12735: Urban Systems Modeling. Loss and decisions. instructor: Matteo Pozzi. Lec : Urban Systems Modeling Lec. 11 Loss and decisions

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1 1735: Urban Systems Modeling Lec. 11 Loss and decisions instructor: Matteo Pozzi 1

2 outline introduction example of decision under uncertainty attitude toward risk principle of minimum expected loss Bayesian prospective on decision making pre posterior analysis: Value of Information

3 overview of risk analysis and decision making Bayesian updating observations inspection scheduling, sensor placement Value of information prior data and analysis probabilistic model risk analysis decision making model selection simulations, scenario analysis, utility/loss theory 3

4 introduction Modern decision theory was developed by Ramsey (1931), Von Neumann and Morgenstern (1944), Raiffa and Schlaifer (1961). Axioms of rational decision, principle of maximum expected utility (MEU). Applications: Economics ecision support systems Planning and active control Artificial intelligence (do the right thing!) Game theory. 4

5 deterministic examples onsider the management of a structure. two actions: o Nothing, Repair consequences are in term of cost. osts are: K$, 1K$. objective is to minimize costs. optimal action and cost are argmin, min, repair 1K$ many actions: argmin optimal action and cost are min argmin continuous domain: min 5

6 decision under uncertainties suppose costs are uncertain, for each action. x A, for each action, the corresponding cost is a distribution. How to compare distributions? expected cost: value distribution p(a) [K$] a 1 a if agent is risk neutral: argmin min argument for minimizing expected cost: in the long run, the expected (mean) cost is a measure of the average loss. Risk neutral agents aim to minimize long term cost. 6

7 decision making under uncertainty: principles lottery: :, ;, ; ;, : outcome [cost], : its probability. lotteries include sure things: 1, 1 an agent has preferences among lotteries: : is preferred to ~: lotteries and are indifferent : is preferred to or lotteries and are indifferent the behavior of an agent is rational only if preferences fulfill these axioms: axioms: Orderability: ~ Transitivity:, lotteries,,, probabilities, ontinuity: :, ; 1, ~ Monotonicity:, ; 1,, ; 1, Substitutability: ~, ; 1, ~, ; 1, ecomposability:, ; 1,, ; 1, ~, ; 1,; 1 1, 7

8 decision making under uncertainty: min. expected loss lottery: :, ;, ; ;, : outcome [cost], : its probability. lotteries include sure things: 1, 1 an agent has preferences among lotteries: : is preferred to ~: lotteries and are indifferent : is preferred to or lotteries and are indifferent the behavior of a rational agent can be described by loss function: : lottery loss ~ the rational agent minimizes expected loss expected loss: function: depends on attitude towards risk. 8

9 attitude towards risk 1.5 1, $.5K sure thing.5,;.5, $1K Expected loss: $1K 1 1 risk-seeking risk-neutral risk-averse 1 1 $1K 1 $.5K Loss.5 agents: $.5K 1 $1K : risk averse: $.5K 1 $1K : risk neutral: ~ [K$] $.5K 1 $1K : risk seeking: raonal agent s preferences loss function loss function is not unique, : function is equivalent to function. sure thing rate of conversion 9

10 attitude toward risk and transformation of RVs ~ln, p L p(la 1 ) 5 p(la ) L 4 3 L = L risk averse : 1 p p(a 1 ) p(a ) ~ln, 1

11 example of attitude toward risk 5 4 a 1 two actions, same expected costs. p(a) 3 a risk neutral: ~ [K$] p(la) a 1 a L 4 3 risk averse : L 1 [K$] p(la) a 1 a L risk seeking : L 1 [K$] 11

12 loss elicitation and loss related to death loss function derives by agent s preferences, and can be identified by posing questions to the agent. assign loss to the best case scenario, loss 1 to the worst case scenario ;, find probability : ~, ; 1,. In other words, find the probability that is indifferent respect to the two lotteries.. micromort: one in a million chance of death. Several studies across a range of people have shown that a micromort is worth about $ in 198 dollars, or under $5 in today s dollars. We can consider a utility curve whose X axis is micromorts. As for monetary utility, this curve behaves differently for positive and negative values. or example, many people are not willing to pay very much to remove a risk of death, but require significant payment in order to assume additional risk. Koller and riedman, Probabilistic Graphical Models Principles and Techniques. 1

