g 2 - Ari Rabl

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Transcript g 2 - Ari Rabl

Uncertainty of External Costs
TRADD, part 4
Ari Rabl, ARMINES/Ecole des Mines de Paris, November 2013
Major sources of uncertainty:
•Modeling of environmental pathways (atmospheric
models etc)
•Exposure-response functions
•Monetary valuation
Statistical analysis
Reference:
•Spadaro JV and Rabl A 2008. “Estimating the Uncertainty of Damage Costs of
Pollution: a Simple Transparent Method and Typical Results”. Environmental Impact
Assessment Review, vol. 28 (2), 166–183.
1
Sources of uncertainty
i) data uncertainty
e.g. slope of a dose-response function, cost of a day of restricted activity, and
deposition velocity of a pollutant;
ii) model uncertainty
e.g. assumptions about causal links between a pollutant and a health impact,
assumptions about form of a dose-response function (e.g. with or without
threshold), and choice of models for atmospheric dispersion and chemistry;
iii) uncertainty about policy and ethical choices
e.g. discount rate for intergenerational costs, and value of statistical life;
iv) uncertainty about the future
e.g. the potential for reducing crop losses by the development of more resistant
species;
v) idiosyncrasies of the analyst
e.g. interpretation of ambiguous or incomplete information, and human error.
The difficulties begin with trying to prepare this list: the distinction between
these sources is not always clear.
2
Appropriate analysis
For data and model uncertainties:
analysis by statistical methods, combining the component
uncertainties over the steps of the impact pathway, to obtain
formal confidence intervals around a central estimate.
For ethical choice, uncertainty about the future, and
subjective choices of the analyst:
sensitivity analysis, indicating how the results depend on
these choices and on the scenarios for the future.
For human error:
be careful and guard against overconfidence.
3
Difficulties
Quantifying the sources of uncertainty in this field is problematic
because of a general lack of information.
Usually one has to fall back on subjective judgment, preferably by
the experts of the respective disciplines.
The uncertainties due to strategic choices of the analyst, e.g. which
dose-response functions to include, are difficult to take into account
in a formal uncertainty analysis.
 the comprehensive uncertainties can be much larger than the
ones that have been quantified (uncertainties due to data and
parameters).
4
Uncertainty  variability
Don’t confuse uncertainty and variability of impacts!
Both can cause estimates to change,
but in very different ways and for totally different reasons:
Uncertainty: insufficient knowledge at the present time,
future estimates may be different when we know more.
Variability: damage cost can vary with the type of source
(where, ground level or tall stacks, …).
Damage cost per kWh are proportional to the emissions and vary
with the technologies used.
These variations are independent of the uncertainties.
5
Variability of damage with site
D/Duni
6
YOLL/1000 t
Porcheville (Paris)
10
Loire-sur-Rhone (Lyon)
5
Albi (Toulouse)
Martigues (Marseille)
4
Cordemais (Nantes)
3
Duni
5
2
1
0
0
50
100
150
Stack Height [m]
200
0
250
An example of dependence on site and on height of source for a primary pollutant: damage D
from SO2 emissions with linear ER function, for five sites in France, in units of Duni for uniform
world model Eq.10 with  = 105 persons/km2 (the nearest big city, 25 to 50 km away, is indicated
in parentheses). The scale on the right indicates YOLL/yr (years of life lost) by acute mortality
from a plant with emission 1000 ton/yr. Plume rise for typical power plant conditions is
accounted for.
6
Variability of damage cost, due to changed emissions
Damage costs of fossil power plants (€/kg of ExternE [2008]). Also shown are the damage
costs for coal and oil plants in France during the mid nineties.
For comparison: market price in France was about 7 €cents/kWh during the mid nineties,
and 11 €cents/kWh in 2011.
7
Probability distributions of parameter values
The most important characteristics are
the mean  and the standard deviation 
Example:
the frequent case of a
68% Confidence interval
0.4
normal (=gaussian) distribution
with  = 0 and  = 1
0.3
Confidence intervals
68% [ - ,  + ]
95% [ - 2,  + 2]
0.2
0.1
-3
-2
-1
1
2
3
8
Combination of errors for a general function of terms
For a sum the standard deviation , and for a product the standard
geometric deviation g, can be calculated exactly, regardless how
wide the distributions of the individual terms. Furthermore, the
entire distribution is often approximately normal (for sums) or
lognormal (for products).
 Explicit closed form solution!
But for general functions y = f(x1, x2, …, xn) a closed form solution
can be obtained only in the limit of small uncertainties (narrow
2
2
distributions)
2
s y2 =
¶f
¶f
¶f
s x12 +
s x 2 2 +...
s xn 2
¶x1
¶x 2
¶xn
That is not appropriate for the large uncertainties of external costs.
General solution via Monte Carlo calculation, i.e.
perform a very large number of numerical simulations,
each calculating the result y for a specific choice of {x1,
x2, … xn}, and look at the resulting distribution of the y.
