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29 January 2015

… since we are dealing with complex and intricate matters it is necessary to build models of the problem

29 January 2015

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. In this exercise we will try to address the burning question regarding dope in the cycling sport somewhat closer. The main question is whether the bare knowledge of a doping test may reveal if a competitor has used dope.

Let us assume that the EPO test is able to detect the use of EPO with a probability of 95% (this number is often cited in newspapers). Moreover, let us assume that Pantani just has been tested positive in the EPO test. Our question is then, what is the probability that he is using EPO – or more precisely, what is the probability that Pantani has used EPO given that he was tested positive? No, the probability is not 95%, since we need information on the probability for a false positive test, and the prior probability that Pantani did use EPO.

In the newspapers nothing has been said regarding the false positive test, i.e., the probability of a positive EPO test given Pantani did not use EPO. Let us assume that this probability is 15%. Next we need to know the probability of the Giro d’Italia participants are using EPO (we cannot just say that all participants are using EPO, because then we did not need the testing). Let us assume that 10% are using EPO.

29 January 2015

1. Why is the cost of the cargo damage dependent on whether or not the damage is repairable? Check the costs at the end of the branch 1Y-2Y-3Y-4N-5Y – 6Y/N.

2. There is not assigned any cargo damage cost to the branch 1Y-2N-3Y-4N, which is wrong. Compare to the branch discussed above.

3. If we check the branch 1Y-2Y-3Y-4N we see that the expected structural damage is E[L]=0.360,000 + 0.7420,000 = 312,000. This branch is comparable to the branch 1Y-2N-3Y-4N (the difference is in whether or not the damage is detected) where the expected structural damage is E[L]=0.2240,000 + 0.8240,000 = 240,000. Why does the effect of damage detection increase the structural damage costs by 30%?

4. In the branch 1N-2N-3Y-4N-5N it is seen that there is assigned a large cargo damage cost to this case when the cargo is not sensitive to humidity. Why? In the branches above this was not the case.

5. Does “Is vessel on voyage?” mean at sea? If “Is vessel on voyage=No” mean that the vessel is at harbour then it is surprising to see that 1 out of 16 lost bulk carriers due to “Damage to Hatchway Watertight Integrity on Bulk Carriers” are lost in harbours. ( P[Loss at sea]= 3.44E-4+8.03E-4+3.28E-5+7.64E-5=1.26E-3 and P[Loss in harbour] = 2.34E-5+5.46E-5 = 7.80E-5). Can this be verified?

The cost of cargo damage is given with a precision that does not reflect the uncertainty in the assessment of the costs.