13 syntax of influence diagrams chance: random variables decision: these are under control of the agent, and no inference is to be done. utility/loss: it a deterministic function of its parents, which are random vars. and decision vars. loss variables define the function that has to be minimize, varying decision. loss can be defined in terms of expected value:,,, 13

14 example of influence diagram scenario x 1 stiffness material x strength Set of random variables, defined by conditional (in)dependence. joint probability: load x 3 x 4 x 5 demand stress x 7 x 6 x 8 damage task: prediction conditional prediction predicted loss, for each action optimal action and loss A L conditional optimal action and loss action loss chance decision utility/loss 14

15 simplest maintenance problem A state action N: do Nothing R: Repair S L : ailure U: Undamaged loss pay off (cost) matrix, U N R E[L] P N Repair o Nothing Optimal Repair / : optimal threshold o Nothing Repair N R min, accepting the risk not accepting the risk prior optimal loss 15

16 with perfect information A state action N: do Nothing R: Repair S L : ailure U: Undamaged loss pay off (cost) matrix, U N R E[L] Repair Repair o o Nothing Nothing Optimal Optimal Opt. Obs P N Repair / : optimal threshold obser. state Undam. amaged o Nothing Repair N, U R, EVPI P expected value of perfect information loss with PI obs. [failure always avoided] obs. min, 1 16

17 with perfect information: example A state action N: do Nothing R: Repair S L : ailure U: Undamaged loss pay off (cost) matrix, 3% U N 1 R E[L] Repair Repair o o Nothing Nothing Optimal Optimal Opt. Obs % P N Repair / : optimal threshold obser. state Undam. amaged o Nothing Repair N, U R, N 3 R expected value of perfect information loss with PI obs..3.6 obs

18 with imperfect information N R state A action R S L sensor outcome loss prob. of measures Y S : Silence A : Alarm U S S U S A Bayes formula A U : prob. of alse Alarm S : prob. of alse Silence A U A optimal decision is affected by sensor outcomes: A: N, A A S: N, S S R, A R, S U S 1 A 1 perfect sensor U S 1 A 1 A min A, S min S, 18

19 with continuous variables p d, c ln 1.3 demand d state S capacity c / p(d) p(c) ln 1.58% 19

20 with continuous variables p p(d) p(d) p(c) p(c) p(yc) 5 y 1 15 d, c demand d capacity c y.58% ~ln, state S /

21 with continuous variables p p(d) p(c) p(yc) p(cy) 5 y 1 15 d, c demand d capacity c y.58% 1.55% Bayes rule: ~ln, VoI [K$] 4 o Nothing Repair Value of Information perfect information no info. A state action S L / loss, U N R $1M c 1 1 $1K sensor precision 1

22 another example: rehabilitarion level R r = R + R + e r Resistance after retrofitting load L R r R R initial Resistance Measure Action: select a nominal value of refrofitting M structural ondition : amaged if L>R r U: Undamaged if R r L X ost for rehabilitation = R ost related to failure 1 = + 1 U

23 selecting the rehabilitation level R [KN] [K$] * [K$] R [KN] 3 strong rehabilitation nominal R [KN] 1 weak rehabilitation 15 strong structure R [KN] 1 5 weak structure

24 selecting the rehabilitation level.1 P.5 prior R [KN] [K$] R [KN] R [KN]

25 selecting the rehabilitation level.1 P.5 prior R [KN] [K$] R [KN] R [KN] optimal action full cost R [KN] rehabilitation cost cost related to risk * [K ] 5

26 selecting the rehabilitation level.1 P.5 prior R [KN] [K$] M R [KN] R [KN] * [K ] R [KN] 6

27 selecting the rehabilitation level.1 18 P.5 prior posterior R [KN] [K$] M R [KN] R [KN] * [K ] R [KN] 7

28 selecting the rehabilitation level.1 18 P.5 prior posterior R [KN] [K$] M R [KN] R [KN] optimal action 6 4 * [K ] R [KN] 8

29 link between decision making and probability of failure U probability of failure, N L R L R L R expected loss doing nothing, S reliability problem: joint probability 6 limit state function x 4 SU x 1 compute P 9

30 expected value of perfect information, without observing, observing argmin argmin, min min,, min, min, : Jensen's inequality: convex function : 3

