Uncertainty in the Mean for Example Emission Rate


For a second example, suppose that one wishes to estimate the uncertainty in national annual emissions for a particular category, such as emissions from gasoline-fuelled passenger automobiles. The rate of emission varies from one individual vehicle to another, illustrated by the inter-unit variability shown in Figure 3.4(a). Because the distribution for inter-vehicle variability is estimated from a small, finite sample of data that could be subject to random sampling error, there is uncertainty regarding what the true but unknown population distribution for inter-vehicle variability might be, as suggested in Figure 3.4(b). There is also intra-unit variability in emissions over time for any particular vehicle. However, for purposes of the national annual estimate, the focus is on the combined contribution of all such vehicles to total emissions during a year long time frame. In this case, we are not interested in the range of inter-vehicle variability, but rather in the range of uncertainty for the average emission rate among all such vehicles (e.g., Figure 3.4(c)). Often, the range of uncertainty is substantially less than that for inter-vehicle (or, more generally, inter-unit) variability (e.g., Frey and Zheng, 2002). Therefore, when the objective of an analysis requires that the assessment be based upon uncertainty in the mean, rather than variability among individual units, it is important to properly focus the analysis on the former. Failure to do so can lead to a misleading over-estimate of the range of uncertainty.

In the case of continuous monitoring of point emissions, or a periodic sampling scheme that captures typical activity patterns, there may be adequate and representative empirical data upon which to base an estimate of uncertainty in mean annual emissions. For example, if there are several years of such data, then the average annual emissions over several years can be quantified, and the distribution of annual emissions from year-to-year can be used to assess a 95 percent confidence interval in the annual average. Provided that the annual average is based upon data from many individual categories, it is unlikely that there will be correlation of errors between years. This has implications for estimation of uncertainty in trends, as discussed in Section 3.3, Uncertainty and Temporal Autocorrelation. However, for diffuse categories, such as agricultural crops, there could be high autocorrelations if they are determined by climate, and this could affect the representativeness of the data for a particular assessment purpose.

Where continuous emission measurements are not available, there may be periodic emission measurements available from which to estimate uncertainty. If these measurements can be linked to representative activity data, which of course is crucial, then it is possible to determine a site-specific emission factor, together with an associated PDF to represent annual emissions. This can be a complex task. To achieve representativeness it may be necessary to partition (or stratify) the data to reflect typical operating conditions. For example:

• Start-up and shut down can give different emission rates relative to activity data. In this case, the data should be partitioned, with separate emission factors and probability density functions derived for steady state, start-up and shut down conditions.

• Emission factors can depend on load. In this case, the total emissions estimation and uncertainty analysis may need to be stratified to take account of load, expressed, for example, as percentage of full capacity. This could be done by regression analysis and scatter plots of the emission rate against likely controlling variables (e.g., emissions versus load) with load becoming part of the activity data needed.

• Measurements taken for another purpose may not be representative. For example, methane measurements made for safety reasons at coal mines and landfills may not necessarily reflect total emissions because they may have been made only when methane emissions were suspected of being high, as a compliance check. In such cases, the ratio between the measured data and total emissions should be estimated for the uncertainty analysis.

• Systematic short-term measurements might not adequately sample episodic events (such as rainfall) that initiate large fluxes of short duration that may nevertheless account for a major fraction of annual emissions. If the sampling strategy misses a significant proportion of these events, then the annual average emission estimate could be substantially biased. Nitrous oxide emissions from agricultural soils can fall into this class.

If the data sample size is large enough, standard statistical goodness-of-fit tests can be used, in combination with expert judgement, to help in deciding which PDF to use for describing variability in the data (partitioned if necessary) and how to parameterise it. However, in many cases, the number of measurements from which to make an inference regarding uncertainty will be small. Theoretically, as long as there are three or more data points, and they are a random representative sample of the variable of interest, it is possible to apply statistical techniques to estimate the values of the parameters of many two-parameter distributions (e.g., normal, lognormal) that can be used to describe variability in the data set (Cullen and Frey, 1999, pp. 116-117). While it is commonly perceived that one must have approximately 8 or 9 data points, and preferably more, as the basis for fitting a distribution to data, the more fundamental and key assumption that must be made in order to fit a distribution to data is that the data are a random, representative sample. If this assumption is valid, then the sample size influences the width of the confidence intervals for any statistic estimated from the sample. As a matter of preference, many analysts may prefer to have a minimum sample size, but this preference is not related to the key issue of representativeness. Data do not become more representative only because of an increase in sample size.

With small sample sizes, there will be large uncertainties regarding the parameter estimates that should be reflected in the quantification of uncertainty for use in the inventory. Furthermore, it is typically not possible to rely on statistical methods to differentiate goodness-of-fit of alternative parametric distributions when sample sizes are very small (Cullen and Frey, 1999, pp. 158-159). Therefore judgement will be required in selecting an appropriate parametric distribution to fit to a very small data set. In situations where the coefficient of variation (standard deviation divided by the mean) is less than approximately 0.3 and is known with reasonable confidence, a normal distribution may be a reasonable assumption (Robinson, 1989). When the coefficient of variation is large and the variable is non-negative, then a positively skewed distribution such as a lognormal may be appropriate. Guidance on the selection of distributions is elaborated in Sections and below.

In cases with large data sets, the uncertainty in the mean can often be estimated as plus or minus 1.96 (or approximately 2) multiples of the standard error, where the standard error is the sample standard deviation divided by the square root of the sample size. This calculation is based on an assumption of a normal distribution. However, in cases of a small number of samples/measurements that will often be the case in determining emission factors, the multiple of 1.96 is replaced with a "coverage factor," referred to as k, that is obtained from the student's t-distribution. For small sample sizes, k is greater than 1.96 for a 95 percent interval, but asymptotically approaches 1.96 as the sample size increases to approximately 30 or more. However, in cases where the uncertainty in the mean is not a symmetric distribution, then numerical methods such as bootstrap simulation can be used instead to obtain the confidence interval for the mean.

Where an annual estimate is based on an average over several years, the uncertainty in the average represents the uncertainty in an average year and not the inter-annual variability. If the objective is to estimate uncertainty in source or sink fluxes for a specific year, then good practice is to make a best estimate of the annual total and to quantify uncertainty associated with the models and data used consistent with the one year time period. If, instead, an averaged annual estimate is used, then the uncertainty in the estimate when applied to a specific year would be described by the inter-annual variability (including measurement errors) relative to the mean, whereas when applied to an average year it would be the confidence interval of the average.

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