,10

10890109

,010,10

10890109

,002,10

1098901

,001,10

10890109

,000,10

10890109


79

Proposals =

[ ]( )

[ ]( )

[ ]( )⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

912

2

812

3

512

4

10,10

10890109

,00,1010891

,00,10108991

1

2

2

We have in the next While-iteration the parent [ ]( )7011 , which begets the offspring

[ ]( )

[ ]( )

( )[ ]( )

( )[ ] n.a.,011,1n.a.,01,10

101098901,01

0112

310

7

reject

reject

×

→

2

1


Proposals =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1012

3

912

2

812

3

512

4

01,10

1098901

,10,10

10890109

,00,1010891

,00,10108991

2

1

2

2

In the next While-iteration we then have as the most probable state vector [ ]( )10012 . We add it together with its probability to the probability sort list, ProbabilitySort =


80

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1012

3

712

3

612

3

412

4

312

5

212

5

112

6

012,10

1098901

,011,10

10890109

,100,10

10890109

,010,10

10890109

,002,10

1098901

,001,10

10890109

,000,10

10890109

as we remove that entry from the proposals list. We have in this While-iteration the parent [ ]( )10012 , which begets no offspring

[ ]( )[ ]( )

( )[ ]( )

( )[ ]rejectreject

reject

011,201,1001

01 10

32 →

So, we have at the end of the While-iteration the updated proposals list

Proposals =

[ ]( )

[ ]( )

[ ]( )⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

912

2

812

3

512

4

10,10

10890109

,00,1010891

,00,10108991

1

2

2

In the next While-iteration we then have as the most probable state vector [ ]( )500 2 . We add it together with its probability to the probability sort list, ProbabilitySort =


81

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

512

4

1012

3

712

3

612

3

412

4

312

5

212

5

112

6

020,10108991

,012,10

1098901

,011,10

10890109

,100,10

10890109

,010,10

10890109

,002,10

1098901

,001,10

10890109

,000,10

10890109

as we remove that entry from the proposals list. We have in this While-iteration the parent [ ]( )500 2 , which begets the offspring

[ ]( )[ ]( )

[ ]( )

[ ]( )12

311

5

10108991,02

n.a.,00n.a.,00

00×

→

1

13

2 reject

reject

So, we have at the end of the While-iteration the updated and replenished proposals list

Proposals =

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1112

3

912

2

812

3

02,10108991

,10,10

10890109

,00,1010891

1

1

2

In the next While-iteration we then have as the most probable state vector [ ]( )9101 . We add it together with its probability to the probability sort list, as we remove that entry from the proposals list:


82

ProbabilitySort =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

912

2

512

4

1012

3

712

3

612

3

412

4

312

5

212

5

112

6

101,10

10890109

,020,10108991

,012,10

1098901

,011,10

10890109

,100,10

10890109

,010,10

10890109

,002,10

1098901

,001,10

10890109

,000,10

10890109

We have in this While-iteration the parent [ ]( )9101 , which begets the offspring

[ ]( )[ ]( )

[ ]( )

( )[ ]( ) n.a.,101,110

10890109,1010

1098901,10

10 1213

12

212

9

reject

×

×

→ 1

2

1



83

Proposals =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1312

1

1212

2

1112

3

812

3

10,10

10890109

,10,10

1098901

,02,10108991

,00,1010891

1

2

1

2

In the next While-iteration we then have as the most probable state vector [ ]( )12102 . We add it together with its probability to the probability sort list, ProbabilitySort =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1212

2

912

2

512

4

1012

3

712

3

612

3

412

4

312

5

212

5

112

6

102,10

1098901

,101,10

10890109

,020,10108991

,012,10

1098901

,011,10

10890109

,100,10

10890109

,010,10

10890109

,002,10

1098901

,001,10

10890109

,000,10

10890109


84

as we remove that entry from the proposals list. We have in this While-iteration the parent [ ]( )12102 , which begets no offspring

[ ]( )[ ]( )

[ ]( )

( )[ ]( )reject

reject

reject

101,21010

10 12 13

2 →

So the updated proposals list at the end of the While-iterations is

Proposals =

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1312

1

1112

3

812

3

10,10

10890109

,02,10108991

,00,1010891

1

1

2

In the next While-iteration we then have as the most probable state vector [ ]( )11021 . We add it together with its probability to the probability sort list,


