session 42_1 peter fries-hansen

Peter Friis-Hansen 12 January 2010

Bayesian Network and its use in risk analysis

Transportforum, 13-14 januari, 2010, Linköbing, Sweden

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Bayesian Network and its use in risk analysis

12 January 2010

2

Structure and materials

Propulsion

Compartmentation

Manoeuvring characteristics

Bridge layout

Quality of crew

+++

Frequency

Consequence

Risk based procedures requires insight deeply into very complex matters

Accidents:

?

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Bayesian Network and its use in risk analysis

12 January 2010

3

Structuring complex systems

REQUIREMENTS

Transparency

Uniformity in modelling complexity

Verifiability of probabilistic modelling

Bayesian Networks bridges the gab between model formulation and analysis

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Bayesian Network and its use in risk analysis

12 January 2010

4

Content

Why Bayesian Networks?

Elements of Bayesian Network

Building Bayesian Networks

Modelling decisions

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Bayesian Network and its use in risk analysis

12 January 2010

5

Introducing Bayesian Networks A Bayesian Network

- is a graphical representation of uncertain quantities- reveals explicitly the probabilistic dependence between the set variables- is designed as a knowledge representation of the considered problem

A BN is a network with directed arcs and no cycles

The nodes represents random variables and/or decisions

Arcs into random variables indicate probabilistic dependence

Causal modelling most effectively does the model building

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Bayesian Network and its use in risk analysis

12 January 2010

What do Bayesian methods offer ?

1. Allows one to learn about causal relationships- this knowledge allow to make predictions in the presence of interventions / observations

2. BN in conjunction with Bayesian statistical techniques facilitate the combination of domain knowledge and data- prior or domain knowledge

3. BN can readily handle incomplete data- missing data

4. Bayesian methods in conjunction with BN and other methods offers efficient methods to avoid over fitting of data

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Bayesian Network and its use in risk analysis

12 January 2010

BN for a set of variables

Battery

Gauge

Fuel

Turnover

Start

p(B)

p(T|B)

p(G|B, F)

p(F)

p(S| F, T)

Directed Acyclic Graph

low, normal, high

none, click, normal

low, normal, high

empty, medium, full

yes, no

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Bayesian Network and its use in risk analysis

12 January 2010

BN - elements

BN for a set of variables consists of:

1. A network structure S that encodes a set of conditional independence assertions about the variables in X

2. A set P of local, conditional probability distributions associated with each variable in X

1. & 2. defines the joint probability distribution for X.

S is a Directed Acyclic Graph (DAG)

Nodes are in one-to-one correspondence with the variables in X

denotes both the stochastic variable and the associated node

denotes the parents to in S

Lack of possible arcs in S encode conditional independence

X { , , }X Xn1

Xi

pai Xi

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Bayesian Network and its use in risk analysis

12 January 2010

Description (nodes)

Probability node (discrete)

Decision node

Utility node

Link / arc

iixP pa| Local probability distribution (conditional)

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Bayesian Network and its use in risk analysis

12 January 2010

Bayesian Network

T

Turnover

- none- click- normal

SStart

- yes- no

P(T) T = none 0.003 T = click 0.001 T = normal 0.996

P(S | T) T = none T = click T = normal S = Yes 0.0 0.02 0.97 S = No 1.0 0.98 0.03

T

tTptTsSpsSp )()|()(

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Bayesian Network and its use in risk analysis

12 January 2010

Missing arcs encode conditional independence

Turnover

T

Gauge

G

P(G)G = not empty 0.995G = empty 0.005

P(T)T = none 0.003T = click 0.001T = normal 0.996

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Bayesian Network and its use in risk analysis

12 January 2010

Bayesian Network Structure: Definition

1. Find the variables of the model

2. Build a DAG that encodes assertions of conditional independence- Given an ordering of the variables

( ,..., )X Xn1

n

iii

n

iiin

iiii

xpxxxpxxp

xpxxxp

11111

11

)|(),....,|(),...,(

)|(),....,|(

pa

pa

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Bayesian Network and its use in risk analysis

12 January 2010

13

Example

Fuel Battery Turnover Gauge Start

p F( ) p B F p B( | ) ( )

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Bayesian Network and its use in risk analysis

12 January 2010

14

Example

Fuel Battery Turnover Gauge Start

p F( ) p B F p B( | ) ( )

p T B F p T B( | , ) ( | )

p G F B T p G F B( | , , ) ( | , )

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Bayesian Network and its use in risk analysis

12 January 2010

Example

Fuel Battery Turnover Gauge Start

p F( ) p B F p B( | ) ( )

p T B F p T B( | , ) ( | )

p G F B T p G F B( | , , ) ( | , )

p S F B T G p S F T( | , , , ) ( | , )

p F B T G S p F p B F p T B F p G F B T p S F B T G

p F p B p T B p G F B p S F T

( , , , , ) ( ) ( | ) ( | , ) ( | , , ) ( | , , , )

( ) ( ) ( | ) ( | , ) ( | , )

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Bayesian Network and its use in risk analysis

12 January 2010

Variable order is important!