31 value of information, without observing, observing,,, argmin argmin, min min,, min, min, : min, min, min, min, 31

32 with imperfect information: recap state R sensor outcome S : Silence A : Alarm A U : prob. of alse Alarm S : prob. of alse Silence S Y U U N R A action L loss S S U S A A U A S 1 A 1 optimal decision is affected by sensor outcomes: A: A min A, S: S min S, obs. A A SS value of information: obs. 3

33 parametrical study *: expected cost w/o sensor 3.5 U * test S 1 A P S P A * PI : expected cost adopting aperfect sensor 33

34 parametrical study *: expected cost w/o sensor VoI PI * test 1.5 VoI P S P A.3.4 y.5 z x.1..3 P S P A.1 x z y * PI : expected cost adopting aperfect sensor VoI = 34

35 area with null VoI *: expected cost w/o sensor.5 osts [K$], R =.5K$, =1K$, P()=.5% *=.5K$ * test P S P A P S x s * truth =1.5$ P A P A =. P S = x s 1 1 A R S R? P( S) =.8% (posterior prob. is lower than prior) null VoI area P( S) = $.8K > R (but the risk is still unbearable) 35

36 the role of the economical framework.5 osts [K$], R =1K$, =1K$, P()=.5% *=5K$ +.5 osts [K$], R =.5K$, =1K$, P()=.5%.5 *=.5K$ P S s P S s * truth =5$ s P A * truth =1.5$ +.5 s P A R = 1K$ R =.5K$ VoI(s 1 ) < VoI(s ) VoI(s 1 ) > VoI(s ) sensor (s ) is better then sensor (s 1 ) sensor (s 1 ) is better then sensor (s ) any metric ranking the sensors without considering the economical issues is inappropriate. 36

37 VoI vs cost of failure 1 R = 1 K$, P() = 5% VoI [K$] P A = %, P S = % P A = 3%, P S = % P A = 3%, P S = 3% [K$] 37

38 VoI vs cost of failure 1 R = 1 K$, P() = 5% 9 VoI [K$] false alarm, but not false silence P A = %, P S = % P A = 3%, P S = % P A = 3%, P S = 3% null-voi range [K$] 38

39 VoI vs cost of failure 1 R = 1 K$, P() = 5% 9 VoI [K$] false alarm, but not false silence P A = %, P S = % P A = 3%, P S = % P A = 3%, P S = 3% null-voi range [K$] null-voi range 39

40 numerical example = 1 K$ R = 1 K$.1.8 load S R 1, P =.96% ott =9.6K$ P [KN] P() =.96% N: do Nothing, * = 9.6 K$ 4

41 numerical example sensor precision [KN] e ) = 1 K$ R = 1 K$.1 load S.8 R 1, P =.96% ott =9.6K$ perfect sensor measuring R no sensor Optimum ost [K$] P [KN] P() =.96% N: do Nothing, * = 9.6 K$ 41

42 numerical example sensor precision [KN] e ) = 1 K$ R = 1 K$.1 load S.8 R 1, P =.96% ott =9.6K$ perfect sensor measuring R no sensor Optimum ost [K$] P cost of a sensor [KN] P() =.96% N: do Nothing, * = 9.6 K$ best sensor perfect no information information suitable sensors sensor precision [KN] 3 1 VoI [K$] 4

43 varying the prior uncertainty P = 1 K$ R = 1 K$ load S R 1, P =.96% ott =9.6K$ R, P =1.% ott =1K$ sensor precision [KN] e ) Optimum ost [K$] [KN] 3 VoI [K$] P() = 1.% R: repair, * = 1 K$ sensor precision [KN] 43

44 varying the prior uncertainty P = 1 K$ R = 1 K$.1 load S.8 R 1, P =.96% ott =9.6K$ R, P =1.% ott =1K$.6 R 3, P =1.5% ott =1K$.4. sensor precision [KN] e ) Optimum ost [K$] [KN] 3 VoI [K$] P() = 1.5% R: repair, * = 1 K$ sensor precision [KN] 44

45 varying the prior uncertainty P = 1 K$ R = 1 K$.1 load S.8 R 1, P =.96% ott =9.6K$ R, P =1.% ott =1K$.6 R 3, P =1.5% ott =1K$ R.4 4, P =% ott =1K$. sensor precision [KN] e ) Optimum ost [K$] [KN] 3 VoI [K$] P() =.5% R: repair, * = 1 K$ sensor precision [KN] 45

