Loss and decisions - CMU · 12735: Urban Systems Modeling Lec. 11 Loss and decisions overview of risk analysis and decision making 3 prior data and analysis probabilistic model risk

Lec. 1112735: Urban Systems Modeling Loss and decisions

12735: Urban Systems Modeling

instructor: Matteo Pozzi

1

Loss and decisions

Lec. 11


outline

2

‐ 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


overview of risk analysis and decision making

3

prior data and analysis

probabilistic model

risk analysis

decision making

observations

utility/loss theory

Bayesian updating

simulations, scenario analysis,

model selection

inspection scheduling,sensor placementValue of information


introduction

4

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

‐ Decision support systems

‐ Planning and active control

‐ Artificial intelligence (do the right thing!)

‐ Game theory.


deterministic examples

5

Consider the management of a structure.

‐ two actions: DoNothing , Repair

‐ consequences are in term of cost . Costs are: 2K$ , 1K$ .

‐ objective is to minimize costs.

‐ optimal action and cost are ∗ argmin , ∗ min , ∗ →

∗ repair∗ 1K$

‐ many actions: →

‐ optimal action and cost are ∗ argmin ∗ min ∗

‐ continuous domain: →∗ argmin ∗ min ∗


decision under uncertainties

6

,

x

CA

suppose costs are uncertain, for each action.

expected cost:

if agent is risk‐neutral:∗ argmin ∗ min ∗

0 0.5 1 1.5 20

1

2

3

4

5

C [K$]

p(Ca

)

a1

a2

for each action, the corresponding cost is a distribution. How to compare distributions?

distributionvalue

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.


decision making under uncertainty: principles

7

lottery: : , ; , ; … ; , : outcome [cost], : its probability.

axioms: Orderability: ≻ ∨ ≻ ∨ ~Transitivity: ≻ , ≻ ⇒ ≻Continuity: ≻ ≻ ⇒ ∃ : , ; 1 , ~Monotonicity: ≻ ⇒ ⟺ , ; 1 , ≽ , ; 1 ,Substitutability: ~ ⇒ , ; 1 , ~ , ; 1 ,Decomposability:

, ; 1 , , ; 1 , ~ , ; 1 , ; 1 1 ,

1lotteries include sure things: 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:

∀lotteries , , ,probabilities ,


decision making under uncertainty: min. expected loss

8

1lotteries include sure things: 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≻ ⇔~ ⇔≽ ⇔

expected loss:the rational agent minimizes expected loss

function: ∙ depends on attitude towards risk.

lottery: : , ; , ; … ; , : outcome [cost], : its probability.


attitude towards risk

9

Expected loss:12 0

12 $1K

12

$0.5K

0 0$1K 1

$0.5K12 $1K : riskaverse: ≻

$0.5K12 $1K : riskneutral: ~

$0.5K12 $1K : riskseeking: ≻

0.5,0; 0.5, $1K1, $0.5Ksure thing

ra onal agent’s preferences → ∃ loss function

∀ 0, ∀ : function is equivalent to function .

agents:

sure thing rate of conversion

loss function is not unique

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

C [K$]

Loss

risk-seekingrisk-neutralrisk-averse


2 3 4 5 6 7 80

0.5

1

1.5

C

p C

C

LL = C2

0 0.05 0.1

10

20

30

40

50

60

pL

L

p(Ca1)

p(Ca2)

p(La1)

p(La2)

attitude toward risk and transformation of RVs

10

risk averse

0: ≻

~ln ,

~ln , 2


0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

L

p(L

a)

0 1 20

1

2

3

4

C [K$]

L

a1

a2

0 0.5 1 1.5 20

1

2

3

4

5

C [K$]

p(Ca

)

a1

a2

example of attitude toward risk

11

risk averse

0: ≻

two actions, same expected costs.

risk neutral: ~

0 0.5 1 1.5 20

2

4

6

8

10

L

p(L

a)

0 1 20

0.5

1

1.5

C [K$]

L

a1

a2

risk seeking

0: ≻


loss elicitation and loss related to death

12

loss function derives by agent’s preferences, and can be identified by posing questions to the agent.‐ assign loss 0 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 $20 in 1980 dollars, or under $50 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. For 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 Friedman, “Probabilistic Graphical Models Principles and Techniques”.


