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1 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.

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2 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.

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3 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.

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4 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).

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5 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.

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6 Variability of damage with site An example of dependence on site and on height of source for a primary pollutant: damage D from SO 2 emissions with linear ER function, for five sites in France, in units of D uni for uniform world model Eq.10 with = 105 persons/km 2 (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.

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7 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.

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8 Probability distributions of parameter values The most important characteristics are the mean and the standard deviation Example: the frequent case of a normal (=gaussian) distribution with = 0 and = 1 Confidence intervals 68% [ - , + ] 95% [ - 2 , + 2 ] 68% Confidence interval

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9 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(x 1, x 2, …, x n ) a closed form solution can be obtained only in the limit of small uncertainties (narrow distributions) 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 {x 1, x 2, … x n }, and look at the resulting distribution of the y.

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10 Combination of errors for a sum of terms If y = x 1 + x 2 + … + x n is a sum of uncorrelated random variables x i, 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 + … + n 2. 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 x i are not normal.

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11 Combination of errors for a product of terms If y = x 1 x 2 … x n, the log of y is a sum ln(y) = ln(x 1 ) + ln(x 2 ) + … + ln(x n ). Let the x i be uncorrelated random variables with probability distributions p i (x i ). Define the geometric mean gi of x i by Then the geometric mean gy of y is given by and it is equal to gy = g1 g2 … gn

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12 Combination of errors for a product of terms, cont’d Now define the geometric standard deviation gi of x i by 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(x i ) 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.

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13 The lognormal distribution To get the lognormal from the normal distribution Change variable u = ln(x). Then du = dx/x and normalization integral becomes which allows interpreting the function 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( g ) or g = exp( ) = geometric standard deviation

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14 The lognormal distribution, cont’d Example, with g = 1 and g = 2 The lognormal distribution is asymmetric, with a long tail and its mean is larger than its median. Its median is equal to g. 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 / g 2, g g 2 ]. gg 68% confid. interval x p(x)

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15 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). x p(x)

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16 Estimation of uncertainties with Uniform World Model the damage cost C per quantity Q of pollutant is a product C = P s ER Q/ v dep. P = “ price ” = unit cost of endpoint (e.g. value of a life year) s ER = slope of exposure-response function = regional average receptor density (with radius of about 1000 km) v dep = depletion velocity These factors are uncorrelated can use simple solution with g For a finer analysis each of the factors p, s ER and v dep 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 distributions becomes uniform.

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17 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 / g 2 and g g 2 Confidence intervals

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18 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. g = 1.5 M£ g = 3.4 Approximately lognormal

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19 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 SO 2 over different surfaces. Approximately lognormal

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20 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 Many studies (e.g. epidemiology) report their errors as 95% confidence intervals, corresponding to two ordinary standard deviations, [ ]. In that case use

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21 Uncertainty, with UWM [ln( gC )] 2 = [ln( gp )] 2 + [ln( gsER )] 2 + [ln( g )] + [ln( gQ )] + [ln( gvdep )] 2 C = P s ER Q/ v dep Note: Only the largest errors make significant contribution

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22 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

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23 Monte Carlo simplified analysis Monte CarloSimplified analysis (UWM) No limit on accuracyApproximate 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 OpaqueTransparent For damage costs the two approaches are complementary

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24 Presentation of uncertainty Damage costs for LCA applications in EU27, €/kg h = stack height

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25 (these values are for LCA applications in Europe) The changes are within the published uncertainty intervals Evolution of damage cost estimates (due to scientific progress, especially epidemiology and monetary valuation)

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26 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/m 3 of SO 2 in smoke stack) level of pollution tax (e.g. €/tonne of SO 2 ) 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

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27 Effect of uncertainties, discrete choices Example where tighter regulation is justified: new directive [EC 2000] for waste incineration Example where tighter regulation is not justified: reduced PM emission limits for cement kilns

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28 Effect of uncertainties, continuous choices, cont’d The cost penalty ratio R versus the error x = D true /D est 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 NO x and CO 2 Cost penalty ratio R =

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29 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, NO x,SO 2 ; somewhat larger for secondary pollutants (especially O 3 ) than for primary pollutants; g around 4 for toxic metals (As, Cd, Cr, Hg, Ni and Pb); g around 5 for greenhouse gases.

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30 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

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31 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 NO x abatement: less than 15% within confidence interval, but gets very large for much larger errors: C true = true damage cost C est = estimated damage cost Thanks to ExternE one can avoid very costly mistakes.

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Lecture II-2: Probability Review

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