## Basis for uncertainty analysis

The chapter uses two main statistical concepts - the probability density function (PDF) and confidence interval defined in the previous section. While this chapter focuses on aspects of uncertainty that are amenable to quantification, there are typically nonquantifiable uncertainties as well. The quantitative uncertainty analysis tends to deal primarily with random errors based on the inherent variability of a system and the finite sample size of available data, random components of measurement error, or inferences regarding the random component of uncertainty obtained from expert judgement. In contrast, systematic errors that may arise because of imperfections in conceptualisation, models, measurement techniques, or other systems for recording or making inferences from data, can be much more difficult to quantify. As mentioned in Section 3.5, Reporting and Documentation, it is good practice for potential sources of uncertainty that have not been quantified to be described, particularly with respect to conceptualisation, models, and data and to make every effort to quantify them in the future.

Good practice requires that bias in conceptualisations, models, and inputs to models be prevented wherever possible, such as by using appropriate QA/QC procedures. Where biases cannot be prevented, it is good practice to identify and correct them when developing a mean estimate of the inventory. In particular, the point estimate that is used for reporting the inventory should be free of biases as much as it is practical and possible. Once biases are corrected to the extent possible, the uncertainty analysis can then focus on quantification of the random errors with respect to the mean estimate.

Good practice requires the use of a 95 percent confidence interval for quantification of random errors. This may also be expressed as a percentage of the central estimate. Where the PDF is symmetrical the confidence interval can be conveniently expressed as plus or minus half the confidence interval width divided by the estimated value of the variable (e.g., ± 10%). Where the PDF is not symmetrical upper and lower limits of the confidence interval need to be specified separately (e.g., -30%,+50%).

If the range of uncertainty for a non-negative variable is small enough relative to the mean value, then the uncertainty often can be described as a symmetric range with respect to a mean value, as shown in Figure 3.3(a). For example, if the mean emissions are 1.0 units, the 2.5th percentile of uncertainty is 0.7 units, and the 97.5th percentile of uncertainty is 1.3 units, then the uncertainty range could be described as 1.0 units ±30%. However, when the relative range of uncertainty is large, and if the uncertainty is regarding a variable that must be nonnegative (such as an emission factor), then the uncertainty range becomes asymmetric with respect to the mean, as shown in Figure 3.3(b). As an example, if the mean emissions are 1.0 units, the 2.5th percentile of uncertainty is 0.5 units, and the 97.5th percentile of uncertainty is 2.0 units, then the range of uncertainty can be described as 1.0 units -50% to +100%. In situations such as the latter, it is often more convenient to summarise uncertainties in a multiplicative, rather than additive, manner. In this particular example, the lower end of the 95 percent probability range is a half the mean, and the upper end is a multiplier of 2 larger than the mean. Such a range is commonly summarised as a "factor of 2." An uncertainty of a "factor of n" refers to a range bounded at the low end by (mean/n) and at the high end by (mean x n). Thus, a factor of 10 uncertainty would have a range of 0.1xmean to 10xmean. The factor 10 uncertainty is also often called "an order of magnitude". Higher powers of 10 are referred to as "orders of magnitude;" for example, a factor of 103 would be referred to as three orders-of-magnitude.

Figure 3.3 Examples of symmetric and asymmetric uncertainties in an emission factor

(a) Example of a symmetric uncertainty of ±30% relative to the mean

Figure 3.3 Examples of symmetric and asymmetric uncertainties in an emission factor

Example Emission Factor

(b) Example of an asymmetric uncertainty of -50% to +100% relative to the mean, or a factor of two

Example Emission Factor