Start Gauge Turnover Battery Fuel

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Bayesian Network and its use in risk analysis

12 January 2010

Causal knowledge simplifies the construction

Battery

Gauge

Fuel

Turnover

Start

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Bayesian Network and its use in risk analysis

12 January 2010

Conditional independence simplifies Probabilistic Inference

Battery

Gauge

Fuel

Turnover

Start

g

s

f

p F f S s G g

p f b t g s

p f b t g sb t

b f t

( | , )

( , , , , )

( , , , , ),

, ,

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Bayesian Network and its use in risk analysis

12 January 2010

“Explaining Away”

Turnover

Start

Fuel

If the car does not start, hearing the engine turn overmakes no fuel more likely.

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Bayesian Network and its use in risk analysis

12 January 2010

20

Did Marco Pantani use EPO? Just before the last lap in the Giro d’Italia in 1999, the Italian Marco Pantani was

excluded from the race because of a positive EPO doping test. Marco Pantani was leading the race when he was excluded.

Question: does the bare fact of a positive EPO test reveal his quilt?

Assumptions:

The EPO test is able to detect the use of EPO with a probability of 95%

False positive test: Let us assume that this probability is 15%.

Probability of riders are using EPO: say, 10% are using EPO.

HUGIN

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Bayesian Network and its use in risk analysis

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21

Max propagation – what is it ?

That configuration in the joint probability distribution that has the largest value

Identical to the ”FORM design point” in x-space

Identical to finding the dominant cut set for fault trees

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Bayesian Network and its use in risk analysis

12 January 2010

Maximise expected utility

Party Location- outdoor- indoor

Utility

Weatherindoors

outdoors

.7

.3

.7

.3

dry

rain

dry

rain

50

60

100

0

EU[indoors] = 0.7 (50) + 0.3 (60) = 53EU[outdoors] = 0.7 (100) + 0.3 (0) = 70 select “outdoor”

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Bayesian Network and its use in risk analysis

12 January 2010

Time critical decisions

System StateH, to

E1 E2 En

Action A, tDuration ofProcess

Utility

EU A t p H E u A H ti j ij

n

j[ , ] ( | , ) ( , , )

1

probability of hypothesises for the different system statesgiven observations E and background information

time dependent utility as a function ofaction A and system state H

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Bayesian Network and its use in risk analysis

12 January 2010

25OOW acts

OOW radar

OOW visual Radar freqLooking freq

Alarm transfer

OOW Task

Bridge

Stress level

OOW trainingOther alarms

Time for radarTime for visual Maneuv. time

Traff ic intensitVessel speed

Radar time

Visual time

Obj. rel. speed

Radar dist.

Visual dist.

Object type

VisibilityDay light

Radar statusWeather

Speed reducti

Basic network for navigator reacting in time

Navigational route

Vessel Object

Visual distance

Time for detection

Minimum distance to avoidcritical situation

v1

v2

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Bayesian Network and its use in risk analysis

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Including the time aspect - DBN

a1(5)a1(4)a1(3)a1(2)a1(1)

SIF1(5)SIF1(4)SIF1 (3)SIF1 (2)SIF1(1)

Seastate5Seastate4Seastate3Seastate2Seastate1

Initial a_1

Model unc.

Fatigue modelling

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Bayesian Network and its use in risk analysis

12 January 2010

27

Fatigue inspection planningModel uncer

a(0)

Seastate(2)Seastate(4)

a(02) a(04)

CI(4)

CR(4)

CI(2)

CR(2)

Inspect(04)Inspect(02)

InspRes(04InspRes(02

CF(2)

a_rep(04)a_rep(02)

PF(02) PF(04)

CF(4) CF(6)

PF(06)

a_rep(06)

InspRes(06

Inspect(06)

CR(6)

CI(6)

a(06)

Seastate(6)

dPF(0-2) dPF(2-4) dPF(4-6)

PF(0)

CF(0)

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Bayesian Network and its use in risk analysis

12 January 2010

Where to get more information ?

HUGIN expert AS

www.hugin.com

Association for Uncertainty in Artificial Intelligence

www.auai.org

Microsoft Decision Group

www.research.Microsoft.com/research/dtg

Bibliography

www-users.cs.york.ac.uk/~sara/reference/biblios/

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Bayesian Network and its use in risk analysis

12 January 2010

29

Two line Transformer station subjected to earth quake

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Bayesian Network and its use in risk analysis

12 January 2010

30

Modelling the disconnect switch

ZiVarYVar

YVarDSjDSi

),(

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Bayesian Network and its use in risk analysis

12 January 2010

31

Bulk carrier safety: MSC74/INF.15, 2001

?