syntax of influence diagrams

13

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:


example of influence diagram

14

x1

x3

scenario

x4

x2

x6

x5

x8x7

material

strength

damagedemand

stiffness

load

stress

LA

action losschance decision utility/loss

joint probability:

prediction – conditional prediction

predicted loss, for each action

optimal action and loss

conditional optimal action and loss

task:

Set of random variables, defined by conditional (in)dependence.


simplest maintenance problem

15

LA

S

state

action loss

,U F

N 0

RN: do NothingR: Repair

F: FailureU: Undamaged

∗ min F ,

→ |R not accepting the risk

prior optimal loss

accepting the risk→ |N F

pay‐off (cost) matrix

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

PF

E[L

]

RepairDo NothingOptimal

RepairDN

/ : optimal threshold

D

Do Nothing

Repair


with perfect information

16

LA

S

state

action loss

,U F

N 0

R

Do Nothing

Repair

N: do NothingR: Repair


∗|obs. F

→ |R, D

[failure always avoided]

→ |N, U 0


0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

PF

E[L

]


RepairDN


Undam.

Damagedobser. state

D

loss with PI

∗ ∗|obs. min F , 1 F

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

PF

E[L

]

RepairDo NothingOptimalOpt. Obs.

D

expected value of perfect information

0 0.2 0.4 0.6 0.8 1012

PFE

VP

I


with perfect information: example

17

LA

S

state

action loss

,U F

N 0 10

R 2 2

Do Nothing

Repair

N: do NothingR: Repair


∗|obs. 20.3 0.6

→ |R, D 2

→ |N, U 0


0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

PF

E[L

]


RepairDN


Undam.

Damagedobser. state

D

loss with PI

∗ ∗|obs. 2 0.6 1.4

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

PF

E[L

]

RepairDo NothingOptimalOpt. Obs.

D


D 30%

30%

∗ |R 2|N 3


with imperfect information

18

LA

state

action loss

NR

S YU F

S S|U S|F

A A|U A|F

|

S : SilenceA : Alarm S|F :prob. of False Silence

A|U : prob. of False Alarmsensor outcome

FR

U F

S 1

A 1

U F

S 1 0

A 0 1

perfect sensor

0 0

A: |N, A F A|R, A

S: |N, S F S|R, S

∗|A min F A , ∗|S min F S ,

optimal decision is affected by sensor outcomes:prob. of measures Bayes’ formula


with continuous variables

19

cd

Sstate

demand capacity

0.58%

0 5 10 150

0.1

0.2

0.3

0.4

d , c

p

p(d)p(c)

/

1021.3

lnln


0 5 10 150

0.1

0.2

0.3

0.4

d , c

p

p(d)p(c)

0 5 10 150

0.1

0.2

0.3

0.4

d , c

p

p(d)p(c)p(yc)


20

cd

Sstate

demand capacity

y

y

0.58%

/

~ln 0,


0 5 10 150

0.1

0.2

0.3

0.4

d , c

p

p(d)p(c)p(yc)p(cy)


21

LA

cd

S

y

state

demand

action loss

capacity

| 1.55%Do NothingRepair

y

0.58%

10-1 100 1010

2

4

c

VoI

[K$]

sensor precision

perfect information no info.

Value of Information

/

Bayes’ rule: ∝ ∙

,U F

N 0

R

~ln 0,

$1M$10K


another example: rehabilitarion level

L

R M

X

C1

structural Condition

R Action: select a nominal value of refrofitting

U F0 CF

Cost related to failure

load

initial Resistance Measure

Rr

Resistance after retrofitting

C2

C

Cost for rehabilitationC2 = R·

C = C2 + C1

if L>Rrif Rr≥L

F: Damaged U: Undamaged

Rr = R + R + er

22


selecting the rehabilitation level

0

10

20

3050

100

150200

0

200

400

600

800

1000

R [KN]nominal R [KN]

C*

[K$]

2 510

20

20

50

50

100

100100200

200200

400

400

400

600

600

600

800

800

8001000

R [KN]

R [K

N]

C [K$]

0 5 10 15 20 25 30

60

80

100

120

140

160

180

200

strongstructure

weakstructureweak

rehabilitation

strongrehabilitation

23



25

10

20

20

50

50

100

100100

200

200200

400

400

400

600

600

600

800

800

8001000

R [KN]