46 varying the prior resistance P = 1 K$ R = 1 K$.1 load S.8 R 1, P =4.8% ott =1K$.6.4 sensor precision [KN] e ) Optimum ost [K$] [KN] P() = 4.81% R: repair, * = 1 K$ VoI [K$] sensor precision [KN] 46

47 varying the prior resistance P = 1 K$ R = 1 K$ load S R 1, P =4.8% ott =1K$ R, P =1.3% ott =1K$ sensor precision [KN] e ) Optimum ost [K$] [KN] P() = 1.8% R: repair, * = 1 K$ VoI [K$] sensor precision [KN] 47

48 varying the prior resistance P = 1 K$ R = 1 K$.1 load S.8 R 1, P =4.8% ott =1K$ R, P =1.3% ott =1K$.6 R 3, P =.63% ott =6.3K$.4 sensor precision [KN] e ) Optimum ost [K$] [KN] P() =.63% N: do Nothing, * = 6.3 K$ VoI [K$] sensor precision [KN] 48

49 varying the prior resistance P = 1 K$ R = 1 K$.1 load S.8 R 1, P =4.8% ott =1K$ R, P =1.3% ott =1K$.6 R 3, P =.63% ott =6.3K$ R.4 4, P =.16% ott =1.6K$ sensor precision [KN] e ) Optimum ost [K$] [KN] P() =.16% N: do Nothing, * = 1.6 K$ VoI [K$] sensor precision [KN] 49

50 a linear log normal model seismic intensity i two-span ordinary bridge log normal log space normal p(i) p(i) seismic intentesity, i [m/s ] longitudinal section cross section seismic intentesity, i [m/s ] 5

51 a linear log normal model structural response seismic intensity i p(i) L Equal displacement approximation: 1 k.5 log log log seismic intentesity, i [m/s ] L cov: 3% 51

52 a linear log normal model structural response damage seismic intensity i fragility curves P ( s s i I ) [mm] s II =cracking.8 s III =yielding.6 s IV =spalling s V =ultimate.4. 5

53 a linear log normal model structural response damage seismic intensity i L A loss available actions: o Nothing Reduce operation lose the structure N RT L L(A,d) [K$] loss matrix I II III IV V d 53

54 a linear log normal model structural response damage seismic intensity i L loss measure of seismic intensity A usgs shakemap

55 a linear log normal model structural response damage seismic intensity i L loss y o A measure of seismic intensity measure of monitoring system a a 1 bridge cross section

56 a linear log normal model structural response damage seismic intensity i L loss y o measure of seismic intensity measure of monitoring system visual inspection y d A

57 a linear log normal model structural response damage seismic intensity i L loss y o measure of seismic intensity measure of monitoring system visual inspection y d A task: evaluating the benefit of getting y o, given y i, depending on the availability of y d. 57

58 visual inspection i L y o y d A 58

59 the value of information (VoI) L VI y eq equivalent precision displ. eq 1.8 A E[L*] [K$] w/o VI with VI P.I. on d.6 o i intensity 59

60 the value of information (VoI) L VI y eq equivalent precision A eq 54% 5% E[L*] [K$] w/o VI with VI P.I. on d displ. o o = 5% i = 4% i intensity VoI [K$] eq 6

61 Lec : Urban Systems Modeling Loss and decisions spatially distributed infrastructure system 61 demand capacity condition state failure iff demand > capacity

62 Lec : Urban Systems Modeling Loss and decisions spatially distributed infrastructure system 6 assume we can locally measure demand, capacity. we can infer consequences on the entire system.

63 spatially distributed phenomena example: ambient temperature field Gaussian process: ;, A. Krause. Optimizing Sensing Theory and Applications, Ph.. Thesis, MU, 8. 63

64 spatially distributed phenomena example: ambient temperature field Gaussian process: ;, processing information: local measurements update the field in the surrounding area A. Krause. Optimizing Sensing Theory and Applications, Ph.. Thesis, MU, 8. 64

65 spatially distributed phenomena Temperature () Position (m) example: ambient temperature field Gaussian process: ;, processing information: local measurements update the field in the surrounding area A. Krause. Optimizing Sensing Theory and Applications, Ph.. Thesis, MU, 8. 65