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Safeguarding life, property and the environment

www.dnv.com

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Bayesian Network and its use in risk analysis

12 January 2010

What is a complex system ?

Complex: A whole made up of dissimilar parts or parts of intricate relationship

Consisting of interconnected or interwoven parts; composite

Intricate: having a complicated organisation, with many parts or aspects difficult to follow or grasp

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Bayesian Network and its use in risk analysis

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36

Propagation in Bayesian Network

U grows exponentially with number of variables and states – for binary O(2N)

Calls for efficient algorithm

JUNCTION TREE- The nodes of the junction tree are sets of variables called cliques- Links are separators, which is the intersection of the adjacent cliques

Separators

CliquesUP ][

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Bayesian Network and its use in risk analysis

12 January 2010

37

Triangulated graph and junction tree

1

2

3

4

5

6 145

456

345

235

45

45

35

1

2

3

4

5

6

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Bayesian Network and its use in risk analysis

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38

Learning

Learning probability distributions- Uses EM algorithm

- Log likelihood optimisation reformulated to nested optimisation- Assures better and faster convergence

- Beta distribution- Dirichlet distribution

Learning the structure – more ambitious- Priors for all structures

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Bayesian Network and its use in risk analysis

12 January 2010

System knowledge and data

X1 X2

X4X3

Prior Network

Sample size

Data

X1 X2 X3 X4x1 : T F T Tx2 : F T T F……xn : T T F F

X1 X2

X4X3

Priors for all structures

Learned structure

http://b-course.cs.helsinki.fi/

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Bayesian Network and its use in risk analysis

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40

Interpretation of critical situation

Navigational route

Vessel Object

Visual distance

Time for detection

Minimum distance to avoidcritical situation

Legend:

v1

v2

Considerations: •Visual detection•Radar detection •Dependency of weather•Correlation among variables•Perception and assessment of situation

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Bayesian Network and its use in risk analysis

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41

Description of the critical situation

“During the watch the considered vessel is on collision course with an object. Moreover, machinery and steering gear are functioning.”

“Does the Officer On the Watch react in time so that the collision is avoided?”

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Bayesian Network and its use in risk analysis

12 January 2010

42OOW acts

OOW radar

OOW visual Radar freqLooking freq

Alarm transfer

OOW Task

Bridge

Stress level

OOW trainingOther alarms

Time for radarTime for visual Maneuv. time

Traff ic intensitVessel speed

Radar time

Visual time

Obj. rel. speed

Radar dist.

Visual dist.

Object type

VisibilityDay light

Radar statusWeather

Speed reducti

Basic network

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Bayesian Network and its use in risk analysis

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43

Concept of the conventional bridge Conventional bridge is a modern bridge

A rating lookout will (in principle) be present on the bridge from sunset to sunrise

Calling of a rating lookout during daytime if conditions causes solo watch keeping being unsafe

- conditions of weather, - visibility, - proximity of dangers to navigation, - traffic situation

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Bayesian Network and its use in risk analysis

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44

Speed reducti

Weather Radar status

Day light

Visibility

Object type

Visual dist.

Radar dist.

Obj. rel. speed

Visual time

Radar time

Vessel speed Traff ic intensit

Maneuv. timeTime for visual Time for radar

Other alarms OOW training

Stress level

Bridge

OOW Task

Alarm transfer

Looking freqRadar freqOOW visual

OOW radar

OOW acts

Rating freq

Rating visual

Rating task

Rating pres

Rating inform

Rating Call

Conventional bridge

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Bayesian Network and its use in risk analysis

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Results - conventional bridge

Case P[OOW not acting]

Day and night 0.00270

Daylight 0.00330

Darkness 0.00209

Object P[OOW not acting] (Day and night

P[OOW not acting] (Daylight)

P[OOW not acting] (Darkness)

Large vessel 0.00193 0.00264 0.00122

Small vessel 0.00191 0.00267 0.00116

Floating object 0.773 0.649 0.898

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Bayesian Network and its use in risk analysis

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46

Comparison with observations

log-log plot of probability of no action

y = -1.0792x - 1.7219

R2 = 0.9743-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

-0.5 0 0.5 1 1.5

log(Visibility)

log

(P)

Log P

Linear (Log P)

Japanese observations in the period from 1966 to 1971 reveals aproportionality between risk for collision and visibility r :

6.1Risk r

Causes ? • Improved radar technology• Difference in causes for low visibility in DK and Japan

Obtained factor is -1.1

Speed reducti

Weather Radar status

Day light

Visibility

Object type

Visual dist.

Radar dist.

Obj. rel. speed

Visual time

Radar time

Vessel speed Traff ic intensit

Maneuv. timeTime for visual Time for radar

Other alarms OOW training

Stress level

Bridge

OOW Task

Alarm transfer

Looking freqRadar freqOOW visual

OOW radar

OOW acts

Rating freq

Rating visual

Rating task

Rating pres

Rating inform

Rating Call