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Beyond ProbabilityA pragmatic approach to uncertainty quantification in engineering
Scott Ferson, Applied Biomathematics, [email protected]
NASA Statistical Engineering Symposium, Williamsburg, Virginia, 4 May 2011
2010 Applied Biomathematics
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Wishful thinking
Using inputs or models because they are
convenient, or because you hope theyre true
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Kansai International Airport
30 km from Kobe in Osaka Bay
Artificial island made with fill
Engineers told planners itd sink [6, 8] m
Planners elected to design for 6 m
Its sunk 9 m so far and is still sinking
(The operator of the airport denies these media reports)
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Variability = aleatory uncertainty
Arises from natural stochasticity
Variability arises fromspatial variation
temporal fluctuations
manufacturing or genetic differences
Not reducible by empirical effort
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Incertitude = epistemic uncertainty
Arises from incomplete knowledge
Incertitude arises fromlimited sample size
mensurational limits (measurement error)
use of surrogate data
Reducible with empirical effort
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Suppose
A is in [2, 4]
B is in [3, 5]
What can be said about the sumA+B?
4 6 8 10
The right answer for
engineering is [5,9]
Propagating incertitude
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They must be treated differently
Variability should be modeled as randomness
with the methods of probability theory
Incertitude should be modeled as ignorance
with the methods of interval analysis
Imprecise probabilities can do both at once
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Incertitude is common in engineering
Periodic observationsWhen did the fish in my aquarium die during the night?
Plus-or-minus measurement uncertaintiesCoarse measurements, measurements from digital readouts
Non-detects and data censoringChemical detection limits, studies prematurely terminated
Privacy requirementsEpidemiological or medical information, census data
Theoretical constraintsConcentrations, solubilities, probabilities, survival rates
Bounding studiesPresumed or hypothetical limits in what-if calculations
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Wishful thinking
Pretending you know the
Value
Distribution function
Dependence
Model
when you dont is wishful thinking
Uncertainty analysis makes a prudent analysis
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Wishfulthinking
Prudentanalysis
Failure
Success
Dumb luck
NegligenceHonorable
failure
Good
engineering
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Traditional uncertainty analyses
Interval analysis
Taylor series approximations (delta method) Normal theory propagation (ISO/NIST)
Monte Carlo simulation
Stochastic PDEs
Two-dimensional Monte Carlo
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Untenable assumptions
Uncertainties are small
Distribution shapes are known Sources of variation are independent
Uncertainties cancel each other out
Linearized models good enough
Underlying physics is known and modeled
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Need ways to relax assumptions
Hard to say what the distribution is precisely
Non-independent, orunknown dependencies Uncertainties that may not cancel
Possibly large uncertainties
Model uncertainty
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Probability bounds analysis (PBA)
Sidesteps the major criticisms
Doesnt force you to make any assumptionsCan use only whatever information is available
Bridges worst case and probabilistic analysis
Distinguishes variability and incertitude
Acceptable to both Bayesians and frequentists
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Probability box (p-box)
0
1
1.0 2.0 3.00.0 X
Cumulativeprobability
Interval bounds on a cumulative distribution function
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Uncertain numbers
Nota uniform
distribution
Cumulativeprobability
0 10 20 30 400
1
10 20 30 400
1
10 20 300
1
Probability
distribution
Probability
box Interval
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Uncertainty arithmetic
We can do math on p-boxes
When inputs are distributions, the answersconform with probability theory
When inputs are intervals, the results agreewith interval (worst case) analysis
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Calculations
All standard mathematical operations Arithmetic (+, , , , ^, min, max)
Transformations (exp, ln, sin, tan, abs, sqrt, etc.)
Magnitude comparisons (, , )
Other operations (nonlinear ODEs, finite-element methods)
Faster than Monte Carlo
Guaranteed to bound the answer
Optimal solutions often easy to compute
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Probability bounds analysis
Special case of imprecise probabilities
Addresses many problems in risk analysisInput distributions unknown
Imperfectly known correlation and dependency
Large measurement error, censoringSmall sample sizes
Model uncertainty
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Better than sensitivity analysis
Unknown distribution is hard for sensitivity
analysis since infinite-dimensional problem
Analysts usually fall back on a maximum
entropy approach, which erases uncertainty
rather than propagates it
Bounding seems very reasonable, so long as
it reflects all available information
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Example: uncontrolled fire
F=A &B & C&D
Probability of ignition source
Probability of abundant fuel presence
Probability fire detection not timely
Probability of suppression system failure
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Imperfect information
CalculateA&B&C&D, with partial information:
As distribution is known, but not itsparameters
Bs parameters known, but not its shape
C has a small empirical data set
D is known to be aprecise distribution
Bounds assuming independence?