9
Combination of errors for a sum of terms
If y = x1 + x2 + … + xn
is a sum of uncorrelated random variables xi,
each with mean i and standard deviation i,
the uncertainty distribution of y has mean
 = 1 + 2 + … + n,
and standard deviation y given by
 y 2 =  1 2 +  2 2 + … +  n2 .
That result is general, but for confidence intervals one
also needs the distribution.
In practice the distribution of y is often close to normal
even for very small n if the individual distributions
(especially those with large widths) are not too far from normal.
Justified by central limit theorem of statistics:
In the limit n, the distribution of y approaches a
normal distribution, even if the distributions of the
individual xi are not normal.
10
Combination of errors for a product of terms
If y = x1 x2 … xn,
the log of y is a sum
ln(y) = ln(x1) + ln(x2) + … + ln(xn).
Let the xi be uncorrelated random variables
with probability distributions pi(xi).
Define the geometric mean gi of xi by
ln(mgi )= ò pi (x i )ln(x i )dx i
Then the geometric mean gy of y is given by
ln(mgy )= ò py (x y )ln(y)dy
and it is equal to
gy = g1 g2 … gn
11
Combination of errors for a product of terms, cont’d
Now define the geometric standard deviation gi of xi by
[ln(s gi )]2 = ò pi (x i )[ln(x i )-ln(mgi )]2 dx i
Then the geometric standard deviation gy of y is given by
[ln(gy)]2 = [ln(g1)]2 + [ln(g2)]2 + … + [ln(gn)]2
That result is general, but for confidence intervals one
also needs the distribution.
In practice the distribution of ln(y) is
often approximately normal, if the distributions of the individual
ln(xi) are not too far from normal.
A variable whose log has a normal distribution is called lognormal.
The distribution of a product is often approximately lognormal.
12
The lognormal distribution
To get the lognormal from the normal distribution
Change variable u
é (u-m)2 ù
1
pn (u)=
expê2 ú
2
s
s 2p
ë
û
= ln(x). Then du = dx/x and normalization integral becomes
1= ò -¥ pn (u) du = ò 0
¥
¥
pn (ln(x))
dx
x
which allows interpreting the function
é (ln(x)-m )2 ù
pn (ln(x))
1
pln( x) (x)=
=
expêú
2
x
2
s
xs 2p
ë
û
as the probability density of a new distribution between 0 and .
This is the lognormal distribution, with
 = ln(g) or g = exp() = geometric mean
and
ln(mg )= ò p(x)ln(x)dx
 = ln(g) or g = exp() = geometric standard deviation
13
[ln(s g )]2 = ò p(x)[ln(x)-ln(
mg )]2 dx
The lognormal distribution, cont’d
The lognormal distribution is asymmetric, with a long tail
and its mean is larger than its median.
Its median is equal to g. p(x)
g
0.7
0.6
Example,
with g = 1 and g = 2
0.5
0.4
0.3
68% confid.
interval
0.2
0.1
0.5
1
1.5
2
2.5
3
When plotted vs ln(x) it looks just like an ordinary normal.
For confidence intervals, note that
68% of the distribution is in the interval [g/ g, g g]
and
95% of the distribution is in the interval [g/ g2, g g2] .
3.5
x
4
14
Lognormal Distribution, cont’d
Probability density of lognormal distribution with g = 1 and g = 3. Mean
 = 1.83. The arrows indicate the 68% confidence interval (1 g interval).
p(x)
x
15
Estimation of uncertainties with Uniform World Model
the damage cost C per quantity Q of pollutant is a product
C = P sER  Q/vdep .
P = “price” = unit cost of endpoint (e.g. value of a life year)
sER = slope of exposure-response function
= regional average receptor density (with radius of about 1000 km)
vdep = depletion velocity
These factors are uncorrelated  can use simple solution with g
For a finer analysis each of the factors p, sER and vdep can be broken
up into separate factors to account for their respective uncertainty
sources.
Comparison of UWM with detailed site specific calculations for about a hundred installations in
many countries of Europe, as well as China, Thailand and Brazil: UWM is so close to the average
that it can be recommended for typical damage costs for emissions from tall stacks (>~ 50 m); for
specific sites the agreement is usually within a factor of two to three.
Note: typical values = average over emission sites, equivalent to averaging over receptor
16
distributions   becomes uniform.
Confidence intervals
Calculation is approximately multiplicative
 distribution of errors is approximately lognormal
(unless the contributions with the largest g have a distribution that is very
different from lognormal)
characterized by geometric standard deviation g
Multiplicative confidence intervals
about geometric mean g (smaller than ordinary mean )
68% between g/g and g g
95% between g/g2 and g g2
17
Uncertainty of the components, cont’d
Example of lognormal distribution for monetary valuation: “value of
statistical life”, in £1990, in 78 studies reviewed by Ives, Kemp and
Thieme [1993], histogram and lognormal fit.