R [K

N]

C [K$]

0 5 10 15 20 25 30

60

80

100

120

140

160

180

20000.050.1

60

80

100

120

140

160

180

200

PDF

R [K

N]

prior

24



25

10

20

20

50

50

100

100100

200

200200

400

400

400

600

600

600

800

800

8001000

R [KN]

R [K

N]

C [K$]

0 5 10 15 20 25 30

60

80

100

120

140

160

180

20000.050.1

60

80

100

120

140

160

180

200

PDF

R [K

N]

prior

0 5 10 15 20 25 300

20

40

60

80

100

C*

[K€]

R [KN]

full costrehabilitation cost

cost related to riskoptimal action

25



2 510

2 0

20

50

50

100

100100200

200200

400

400

400

600

600

600

800

800

8001000

R [KN]

R [K

N]

C [K$]

0 5 10 15 20 25 30

60

80

100

120

140

160

180

20000.050.1

60

80

100

120

140

160

180

200

PDF

R [K

N]

prior

0 5 10 15 20 25 300

20

40

60

80

100

C*

[K€]

R [KN]

M

26



25

10

20

20

50

50

100

100100200

200200

400

400

400

600

600

600

800

800

8001000

R [KN]

R [K

N]

C [K$]

0 5 10 15 20 25 30

60

80

100

120

140

160

180

20000.050.1

60

80

100

120

140

160

180

200

PDF

R [K

N]

0 5 10 15 20 25 300

20

40

60

80

100

C*

[K€]

R [KN]

priorposterior

M

27



2 510

2 0

20

50

50

100

100100200

200200

400

400

400

600

600

600

800

800

8001000

R [KN]

R [K

N]

C [K$]

0 5 10 15 20 25 30

60

80

100

120

140

160

180

20000.050.1

60

80

100

120

140

160

180

200

PDF

R [K

N]

0 5 10 15 20 25 300

20

40

60

80

100

C*

[K€]

R [KN]

priorposterior

optimal action

M

28


link between decision making and probability of failure

29

,

U F

N 0 LF

R LR LR

F

x1

x 2

0 2 4 6 8 100

2

4

6

8

10

P F

limit state functionjoint probability

compute

S F

S U

probability of failure

,1 0

expected loss doing nothing

reliability problem:



30

,∗ argmin ∗ min ∗

∗ argmin , ∗ min , ∗ ,

∗ ∗∗ ∗

min , min , 0 ∀ : ∗ → 0

,

without observing observing

Jensen's inequality: ∀convexfunction :


value of information

31

, , ,| ,

,∗ argmin ∗ min ∗

∗ argmin , ∗ min , ∗ ,

∗ ∗∗ ∗

min , min | ,

min | , min | , min , min ,

without observing observing

∀ : ∗ → 0


with imperfect information: recap

32

LA

state

action loss

NR

S YU F

S S|U S|F

A A|U A|F

|

S : SilenceA : Alarm S|F :prob. of False Silence

A|U : prob. of False Alarmsensor outcome

FR

U F

S 1

A 1

A: S:∗|A min F A , ∗|S min F S ,

optimal decision is affected by sensor outcomes:

∗|obs. ∗|A A ∗|S S

∗ ∗|obs.value of information:


parametrical study

33

U F

S 1

A 1

00.1

0.20.3

0.40.5

00.1

0.20.3

0.40.5

0

0.5

1

1.5

2

2.5

3

PFAPFS

C* te

st

C*PI : expected cost adopting a perfect sensor

C*: expected cost w/o sensor


00.1

0.20.3

0.40.5

00.1

0.20.3

0.40.5

0

0.5

1

1.5

2

2.5

3

PFAPFS

C* te

st

00.1

0.20.3

0.40.5

00.1

0.20.3

0.40.5

0

0.5

1

1.5

2

2.5

3

PFAPFS

VoI

parametrical study

xy

z z

x y

VoI = 0

VoIPI

34

C*PI : expected cost adopting a perfect sensor



0.5

0.5

1

1

1

1.5

1.5

1.5

1.5

2

2

2

2

2.5

2.5

2.5

2.5

PFA

PFS

Costs [K$], CR=2.5K$, CF=1000K$, P(F)=0.5%

C*truth=12.5$+

C*=2.5K$+

xs1

xs2

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

area with null‐VoI

+

PFA = 0.20PFS = 0.45

A→ RS→ ?

