## Approach 1 details of the equations for trend uncertainty

The following steps show how to calculate trend uncertainty using Types A and B sensitivities (see also Section 3.2.3.1).

1) The method for assessing level uncertainty in year Y assumes that categories and gases are uncorrelated, or are aggregated until the aggregated categories can be treated as uncorrelated.

2) The uncertainty in the trend in total emissions from the country (the quantity at the foot of Column M) is estimated as:

where UT is the uncertainty in the trend in total emissions from the country and U, is the uncertainty introduced into UT by the category , and gas.

3) We take

Ui where UEi is the uncertainty introduced into Ui by the uncertainty associated with the emission factor of the category i and gas, and UAi is the uncertainty introduced into U by the uncertainty associated with the activity data of the category i and gas.

4) We know from Columns E and F what the uncertainties related to activity data and emission factors for the category i and gas are in percentage terms, but we do not yet know how these uncertainties affect the trend in the total emissions, which is what we need for UEi and UAi. For this we write

Where Ai is the Type A sensitivity associated with the category i and gas, and uei the percentage uncertainty associated with the emission factor in Column F, and Bt is the Type B sensitivity associated with the category i and gas, and uai the percentage uncertainty associated with the activity data in Column E. Essentially Type A and Type B sensitivities are elasticities relating respectively a percentage difference that is self-correlated between the base year and year Y, and one which is uncorrelated, to the percentage change in total emissions. The method allows for this assumption to be inverted, or for both emission factor and activity data to be self-correlated between years, or for neither to be self-correlated.

5) The Type A and Type B sensitivities are calculable from formulae for the trend in terms of sums over categories and gases in the base year and in year Y. The additional factor of V2 is introduced because an uncorrelated uncertainly might affect either the base year or the year Y. The current formulation assumes for Type B sensitivity that the emissions in year Y are not too different from those in the base year; if this were not the case we would have to introduce separate consideration of the base year and year Y for the uncorrelated uncertainties, rather than using the V2 factor.

DERIVATION OF TYPE A SENSITIVITY

The trend can be written as (assuming that 1990 is a base year):

If the category i and gas is increased by 1 percent throughout (consistent with the assumption that Type A sensitivity captures the effect of uncertainties which are correlated between years) the trend becomes:

and the sensitivity Ai becomes:

This is the same as the expression given for the Type A sensitivity in Note B on page 6.18 of the GPG2000. TYPE B SENSITIVITY

The Type B sensitivity we assume that the category i and gas is increased by 1 percent in yeary only. In this case the trend becomes:

So the sensitivity Bt becomes:

y 1990

All the terms on the numerator cancel out between the brackets except for 0.01 e, y which becomes e,y when multiplied by 100. So the expression for B,- simplifies to y Column J on page 6.16 of the GPG2000.

N which is the expression at the top of

E ei,1990

3.7.3 Dealing with large and asymmetric uncertainties in the results of Approach 1

This section provides guidance on how to correct for biases in large estimates of uncertainty from Approach 1 and how to convert the uncertainty ranges into asymmetric 95 percent probability ranges based upon a lognormal distribution.

Correction of uncertainty estimate for large uncertainties: The approximate error propagation method of Approach 1 produces an estimate of the uncertainty half range (U), expressed as a percentage relative to the mean, of the inventory results. As the uncertainty in the total inventory uncertainty becomes larger, the error propagation approach systematically underestimates the uncertainty unless the model is purely additive. However, most inventories are estimated based upon the sum of terms, each of which is a product (e.g., of emission factors and activity data). The error propagation approach is not exact for such multiplicative terms. Results from empirical studies show that in some cases uncertainty estimated using Approach 1 could be underestimated; the analyst could use a correction factor, for example that proposed in Frey (2003). Frey (2003) evaluated the performance of an analytical approach for combining uncertainty in comparison to a Monte Carlo simulation with large sample sizes for many cases involving different ranges of uncertainty for additive, multiplicative, and quotient models. Error propagation and Monte Carlo simulated estimates of the uncertainty half-range of the model output agreed well for values of less than 100 percent. As the uncertainty in the total inventory increased to higher levels, there was a systematic under-estimation of uncertainty in the total inventory by the error propagation approach. The relationship between the simulated and error propagation estimates was found to well-behaved. Thus, a correction factor was developed from the comparison that is applicable if U for the total inventory uncertainty is large (e.g., greater than 100 percent) and is given by:

Equation 3.3 Correction factor for uncertainty half-range

Note: Use if U > 100% and if the model contains multiplicative or quotient terms Not necessarily reliable for U > 230% Not necessary for models that are purely additive.