Without any assumption about dependence?
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A = {lognormal, mean = [.05,.06], variance = [.0001,.001])
B = {min = 0, max = 0.05, mode = 0.03}
C= {sample data = 0.2, 0.5, 0.6, 0.7, 0.75, 0.8}
D = uniform(0, 1)
0 0.1 0.2 0.30
1
A
0 0.02 0.04 0.060
1
B
0 1
0
1
D
0 1
0
1
CCDF
CDF
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Resulting answers
0 0.02 0.04 0.06
0
10 0.01 0.02
0
1
Cumulativeprobability
Cumulativepro
bability
All variables
independent
Make no assumption
about dependencies
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Summary statisticsIndependent
Range [0, 0.011]
Median [0, 0.00113]
Mean [0.00006, 0.00119]
Variance [2.9 10 9, 2.1 10 6]
Standard deviation [0.000054, 0.0014]
No assumptions about dependence
Range [0, 0.05]Median [0, 0.04]
Mean [0, 0.04]
Variance [0, 0.00052]
Standard deviation [0, 0.023]
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How to use the results
When uncertainty makes no difference(because results are so clear), bounding gives
confidence in the reliability of the decision
When uncertainty swamps the decision
(i) use other criteria within probability bounds, or
(ii) use results to identify inputs to study better
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Justifying further empirical effort
If incertitude is too wide for decisions, and
bounds are best possible, more data is needed
Strong argument for collecting more data
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Advantages
Computationally efficient
No simulation or parallel calculations needed
Fewer assumptions
Not just different assumptions,fewerof them
Distribution-free probabilistic risk analysis
Rigorous results
Built-in quality assurance
Automatically verified calculation
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Disadvantages
P-boxes dont say what outcome is most likely
Hard to get optimal bounds on non-tail risks
Some technical limits (e.g., sensitive to
repeated variables, tricky with black boxes)
A p-box may not express the tightest possible
bounds given all available information
(although it often will)
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Software
UC add-in for Excel (NASA, beta 2011)
RAMAS Risk Calc 4.0 (NIH, commercial)
Statool (Dan Berleant, freeware)
Constructor (Sandia and NIH, freeware)
Pbox.r library for R
PBDemo (freeware)
Williamson and Downs (1990)
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Diverse applications
Superfund risk analyses
Conservation biology extinction/reintroduction
Occupational exposure assessment
Food safety
Chemostat dynamics
Global climate change forecasts
Safety of engineered systems
Engineering design
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Case study:
Spacecraft design undermission uncertainty
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Mission
Deploy satellite carrying a large optical sensor
W
L
D
X Y
Z
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Wertz and Larson (1999) Space Mission Analysis and Design (SMAD). Kluwer.
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Attitude control
Command data systems
Configuration
CostGround systems
Instruments
Mission design
Power
Program management
Propulsion
Science
Solar arraySystems engineering
TelecommunicationsSystem
TelecommunicationsHardware
Thermal control
Typical subsystems
Attitude control
Command data systems
Configuration
CostGround systems
Instruments
Mission design
Power
Program management
Propulsion
Science
Solar arraySystems engineering
TelecommunicationsSystem
TelecommunicationsHardware
Thermal control
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Demonstrations
Calculations within a single subsystem (ACS)
Calculations within linked subsystems
Attitudecontrol
Power
Solararray
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Attitude control subsystem (ACS)
3 reaction wheels Design problem: solve forh
Required angular momentum
Needed to choose reaction wheels
Mission constraints
torbit = 1/4 orbit time
slew = max slew angle
tslew
= min maneuver time
Inputs from other subsystems
I,Imax,Imin = inertial moment Depend on solar panel size, which
depends on power needed, so on h
h tot torbit
slew
4 slew
tslew2
I
tot slew dist
dist g sp m a
g
3
2 RE H3
Imax Imin sin 2
sp LspFs
cAs 1 q cos i
m
2MD
RE H3
a
1
2La CdAV2
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Attitude control input variablesSymbol Unit Variable Type Value SMAD
Cd unitless Drag coefficient p-box range=[2,4]
mean=3.13
3.13
La m Aerodynamic drag torque moment p-box range=[0,3.75]
mean=0.25
0.25
Lsp m Solar radiation torque moment p-box range=[0,3.75]
mean=[0.25]
0.25
Dr A m2 Residual dipole interval [0,1] 1
i degrees Sun incidence angle interval [0,90] 0
kg m3
Atmospheric density interval [3.96e-12,9.9e-11] 1.98e-11
degrees Major moment axis deviation from nadir interval [10,19] 10
