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Session 42_1 Peter Fries-Hansen
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Transcript of 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
© Det Norske Veritas AS. All rights reserved.
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:
?
© Det Norske Veritas AS. All rights reserved.
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
© Det Norske Veritas AS. All rights reserved.
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
© Det Norske Veritas AS. All rights reserved.
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
© Det Norske Veritas AS. All rights reserved.
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
© Det Norske Veritas AS. All rights reserved.
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
© Det Norske Veritas AS. All rights reserved.
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
© Det Norske Veritas AS. All rights reserved.
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)
© Det Norske Veritas AS. All rights reserved.
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 )()|()(
© Det Norske Veritas AS. All rights reserved.
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
© Det Norske Veritas AS. All rights reserved.
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
© Det Norske Veritas AS. All rights reserved.
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( | , , ) ( | , )
© Det Norske Veritas AS. All rights reserved.
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
( , , , , ) ( ) ( | ) ( | , ) ( | , , ) ( | , , , )
( ) ( ) ( | ) ( | , ) ( | , )
© Det Norske Veritas AS. All rights reserved.
Bayesian Network and its use in risk analysis
12 January 2010
Variable order is important!
Start Gauge Turnover Battery Fuel
© Det Norske Veritas AS. All rights reserved.
Bayesian Network and its use in risk analysis
12 January 2010
Causal knowledge simplifies the construction
Battery
Gauge
Fuel
Turnover
Start
© Det Norske Veritas AS. All rights reserved.
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
( | , )
( , , , , )
( , , , , ),
, ,
© Det Norske Veritas AS. All rights reserved.
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.
© Det Norske Veritas AS. All rights reserved.
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
12 January 2010
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
© Det Norske Veritas AS. All rights reserved.
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
12 January 2010
26
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)
© Det Norske Veritas AS. All rights reserved.
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/
© Det Norske Veritas AS. All rights reserved.
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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Bayesian Network and its use in risk analysis
12 January 2010
32
Safeguarding life, property and the environment
www.dnv.com
© Det Norske Veritas AS. All rights reserved.
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
© Det Norske Veritas AS. All rights reserved.
Bayesian Network and its use in risk analysis
12 January 2010
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
12 January 2010
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
12 January 2010
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
12 January 2010
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?”
© Det Norske Veritas AS. All rights reserved.
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
12 January 2010
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
12 January 2010
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
12 January 2010
45
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
12 January 2010
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