P(F|S) = 0.28% (posterior prob. is lower than prior)

CF∙P(F|S) = $2.8K > CR (but the risk is still unbearable)

null‐VoI area

R

35

00.1

0.20.3

0.40.5

00.1

0.20.3

0.40.5

0

0.5

1

1.5

2

2.5

3

PFAPFS

C* te

st



the role of the economical framework

0.5 1

1

1.5

1.5

2

2

2

2.5

2.5

2.53

3

3

3.5

3.5

3.5

4

4

4

4.5

4.5

4.5

5

5

5

5

Costs [K$], CR=10K$, CF=1000K$, P(D)=0.5%

PFA

PFS

C*truth=50$+

C*=5K$+

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.5

0.5

1

1

1

1.5

1.5

1.5

1.5

2

2

2

2

2.5

2.5

2.5

2.5

Costs [K$], CR=2.5K$, CF=1000K$, P(D)=0.5%

PFA

PFS

C*truth=12.5$+

C*=2.5K$+

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

CR = 10K$ CR = 2.5K$

s2

s1

s2

s1

VoI(s1) < VoI(s2) VoI(s1) > VoI(s2)sensor (s2) is better then sensor (s1) sensor (s1) is better then sensor (s2)

any metric ranking the sensors without considering the economical issues is inappropriate.

36


VoI vs cost of failure

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

100

CF [K$]

VoI

[K$]

CR = 100 K$, P(D) = 5%

PFA = 0% , PFS = 0%

PFA = 30% , PFS = 0%

PFA = 30% , PFS = 30%

37



0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

100

CF [K$]

VoI

[K$]

CR = 100 K$, P(D) = 5%

PFA = 0% , PFS = 0%

PFA = 30% , PFS = 0%

PFA = 30% , PFS = 30%

null-VoI range

false alarm, but not false silence

38



0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

100

CF [K$]

VoI

[K$]

CR = 100 K$, P(D) = 5%

PFA = 0% , PFS = 0%

PFA = 30% , PFS = 0%

PFA = 30% , PFS = 30%

false alarm, but not false silence

null-VoI range null-VoI range

39


numerical example

20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F

load SR1, PF=0.96% Cott=9.6K$

CF = 1’000 K$ CR = 10 K$

P(F) = 0.96% N: do Nothing, C* = 9.6 K$

40


numerical example

20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F

10-1 100 101 102

5

6

7

8

9

10

11

sensor precision [KN]

Opt

imum

Cos

t [K

$]


perfect sensormeasuring R no sensor

CF = 1’000 K$ CR = 10 K$

P(F) = 0.96% N: do Nothing, C* = 9.6 K$

e)

41


numerical example

20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F

10-1 100 101 102

5

6

7

8

9

10

11


Opt

imum

Cos

t [K

$]

10-1 100 101 1020

1

2

3

4

5


VoI

[K$]


perfect information

no information

CF = 1’000 K$ CR = 10 K$

P(F) = 0.96% N: do Nothing, C* = 9.6 K$

cost of a sensor

suitable sensors

best sensor

e)

perfect sensormeasuring R no sensor

42


varying the prior uncertainty

20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F


R2, PF=1.2% Cott=10K$

10-1 100 101 102

5

6

7

8

9

10

11


Opt

imum

Cos

t [K

$]

10-1 100 101 1020

1

2

3

4

5


VoI

[K$]

CF = 1’000 K$ CR = 10 K$

P(F) = 1.20% R: repair, C* = 10 K$

e)

43



20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F

10-1 100 101 102

5

6

7

8

9

10

11


Opt

imum

Cos

t [K

$]

10-1 100 101 1020

1

2

3

4

5


VoI

[K$]


R2, PF=1.2% Cott=10K$

R3, PF=1.5% Cott=10K$

CF = 1’000 K$ CR = 10 K$

P(F) = 1.50% R: repair, C* = 10 K$

e)

44



20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F

10-1 100 101 102

5

6

7

8

9

10

11


Opt

imum

Cos

t [K

$]

10-1 100 101 1020

1

2

3

4

5


VoI

[K$]