q unitless Surface reflectivity interval [0.1,0.99] 0.6
Imin kg m2 Minimum moment of inertia interval [4655] 4655
Imax kg m2 Maximum moment of inertia interval [7315] 7315
m3 s-2 Earth gravity constant point 3.98e14 3.98e14
A m2 Area in the direction of flight point 3.752 3.752
RE km Earth radius point 6378.14 6378.14H km Orbit altitude point 340 340
Fs W m-2 Average solar flux point 1367 1367
slew degrees Maximum slewing angle point 38 38
c m s-1 Light speed point 2.9979e8 2.9979e8
M A m2 Earth magnetic moment point 7.96e22 7.96e22
tslew s Minimum maneuver time point 760 760
As m2
Area reflecting solar radiation point 3.752
3.752
torbit s Quarter orbit period point 1370 1370
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Coefficient of drag, Cd
Cd (unitless)
1 2 3 4 50
1SMAD
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Aerodynamic drag torque moment,La
La (m)
-1 0 1 2 3 40
1SMAD
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Required angular momentum h
0.02 0.030
0.5
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.5
1
0 200 400 600 8000
0.5
1
distsleworbitth
4655torbit (sec)
1370
= ( + )
h (N m sec) slew (N m) dist (N m)
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Value of information: pinching
Initial result
After pinching(atmospheric density)
(kg m-3
) h (N m sec)
0 200 400 600 800 10000
1
0 10 100
1
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Three linked subsystems
Attitudecontrol
Power
Solararray
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Variables passed iteratively
Minimum moment of inertiaImin
Maximum moment of inertiaImax
Total torque tot
Total powerPtot
Solar panel areaAsa
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Analysis of calculations
Need to check that original SMAD values and all
Monte Carlo simulations are enclosed by p-boxes
Need to ensure iteration through links doesntcause runaway uncertainty growth (or reduction)
Four parallel analyses SMADspoint estimates
Monte Carlo simulation
P-boxes but without linkage among subsystems
P-boxes with fully linked subsystems
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Supports of results
tot (N m)0.0 0.2 0.4 0.6 0.8 1.0
Ptot (W)
1000 1500 2000
Asa
(m2)0 10 20 30 40 50 60
Probability bounds
PBA but unlinkedMonte Carlo simulation
SMADpoint estimates
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Case study findings
Different answers are consistent Point estimates match the SMAD results
P-boxes span the points and the Monte Carlo intervals
Calculations workable No runaway inflation (or loss) of uncertainty
Easier than with Monte Carlo
Practical and interesting results Uncertainty can affect engineering decisions
Reducing uncertainty about (by picking a launch date)
strongly reduces design uncertainty
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Accounting for epistemic and aleatory
uncertainty in early system design
NASA SBIR Phase 2 Final Report
Applied Biomathematics
100 North Country Road
Setauket, NY 11733
Phone: 1-631-751-4350
Fax: 1-631-751-3435
Email: [email protected]
www.ramas.com
Order Number: NNL07AA06C
July 2009
For more information, consult
SBIR project report to NASA,
July 2009
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Uses for probability bounds analysis
Uncertainty propagation
Risk assessment
Sensitivity analysis (for control and study)
Reliability theory
Engineering design
Validation
Decision theory Regulatory compliance
Finite element modeling
Differential equations
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Differential equations
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Uncertainty usually explodes
Time
x
The explosion can be traced
to numerical instabilities
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Uncertainty
Artifactual uncertainty
Too few polynomial terms
Numerical instabilityCan be reduced by a better analysis
Authentic uncertainty
Genuine unpredictability due to input uncertainty
Cannot be reduced by a better analysis
Only by more information, data or assumptions
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Uncertainty propagation
We wantthe prediction to break down if
thats what should happen
But we dont want artifactual uncertainty
Numerical instabilities
Wrapping effectDependence problem
Repeated parameters
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Problem
Nonlinear ordinary differential equation (ODE)
dx/dt=f(x, )
with uncertain and uncertain initial statex0
Information about andx0 comes asInterval ranges
Probability distributions
Probability boxes
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Model
Initial states (bounds)
Parameters (bounds)
VSPODEMark Stadherr et al. (Notre Dame)
List of constants
plus remainder
Taylor models
Interval Taylor series
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Example ODE
dx1/dt= 1x1(1x2)
dx2/dt= 2x2(x11)
What are the states at t= 10?