Where:

U = /-range for uncertainty estimated from error propagation, in units of percent

Fc = Correction factor for analytical estimate of the variance, dimensionless ratio of corrected to uncorrected uncertainty

The empirically-based correction factor produces values from 1.06 to 1.69 as U varies from 100% to 230%. The correction factor is used to develop a new, corrected, estimate of the total inventory uncertainty half-range, Ucorrected, which in turn is used to develop confidence intervals.

Equation 3.4 Corrected uncertainty half-range

Ucorrected =U # FC

Where:

Ucorrected = Corrected '/2-range for uncertainty estimated from error propagation, in units of %

The errors in the analytical estimate of the variance are generally small for uncertainty half-ranges (U) of less than approximately 100 percent. If the correction factor is applied for U > 100% for values of U up to 230%, the typical error in the estimate of U is expected to be within plus or minus 10 percent in most cases. The correction factor will not necessarily be reliable for larger uncertainties because it was calibrated over the range of 10% to 230%.

Calculation of asymmetric confidence intervals for large uncertainties: In order to calculate confidence intervals for the model output based upon only the mean and half-range for uncertainty, a distribution must be assumed. For models that are purely additive, and for which the half range of uncertainty is less than approximately 50 percent, a normal distribution is often an accurate assumption for the form of the model output. In this case, a symmetric uncertainty range with respect to the mean can be assumed. For multiplicative models, or when the uncertainty is large for a variable that must be non-negative, a lognormal distribution is typically an accurate assumption for the form of the model output. In such cases, the uncertainty range is not symmetric with respect to the mean, even though the variance for the total inventory may be correctly estimated from Approach 1. Here, we provide a practical methodology for calculating approximate asymmetric uncertainty ranges based upon the results of error propagation, based upon a methodology developed by Frey (2003). A key characteristic of the 95 percent confidence intervals is that they are approximately symmetric for small ranges of uncertainty and they are positively skewed for large ranges of uncertainty. The latter result is necessary for a non-negative variable.

The parameters of the lognormal distribution can be defined in several ways, such as in terms of the geometric mean and geometric standard deviation. The geometric mean can be estimated based upon the arithmetic mean and the arithmetic standard deviation:

Where:

|g = geometric mean I = arithmetic mean The geometric standard deviation is given by:

Where:

CTg = geometric standard deviation

A confidence interval can be estimated based upon the geometric mean, geometric standard deviation, and the inverse cumulative probability distribution of a standard normal distribution (with a logarithmic transformation):

Where:

Uiow = Lower ^-range for uncertainty estimated from error propagation, in units of %.

Uhigh = Upper Grange for uncertainty estimated from error propagation, in units of %.

To illustrate the use of these equations, consider an example. Suppose the mean is 1.0 and the Grange of uncertainty estimated from error propagation is 100 percent. In this case, the geometric mean is 0.89 and the geometric standard deviation is 1.60. The 95 percent probability range as a percentage relative to the mean is given by the interval from Uiow to Uhigh of Equations 3.7. In the example, the result is -65% to +126%. In contrast, if a normal distribution had been used as the basis for uncertainty estimation, the range would have been estimated as approximately ±100% and there would be a probability of approximately two percent of obtaining negative values. Figure 3.9 illustrates the sensitivity of the lower and upper bounds of the 95 percent probability range, which are the 2.5th and 97.5th percentiles, respectively, calculated assuming a lognormal distribution based upon an estimated uncertainty half-range from an error propagation approach. The uncertainty range is approximately symmetric relative to the mean up to an uncertainty half-range of approximately 10 to 20 percent. As the uncertainty half-range, U, becomes large, the 95 percent uncertainty range shown in Figure 3.9 becomes large and asymmetric. For example, if U is 73 percent, then the estimated probability range is approximately -50% to +100%, or a factor of two.

Figure 3.9 Estimates of asymmetric ranges of uncertainty with respect to the arithmetic mean assuming a lognormal distribution based upon uncertainty half-range calculated from a propagation of error approach

Figure 3.9 Estimates of asymmetric ranges of uncertainty with respect to the arithmetic mean assuming a lognormal distribution based upon uncertainty half-range calculated from a propagation of error approach

100 150

Uncertainty Half-Range (%)

100 150

Uncertainty Half-Range (%)