85

ProbabilitySort =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1112

3

1212

2

912

2

512

4

1012

3

712

3

612

3

412

4

312

5

212

5

112

6

021,10108991

,102,10

1098901

,101,10

10890109

,020,10108991

,012,10

1098901

,011,10

10890109

,100,10

10890109

,010,10

10890109

,002,10

1098901

,001,10

10890109

,000,10

10890109

as we remove that entry from the proposals list. We have in this While-iteration the parent [ ]( )11021 , which begets the offspring

[ ]( )

[ ]( )

( )[ ]( )

( )[ ]( ) n.a.,021,1n.a.,01,20

1010999,02

0212

314

11

reject

reject

×

→

2

1

So the updated proposals list at the end of the While-iterations is


86

Proposals =

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1412

3

1312

1

812

3

02,1010999

,10,10

10890109

,00,1010891

2

1

2

In the next While-iteration we then have as the most probable state vector [ ]( )1210 1 . We add it together with its probability to the probability sort list,

ProbabilitySort =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1312

1

1112

3

1212

2

912

2

512

4

1012

3

712

3

612

3

412

4

312

5

212

5

112

6

110,10

10890109

,021,10108991

,102,10

1098901

,101,10

10890109

,020,10108991

,012,10

1098901

,011,10

10890109

,100,10

10890109

,010,10

10890109

,002,10

1098901

,001,10

10890109

,000,10

10890109


87

as we remove that entry from the proposals list. We have in the next While-iteration the parent [ ]( )1310 1 , which begets the offspring

[ ]( )

[ ]( )

( )[ ]( )

[ ]( )12

16

1215

13

10890109,11

n.a.,1,10010

108991,10

10

1

2

1 reject

×

→


Proposals =

[ ]( )

[ ]( )

[ ]( )

[ ]( )

⎭⎬⎫

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

⎭⎬⎫

⎩⎨⎧ ×

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧ ×

1612

1512

1412

3

812

3

11,10890109

,10,10

108991

,02,1010999

,00,1010891

1

2

2

2

All the remaining proposals have probabilities smaller than 610− , which is why we (arbitrarily) terminate this pend-and-paper run. Note that we build in the (arbitrary) If-statement for crit = 610− into the algorithmic Step 6.a, 6.b, and 6.c, then the above Proposals list would have been empty and the algorithm would have terminated automatically at this point. We give below the rest of the event-tree coverage without the corresponding probabilities.

[ ]( )[ ]( )

[ ]( )

[ ]( )17

8

2000000

001

13

2 reject

reject

→

[ ]( )[ ]( )

( )[ ]( )

( )[ ]( )reject

reject

reject

021,201,2002

02 14

32 →


88

[ ]( )[ ]( )

( )[ ]( )

[ ]( )18

15

121,10010

101

32 reject

reject

→

[ ]( )[ ]( )

( )[ ]( )

( )[ ]( )reject

reject

111,111,1011

11

19

16

21 →

[ ]( )[ ]( )

[ ]( )

( )[ ]( )reject201,12020

20 21

20

17 12

1 →

[ ]( )[ ]( )

( )[ ]( )

( )[ ]( )reject

reject

121,111,2012

12

22

18

21 →

[ ]( )[ ]( )

( )[ ]( )

( )[ ]( )reject

reject

reject

111,211,1011

11 19

32 →

[ ]( )[ ]( )

[ ]( )

( )[ ]( )reject

reject

reject

201,22020

20 20 13

2 →

[ ]( )[ ]( )

( )[ ]( )

[ ]( )24

23

21

211,20020

201

21 reject→

[ ]( )[ ]( )

( )[ ]( )

( )[ ]( )reject

reject

reject

121,211,2012

12 22

32 →

[ ]( )[ ]( )

( )[ ]( )

[ ]( )25

23

221,20020

201

32 reject

reject

→


89

[ ]( )[ ]( )

( )[ ]( )

( )[ ]( )reject

reject

211,121,1021

21

26

24

21 →

[ ]( )[ ]( )

( )[ ]( )

( )[ ]( )reject

reject

221,121,2022

22

27

25

21 →

[ ]( )[ ]( )

( )[ ]( )

( )[ ]( )reject

reject

reject

211,221,1021

21 26

32 →

[ ]( )[ ]( )

( )[ ]( )

( )[ ]( )reject

reject

reject

221,221,2022

22 27

32 →

And we see that the algorithm covers all of the 2733 ==NM leaves in our event tree.

security sensitivity committee deliverable...

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