Count
16
14
Approximately
lognormal
12
10
g = 1.5 M£
g = 3.4
8
6
4
2
0
4
4.5
5
5.5
6
6.5
lg10(Value in £)
7
7.5
8
18
Uncertainty of the components
For Dispersion: deposition velocities.
example: Distribution and lognormal fit of data points in the review
of Sehmel [1980], for dry deposition velocity [in cm/s] of SO2 over
different surfaces.
Approximately
lognormal
19
Uncertainty of the components, cont’d
How to estimate an approximate equivalent geometric
standard deviation g when one only knows mean 
and ordinary standard deviations 
Assume that the one-standard deviation (68% probability) interval
[- ,  + ] corresponds to the interval
[g/g, g g] of the lognormal distribution

m+s
sg »
m -s
Many studies (e.g. epidemiology) report their errors as 95% confidence intervals,
corresponding to two ordinary standard deviations, [- 2 ,  + 2 ] .
In that case use
m+2s
sg » 4
m -2s
20
Uncertainty, with UWM
C = P sER  Q/vdep
[ln(gC)]2 = [ln(gp)]2 + [ln(gsER)]2 + [ln(g)] + [ln(gQ)] + [ln(gvdep)] 2
Note:
Only the largest
errors make
significant
contribution
21
Uncertainty of mortality cost, more detail
[ln(gy)]2 = [ln(g1)]2 + [ln( g2)]2 + … + [ln(gn)]2
Note: Only the largest errors make significant contribution
22
Monte Carlo  simplified analysis
Monte Carlo
No limit on accuracy
Simplified analysis
(UWM)
Approximate
Can handle any
combination of errors
sources and distributions
Only for products (sums);
OK only if the
distributions with the
largest widths are not too
far from lognormal
(normal)
Requires large number of
computer calculations
Simple hand calculation
Opaque
Transparent
For damage costs the two approaches are complementary 23
Presentation
of
uncertainty
Damage costs for
LCA applications
in EU27,
€/kg
h = stack height
24
Evolution of damage cost estimates
(due to scientific progress, especially epidemiology and monetary valuation)
(these values are for
LCA applications in
Europe)
The changes are
within the
published
uncertainty
intervals
25
Effect of uncertainties
depends on specifics of each decision, e.g.
•choice of technology (e.g. coal or nuclear)
•optimal pollution control (e.g. mg/m3 of SO2 in smoke stack)
•level of pollution tax (e.g. €/tonne of SO2)
•replacement of old dirty technologies (e.g. old cars)
•choice of site (e.g. rural or urban)
•green accounting
Key question:
What is life cycle cost penalty to society if decision
based on erroneous damage estimate?
discrete choices: no effect if uncertainty does not change ranking
continuous choices: sensitivity to uncertainty is small near
optimum
26
Effect of uncertainties, discrete choices
0
Example where tighter
regulation is justified:
new directive [EC
2000] for waste
incineration
10
€/t waste
20
30
40
50
Cost
Benefit,
Paris
Benefit,
"urban"
Abatement
PM
Benefit,
"rural"
SO2
0
Example where
tighter regulation is
not justified:
reduced PM emission
limits for cement
kilns
€/kgPM
40
60
20
80
Benefit
PM 20->5
Cost, low
Cost, high
0
0.5
1
1.5
2
€/t clinker
2.5
27
Effect of uncertainties, continuous choices, cont’d
Cost penalty ratio R =
social cost withwrongDest
optimal social cost
The cost penalty ratio R
versus the error x =
Dtrue/Dest in the damage
cost estimate for several
countries, selected to
show extremes as well as
intermediate curves. The
labels are placed in the
same order as the curves.
Dashed lines correspond
to the extrapolated
regions of the cost curves.
Similar results for NOx
and CO2
28
Conclusion on Uncertainties
The uncertainties of damage costs are large,
• typically geometric standard deviations g around 3 to 5
• g around 3 for primary pollutants PM, NOx,SO2;
• somewhat larger for secondary pollutants (especially O3)
than for primary pollutants;
 g around 4 for toxic metals (As, Cd, Cr, Hg, Ni and Pb);
 g around 5 for greenhouse gases.
29
Effect of uncertainties
Some people think that the uncertainties of ExternE estimates are
too large to be useful
However:
1) Better 1/3 x to 3 x than 0 to 
2) What matters is not the uncertainty itself, but the social
cost of a wrong choice:
a) Without cost estimates such costs can be very large, but with
ExternE they can be remarkably small in many if not most cases.
b) For many yes/no choices the uncertainty is small enough not to
affect the answer.
3) Uncertainties can be reduced by a) research and b)
guidelines by decision makers on monetary values 30
Extra social cost due to errors
Without cost estimates such costs can be very large, but with ExternE
they can be remarkably small in many if not most cases.
Example: extra social cost
(relative to optimum with
perfect information) for
NOx abatement: less than
15% within confidence
interval, but gets very large
for much larger errors:
Ctrue = true damage cost
Cest = estimated damage cost
Thanks to ExternE one
can avoid very costly
mistakes.
31