66 optimal sensor placement Temperature () Position (m) example: ambient temperature field Gaussian process: ;, processing information: local measurements update the field in the surrounding area entropy : total uncertainty in the field. : residual uncertainty. A. Krause. Optimizing Sensing Theory and Applications, Ph.. Thesis, MU, 8. 66

67 optimal sensor placement Temperature () Position (m) example: ambient temperature field Gaussian process: ;, processing information: local measurements update the field in the surrounding area entropy : total uncertainty in the field. : residual uncertainty. optimal sensor placement: minimize subject to # sens. = N A. Krause. Optimizing Sensing Theory and Applications, Ph.. Thesis, MU, 8. 67

68 optimal sensor placement Temperature () Temperature () Position (m) Exact Placement Greedy Placement Position (m) example: ambient temperature field Gaussian process: ;, processing information: local measurements update the field in the surrounding area entropy : total uncertainty in the field. : residual uncertainty. optimal sensor placement: minimize subject to # sens. = N combinatorial explosion: candidate location: 1 # of sensors: 1 # of configurations: T greedy algorithm: select best location, add best matching location 68

69 optimal sensor placement Temperature () Temperature () Position (m) Exact Placement Greedy Placement Position (m) example: ambient temperature field Gaussian process: ;, processing information: local measurements update the field in the surrounding area entropy : total uncertainty in the field. : residual uncertainty. optimal sensor placement: minimize subject to # sens. = N combinatorial explosion: candidate location: 1 # of sensors: 1 # of configurations: T greedy algorithm: select best location, add best matching location 69

70 optimal sensor placement Temperature () Temperature () Exact Placement Position (m) Greedy Placement Position (m) example: ambient temperature field Gaussian process: ;, processing information: local measurements update the field in the surrounding area entropy : total uncertainty in the field. : residual uncertainty. optimal sensor placement: minimize subject to # sens. = N combinatorial explosion: candidate location: 1 # of sensors: 1 # of configurations: T greedy algorithm: select best location, add best matching location 7

71 Lec : Urban Systems Modeling Loss and decisions effects on monitoring the system 71 close proximity demand can be similar in close proximity

72 effects on monitoring the system demand capacity can be similar in close proximity similar for similar components similar type close proximity 7

73 Application to Infrastructure Systems n n n regret entropy entropy regret entropy G 1 G G n regret ailure Probability P [ 1 =] ailure Probability P [ =] ailure Probability P [ n =] system metric : approximation: decision making problems are independent for each component 73

74 example of seismic risk: overview fault line nearby locations: highly correlated accelerations structural response T epicenter more distant locations: less correlated ground acceleration example of seismic demand (PGA) example fragility curve Peak Ground Acceleration (g) istance from Epicenter (km) Probability of ailure Strucural emand (PGA) 74

75 example of seismic risk: overview epicenters seismic scenarios based on the relative probabilities of earthquakes along different faults (USGS). measures 1735: Urban Systems Modeling emand Measurements apacity Measurements Ground Acceleration Maximum tolerable displacement Seismometer readings Structural assessment Lec. 11 Loss and decisions 75

76 example of seismic risk: overview infrastructure system infrastructure components in the San rancisco, including 18 bridges and 9 tunnels. ivided among 13 infrastructure types, e.g. steel suspension bridges, concrete girder bridges, and cut and cover tunnels. 1735: Urban Systems Modeling Lec. 11 Loss and decisions 76

77 example of seismic risk: results entropy metric 1 measurements of demand or capacity. results: demand measurements are clustered in the upper Bay area, where the density of components is highest. apacity measurements are spread evenly among different asset types. 77

78 example of seismic risk: results value of information 1 measurements of demand or capacity. results: some similarity with entropy (3 sensor placements out of 1), but components with higher costs are more closely monitored, reflecting the relevant economic factors. 78

79 density of Value of Information density VoI comes from possible different scenarios: this map shows how it is related to epicenter location. Total VoI about $4M 79

80 references Barber, B. (1). Bayesian Reasoning and Machine Learning. ambridge UP. ownloadable from Bishop,. (6). Pattern Recognition and Machine Learning. Springer. Koller,. and riedman, N. (9). Probabilistic Graphical Models Principles and Techniques. the MIT Press. Russell, S. and P. Norvig. (1). Artificial Intelligence: A Modern Approach. Pearson Education. 8