R2, PF=1.2% Cott=10K$

R3, PF=1.5% Cott=10K$

R4, PF=2% Cott=10K$

CF = 1’000 K$ CR = 10 K$

P(F) = 2.05% R: repair, C* = 10 K$

e)

45


varying the prior resistance

20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F

load SR1, PF=4.8% Cott=10K$

10-1 100 101 102

0

2

4

6

8

10

12


Opt

imum

Cos

t [K

$]

10-1 100 101 1020

1

2

3

4

5

6


VoI

[K$]

CF = 1’000 K$ CR = 10 K$

P(F) = 4.81% R: repair, C* = 10 K$

e)

46



20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F

10-1 100 101 102

0

2

4

6

8

10

12


Opt

imum

Cos

t [K

$]

10-1 100 101 1020

1

2

3

4

5

6


VoI

[K$]


R2, PF=1.3% Cott=10K$

CF = 1’000 K$ CR = 10 K$

P(F) = 1.28% R: repair, C* = 10 K$

e)

47



20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F

10-1 100 101 102

0

2

4

6

8

10

12


Opt

imum

Cos

t [K

$]

10-1 100 101 1020

1

2

3

4

5

6


VoI

[K$]


R2, PF=1.3% Cott=10K$

R3, PF=0.63% Cott=6.3K$

CF = 1’000 K$ CR = 10 K$

P(F) = 0.63% N: do Nothing, C* = 6.3 K$

e)

48



20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

[KN]

PD

F

10-1 100 101 102

0

2

4

6

8

10

12


Opt

imum

Cos

t [K

$]

10-1 100 101 1020

1

2

3

4

5

6


VoI

[K$]


R2, PF=1.3% Cott=10K$

R3, PF=0.63% Cott=6.3K$

R4, PF=0.16% Cott=1.6K$

CF = 1’000 K$ CR = 10 K$

P(F) = 0.16% N: do Nothing, C* = 1.6 K$

e)

49


a linear log‐normal model

i

seismic intensity

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

seismic intentesity, i [m/s2]

p(i)

10-2

10-1

100

101

1020

0.05

0.1

0.15

0.2

0.25

0.3

0.35


p(i)

log space

log normal

normal

longitudinal section cross section

two-span ordinary bridge

50



i

seismic intensity

structural response

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35


p(i)

F

1

k

0 L

FL

∙∙

Equal displacement approximation:

log log log

2

cov: 30%

51



100

101

102

103

0

0.2

0.4

0.6

0.8

1

[mm]

P (

s s

i I

)

sII=cracking

sIII=yielding

sIV=spalling

sV=ultimate

i

seismic intensity

structural response damage

fragility curves

52



L

A

i

seismic intensity


loss

loss matrix I II III IV V 10

0

101

102

103

104

d

L(A,

d)

[K

$]

DNRTCL

available actions:Do Nothing

Reduce operation Close the structure

53



L

A

i

seismic intensity


loss

measure of seismic intensity

usgs shakemap



L

A

i

seismic intensity


loss


yo

measure of monitoring system

bridge cross section

a1

a2



L

A

i

seismic intensity


loss


yo


yd

visual inspection



L

A

i

seismic intensity


loss


yo


yd

visual inspection

task:evaluating the benefit of getting yo, given yi, depending on the availability of yd.

57


visual inspection

L

A

i

yo yd

58


i

o

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

the value of information (VoI)

A

LVI

yeq

10-2

10-1

100

101

102

100

110

120

130

140

150

E[L

*]

[K$]

w/o VIwith VIP.I. on d

eq

equivalent precision

displ.

59

intensity


i

o

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

the value of information (VoI)

A

LVI

yeq

10-2

10-1

100

101

102

-10

0

10

20

30

40

50

eq

VoI

[K

$]

10-2

10-1

100

101

102

100

110

120

130

140

150

E[L

*]

[K$]

w/o VIwith VIP.I. on d

i = 40%

o = 5%

eq 54% 5%

equivalent precision

intensity

displ.

60


spatially distributed infrastructure system

61

D CF

D CF

D CF D C

F

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

demand capacity

condition statefailure iff demand > capacity


spatially distributed infrastructure system

62

D CF

D CF

D CF D C

F

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

assume we can locally measure demand , capacity . we can infer consequences on the entire system.