29 January 2015

- 1. Peter Friis-Hansen 12 January 2010 Bayesian Network and its use in risk analysis Transportforum, 13-14 januari, 2010, Linköbing, Sweden
- 2. © 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: ?
- 3. © 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
- 4. © 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
- 5. © 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
- 6. © 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
- 7. © 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
- 8. © 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
- 9. © 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)
- 10. © 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 S Start - 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 )()|()(
- 11. © 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 0.995 G = empty 0.005 P(T) T = none 0.003 T = click 0.001 T = normal 0.996
- 12. © 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 i ii n i iin iiii xpxxxpxxp xpxxxp 11 111 11 )|(),....,|(),...,( )|(),....,|( pa pa
- 13. © 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( | ) ( )=
- 14. © Det Norske Veritas AS. All rights reserved. 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( | , , ) ( | , )=
- 15. © 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 ( , , , , ) ( ) ( | ) ( | , ) ( | , , ) ( | , , , ) ( ) ( ) ( | ) ( | , ) ( | , ) = =
- 16. © 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
- 17. © 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
- 18. © 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 s b t b f t ( | , ) ( , , , , ) ( , , , , ) , , , = = = = ∑ ∑
- 19. © 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 over makes no fuel more likely.
- 20. © 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
- 21. © Det Norske Veritas AS. All rights reserved. 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
- 22. © Det Norske Veritas AS. All rights reserved. Bayesian Network and its use in risk analysis 12 January 2010 Maximise expected utility Party Location - outdoor - indoor Utility Weather indoors outdoors .7 .3 .7 .3 dry rain dry rain 50 60 100 0 EU[indoors] = 0.7 (50) + 0.3 (60) = 53 EU[outdoors] = 0.7 (100) + 0.3 (0) = 70 select “outdoor”
- 23. © Det Norske Veritas AS. All rights reserved. Bayesian Network and its use in risk analysis 12 January 2010 Time critical decisions System State H, to E1 E2 En Action A, t Duration of Process Utility EU A t p H E u A H ti j i j n j[ , ] ( | , ) ( , , )= = ∑ ξ 1 probability of hypothesises for the different system states given observations E and background information ξ time dependent utility as a function of action A and system state H
- 24. © Det Norske Veritas AS. All rights reserved. Bayesian Network and its use in risk analysis 12 January 2010 25 OOW acts OOW radar OOW visual Radar freq Looking freq Alarm transfer OOW Task Bridge Stress level OOW trainingOther alarms Time for radarTime for visual Maneuv. time Traffic intensitVessel speed Radar time Visual time Obj. rel. speed Radar dist. Visual dist. Object type Visibility Day light Radar statusWeather Speed reducti Basic network for navigator reacting in time Navigational route Vessel Object Visual distance Time for detection Minimum distance to avoid critical situation v1 v2
- 25. © Det Norske Veritas AS. All rights reserved. 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
- 26. © Det Norske Veritas AS. All rights reserved. Bayesian Network and its use in risk analysis 12 January 2010 27 Fatigue inspection planning Model 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)
- 27. © 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/
- 28. © 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
- 29. © Det Norske Veritas AS. All rights reserved. Bayesian Network and its use in risk analysis 12 January 2010 30 Modelling the disconnect switch ZiVarYVar YVar DSjDSi + =),(ρ
- 30. © Det Norske Veritas AS. All rights reserved. Bayesian Network and its use in risk analysis 12 January 2010 31 Bulk carrier safety: MSC74/INF.15, 2001 ?
- 31. © Det Norske Veritas AS. All rights reserved. Bayesian Network and its use in risk analysis 12 January 2010 32 Safeguarding life, property and the environment www.dnv.com
- 32. © 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
- 33. © 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 Cliques UP ][
- 34. © Det Norske Veritas AS. All rights reserved. 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
- 35. © Det Norske Veritas AS. All rights reserved. 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
- 36. © Det Norske Veritas AS. All rights reserved. Bayesian Network and its use in risk analysis 12 January 2010 System knowledge and data X1 X2 X4X3 Prior Network α Sample size Data X1 X2 X3 X4 x1 : T F T T x2 : F T T F …… xn : T T F F X1 X2 X4X3 Priors for all structures Learned structure http://b-course.cs.helsinki.fi/
- 37. © Det Norske Veritas AS. All rights reserved. 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 avoid critical situation Legend: v1 v2 Considerations: •Visual detection •Radar detection •Dependency of weather •Correlation among variables •Perception and assessment of situation
- 38. © Det Norske Veritas AS. All rights reserved. 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?”
- 39. © Det Norske Veritas AS. All rights reserved. Bayesian Network and its use in risk analysis 12 January 2010 42 OOW acts OOW radar OOW visual Radar freq Looking freq Alarm transfer OOW Task Bridge Stress level OOW trainingOther alarms Time for radarTime for visual Maneuv. time Traffic intensitVessel speed Radar time Visual time Obj. rel. speed Radar dist. Visual dist. Object type Visibility Day light Radar statusWeather Speed reducti Basic network
- 40. © Det Norske Veritas AS. All rights reserved. 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
- 41. © Det Norske Veritas AS. All rights reserved. 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 Traffic intensit Maneuv. timeTime for visual Time for radar Other alarms OOW training Stress level Bridge OOW Task Alarm transfer Looking freq Radar freqOOW visual OOW radar OOW acts Rating freq Rating visual Rating task Rating pres Rating inform Rating Call Conventional bridge
- 42. © Det Norske Veritas AS. All rights reserved. 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
- 43. © Det Norske Veritas AS. All rights reserved. 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 a proportionality between risk for collision and visibility r : 6.1 Risk − ∝ 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 Traffic intensit Maneuv. timeTime for visual Time for radar Other alarms OOW training Stress level Bridge OOW Task Alarm transfer Looking freq Radar freqOOW visual OOW radar OOW acts Rating freq Rating visual Rating task Rating pres Rating inform Rating Call

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