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15.023J / 12.848J / ESD.128J Global Climate Change: Economics, Science, and Policy
Spring 2008
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Methods
of
Uncertainty
Analysis
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Second level
Third level
Fourth level
Fifth level
15.023 Lecture
9 April 2008
Travis Franck
Eunjee Lee
(material from previous lectures by Mort Webster, Ian Sue Wing, Marcus Sarofim, and Travis Franck)
1
“Persons
to forecast
the
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future shall be considered disorderly
•under
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subdivision
3, styles
section 901 of
•the
Second
level code and liable to a fine
criminal
• Third level
of $250 and/or six months in
• Fourth level
prison.”
• Fifth level
Section 889, New York State Code of Criminal Procedure
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How
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Second
Understanding
level mean vs. median, standard
deviations,
Third level etc.
How tolevel
read PDFs (and make one from a
Fourth
histogram)
Fifth level
The role of uncertainty in climate change
science
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Elements
of
a
Decision
Tree
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Uncertainty Node: Shows different possible
outcomes
foredit
an uncertain
event,
and the respective
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probabilities for each outcome
• Second level
p
• Third level
• Fourth level
1-p
• Fifth Node:
level possible actions at a choice point
Decision
act
BAU
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Example: Biking in the Rain
Figure by MIT OpenCourseWare.
Decision
tree
without
Memory
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Period 1
Decision Outcome
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No connection between periods
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Decision Tree with Memory
Period 1
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Decision Outcome
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Depends on
the path
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Emissions
Scenarios
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Growth
AEEI
Shorthand for
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Emissions
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levelL
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Climate
Scenarios
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Growth
AEEI
Kv
S
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level
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Collapse into RR, HL, LH
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In Homework
Economic
Uncertainty
Faer
L
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Climate Uncertainty
Probability
distribution
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Handling
scientific
uncertainty
in AR4
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• In general, uncertainty ranges for results given in this
for
Policymakers
are 90%
uncertainty intervals
• Summary
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unless stated otherwise, that is, there is an estimated 5%
likelihood level
that the value could be above the range given in
• Second
square brackets and 5% likelihood that the value could be
below that
range. Best estimates are given where available.
• Third
level
Assessed uncertainty intervals are not always symmetric
about the level
corresponding best estimate. Note that a number
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of uncertainty ranges in the Working Group I TAR
corresponded
to 2 standard deviations (95%), often using
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level
expert judgment.
(IPCC, 2007: Summary for Policymakers. In: Climate Change 2007: The Physical Science Basis.
Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on
Climate Change)
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Why
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Uncertainty?
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• To
identify
important
and assumptions
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underlying disagreements
• Second level
• To know where more information is needed
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• To attach a range to model forecasts
•• To
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level attitudes toward risk
understand
•• To
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levelfor learning over time
account
(Attaching uncertainty to predictions might be as
natural as attaching cost/benefit to climate targets)
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ofMaster
Uncertainty
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Parametric Uncertainty
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– Uncertainty in the value of a quantity
Second level
Model or Structural Uncertainty
Third level
– Uncertainty in the form of a model
Fourth
level vs. Quadratic relationship
– e.g. Linear
Fifth
level
Surprise/Indeterminacy
– Don’t know what we don’t know
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Describing an Uncertainty Quantity
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∞
μ x = ∫ xf x ( x)dx = E[ x]
• Mean
−∞
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••
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∞
2
Variance
σ
=
(
x
−
μ
x
)
fx ( x)dx = Var[ x]
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∫−∞
Third level
σ x = Var (x)
Standard
Deviation
Fourth level
Covariance
cov( X 1 , X 2 ) = E [X 1 ∗ X 2 ] (if E[x]=0)
Fifth level
2
x
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Describing
an Uncertain
Quantity
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styleII
•• Mode:
value
(peak)
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• Median: Value of x such that
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– Prob (x<x0) = Prob (x>x0) = 0.50
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Fractile:
The p fractile is the value x0 such that
– Prob (x<x
0) = p
Fourth
level
Probability Density Function (pdf)
Fifth level
– The integral under a portion of the function is the
probability that the event will fall into that range.
• Cumulative Density Function (cdf)
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3
Probability Density Functions
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Median
Mean
1 Standard
Deviation
(67% bounds)
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2 Standard
Deviations
(95% bounds)
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Input
distributions
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Probability Density
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0.5
Probability Density
Probability Density (Values x 10-2)
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1.5
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Ocean Diffusivity (Kv) (cm 2/s)
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2
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Climate Sensitivity (°C)
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0
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Faer (Aerosol Forcing) (W/m 2)
Methods of Input Distribution Acquisition:
Expert Elicitation, Historical Analysis, Modeling Ranges
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1
Correlation
between
Distributions
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Measuring
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Uncertainty
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- Global
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Range
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– Varying
each x from low to high (tornado plots)
Second
level
Joint
Analysis
ThirdParametric
level
– Vary level
lows and highs for multiple variables
Fourth
Montelevel
Carlo Simulation
Fifth
– Use pdfs of inputs to produce pdfs of outputs
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Carlo
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Simulation
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Monte
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Second
Stratified
level
Sampling Methods
– Latinlevel
Hypercube Sampling
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Importance
Fourth
levelSampling Methods
Response
Fifth levelSurface Methods
– Probabilistic Collocation Method
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Simulation
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Example
1: Throwing
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Examplelevel
2:
Second
– Y = X1+ X2
• Third level
– X1 = N(50,20)
• Fourth
– X2 = level
N(40,25)
• Fifth
level Analysis: Y = N(90,32)
– Analytic
– Compare to Monte Carlo samples of size
n=10,100,1000,10000
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Monte Carlo, n=10,100
Figure by MIT OpenCourseWare.
Monte Carlo, n=1000,10000
Figure by MIT OpenCourseWare.
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Webster
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Business As Usual
Policy
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Second level Questions?
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