D C


spatially distributed phenomena

63

Gaussian process: ; ,example: ambient temperature field

A. Krause. “Optimizing Sensing – Theory and Applications”, Ph.D. Thesis, CMU, 2008.



64


processing information:local measurements update the field in the surrounding area



0 5 10 1555

60

65

70

75

80

85

Position (m)

Tem

pera

ture

(F)


65





0 5 10 1555

60

65

70

75

80

85

Position (m)

Tem

pera

ture

(F)

optimal sensor placement

66



entropy : total uncertainty in the field.: residual uncertainty.



0 5 10 1555

60

65

70

75

80

85

Position (m)

Tem

pera

ture

(F)


67



optimal sensor placement:minimize subject to # sens. = N

1




0 5 10 1555

60

65

70

75

80

85

Position (m)

Tem

pera

ture

(F)

Greedy Placement

0 5 10 1555

60

65

70

75

80

85

Position (m)

Tem

pera

ture

(F)

Exact Placement


68




combinatorial explosion:candidate location: 100# of sensors: 10# of configurations: 1.7 10 17T

greedy algorithm:‐ select best location,‐ add best matching location…

2



0 5 10 1555

60

65

70

75

80

85

Position (m)

Tem

pera

ture

(F)

Greedy Placement

0 5 10 1555

60

65

70

75

80

85

Position (m)

Tem

pera

ture

(F)

Exact Placement


69






3



0 5 10 1555

60

65

70

75

80

85

Position (m)

Tem

pera

ture

(F)

Exact Placement

0 5 10 1555

60

65

70

75

80

85

Position (m)

Tem

pera

ture

(F)

Greedy Placement


70






4



effects on monitoring the system

71

close proximity

D CF

D CF

D CF D C

F

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

demand can be similar in close proximityD


effects on monitoring the system

72

close proximity

D CF

D CF

D CF D C

F

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

D CF

similar type

demand can be similar in close proximitycapacity similar for similar components

DC


Application to Infrastructure Systems

73

…

Failure ProbabilityP [F1=0]

entropyregret

Failure ProbabilityP [F2=0]

entropy

regret

Failure ProbabilityP [Fn=0]

entropyregret

G1

D1 C1F1

D2 C2F2

Dn CnFn

G2

Gn

system metric : approximation: decision making problems are independent for each component


example of seismic risk: overview

74

fault line

epicenter

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

Pea

k G

roun

d A

ccel

erat

ion

(g)

Distance from Epicenter (km)

nearby locations: highlycorrelated accelerations

more distant locations: less correlated

example of seismic demand (PGA)

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

Strucural Demand (PGA)

Pro

babi

lity

of F

ailu

re

example fragility curve

T

0

ground acceleration

structural response



75

epicentersseismic scenarios based on the relative probabilities of earthquakes along different faults (USGS).

Demand Measurements

Capacity Measurements

Ground Acceleration

Maximum tolerable displacement

Seismometer readings

Structural assessment

measures



76

infrastructure system infrastructure components in the San Francisco, including18 bridges and 9 tunnels.Divided among 13 infrastructure types,e.g. steel suspension bridges, concrete girder bridges, and cut‐and‐cover tunnels.


example of seismic risk: results

77

entropy metric10 measurements of demand or capacity.

results:demand measurements are clustered in the upper Bay area, where the density of components is highest. Capacity measurements are spread evenly among different asset types.


example of seismic risk: results

78

value of information10 measurements of demand or capacity.

results:some similarity with entropy (3 sensor placements out of 10), but components with higher costs are more closely monitored, reflecting the relevant economic factors.


density of Value of Information

79

densityVoI comes from possible different scenarios: this map shows how it is related to epicenter location.

Total VoIabout $400M


references

80

Barber, B. (2012). Bayesian Reasoning and Machine Learning. Cambridge UP. Downloadable from http://web4.cs.ucl.ac.uk/staff/D.Barber/pmwiki/pmwiki.php?n=Brml.HomePage

Bishop, C. (2006). Pattern Recognition and Machine Learning. Springer.

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models Principles and Techniques. the MIT Press.

Russell, S. and P. Norvig. (2010). Artificial Intelligence: A Modern Approach. Pearson Education.

Documents

Loss and decisions - CMU · 12735: Urban Systems Modeling Lec. 11 Loss and decisions overview of risk analysis and decision making 3 prior data and analysis probabilistic model risk