x0 = (1.2, 1.1)T
1 [2.99, 3.01]
2 [0.99, 1.01]
VSPODE
Constant step size h = 0.1, Order of Taylor model q = 5,
Order of interval Taylor series k= 17, QR factorization
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VSPODE tells how to computex1
1.916037656181642 10
21 + 0.689979149231081 1
120 +
-4.690741189299572 10
22 + -2.275734193378134 1
121 +
-0.450416914564394 12
20 + -29.788252573360062 1
023 +
-35.200757076497972 11
2
2
+ -12.401600707197074 12
2
1
+-1.349694561113611 13
20 + 6.062509834147210 1
024 +
-29.503128650484253 11
23 + -25.744336555602068 1
222 +
-5.563350070358247 13
21 + -0.222000132892585 1
420 +
218.607042326120308 10
25 + 390.260443722081675 1
124 +
256.315067368131281 12 23 + 86.029720297509172 13 22 +15.322357274648443 1
421 + 1.094676837431721 1
520 +
[ 1.1477537620811058, 1.1477539164945061 ]
where s are centered forms of the parameters; 1 = 1 3, 2 = 2 1
Input p boxes
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1 2
p-box
0
1
Pro
bability
0.99 1 1.012.99 3 3.01
precise
Input p-boxes
R lt
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Results
Probability
1.12 1.14 1.16 1.180
1
0.87 0.88 0.89 0.9
x1 x2
p-box
precise
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Still repeated uncertainties
1.916037656181642 10
21 + 0.689979149231081 1
120 +
-4.690741189299572 10
22 + -2.275734193378134 1
121 +
-0.450416914564394 12
20 + -29.788252573360062 1
023 +
-35.2007570764979721
1
2
2 + -12.4016007071970741
2
2
1 +
-1.349694561113611 13
20 + 6.062509834147210 1
024 +
-29.503128650484253 11
23 + -25.744336555602068 1
222 +
-5.563350070358247 13
21 + -0.222000132892585 1
420 +
218.607042326120308 10
25 + 390.260443722081675 1
124 +
256.315067368131281 12 23 + 86.029720297509172 13 22 +15.322357274648443 1
421 + 1.094676837431721 1
520 +
[ 1.1477537620811058, 1.1477539164945061 ]
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Subinterval reconstitution
Subinterval reconstitution (SIR)
Partition the inputs into subintervals
Apply the function to each subintervalForm the union of the results
Still rigorous, but often tighter
The finer the partition, the tighter the unionMany strategies for partitioning
Apply to each cellin the Cartesian product
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Discretizations
2.99 3 3.01
0
1
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0.87 0.88 0.89 0.90
1
1.12 1.14 1.160
1
x1 x2
Contraction from SIR
Probability
Best possible bounds
reveal the authenticuncertainty
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Monte Carlo is more limited
Monte Carlo cannot propagate incertitude
Monte Carlo cannot produce validated results
Though can be checked by repeating simulation
Validated results from distributions can be obtained
by modeling inputs with (narrow) p-boxes and
applying probability bounds analysis
Results converge to narrow p-boxes obtained from
infinitely many Monte Carlo replications
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Results
Probability bounds analysis with VSPODEare useful for bounding solutions ofnonlinear ODEs
They rigorously propagate uncertainty
about in the form of
IntervalsDistributions
P-boxes
Initial states
Parameters
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Paper inAIChE Journal[American
Institute of Chemical Engineers],
February 2011 (on line May 2010)
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PBA relaxes assumptions
Everyone makes assumptions, but not all sets
of assumptions are equal:
Linear Normal Independence
Montonic Unimodal Known correlation
Any function Any distr ibution Any dependence
PBA doesnt require unwarranted assumptions
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Wishful thinking
Analysts often make convenient assumptions
that are not really justified:
1. Variables are independent of one another
2. Uniform distributions model gross incertitude
3. Distributions are stationary (unchanging)4. Distributions are perfectly precisely specified
5. Measurement uncertainty is negligible
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You dont have to think wishfully
A p-box can discharge a false assumption:
1. Dont have to assume any dependence at all
2. An interval can be a better model of incertitude
3. P-boxes can enclose non-stationary distributions4. Can handle imprecise specifications
5. Measurement data with plus-minus, censoring
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Rigorousness
Automatically verified calculations
The computations are guaranteed to enclosethe true results (so long as the inputs do)
You can still be wrong, but the methodwont be the reason if you are
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Take-home messages
Using bounding, you dont have to pretend
you know a lot to get quantitative results
Probability bounds analysis bridges worst
case and probabilistic analyses in a way thats
faithful to both and makes it suitable for use
in early design
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Acknowledgments
Larry Green, LaRC
Bill Oberkampf, Sandia
Chris Paredis, Georgia Tech
NASA
National Institutes of Health (NIH)Electric Power Research Institute (EPRI)
Sandia National Laboratories
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End
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