Probabilistic Modeling of Variability and Uncertainty in Urban Air Toxics Emissions
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Date
2003-12-21
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Abstract
Air toxic emission factor data often contain one or more censored points below a single or multiple detection limits. Such data sets are referred to as "censored." Conventional methods used to deal with censored data sets include removing non-detects, or replacing the censored points with zero, half of the detection limit or the detection limit. However, the estimated means of the censored data set by conventional methods are usually biased. Here, an approach to quantification of the variability and uncertainty of censored data sets is demonstrated. Empirical bootstrap simulation is used to simulate censored bootstrap samples from the original data. Maximum Likelihood Estimation (MLE) is used to fit parametric probability distributions to each bootstrap sample, thereby specifying alternative estimates of the unknown population distribution of the censored data sets. Sampling distributions for uncertainty in statistics such as the mean, median and percentile are calculated. The robustness of the method was tested by application to different degrees of censoring, sample sizes, coefficients of variation and numbers of detection limits. Lognormal, gamma and Weibull distributions were evaluated. The reliability of using this method to estimate the mean is proved. The application of MLE/Bootstrap was compared favorably to results obtained with the non-parametric Kaplan-Meier method, which verify the accuracy of this method.
The MLE/bootstrap method is applied to 16 cases of censored air toxic emission factors, including benzene, formaldehyde, Benzo(a)pyrene, mercury, arsenic, cadmium, total chromium, chromium VI and lead with single or multiple detection limits from coal, fuel oil and/or wood waste external combustion sources. The data differs regarding sample size, censoring degree, inter-unit variability and so on. The proportion of censored values in the emission factor data ranges from 4 to 80 percent. The largest range of uncertainty in the mean was obtained for the external coal combustion benzene emission factor, with a 95 percent probability range of minus 93 to plus 411 percent of the mean.
Probabilistic emission inventories were developed for benzene, formaldehyde, chromium, and arsenic for Houston 1996 emission inventory and for 1, 3-butadiene, mercury, arsenic, benzene, formaldehyde and lead. Parametric distributions for inter-unit variability were fit using maximum likelihood estimation (MLE) and uncertainty in mean emission factors was estimated using parametric bootstrap simulation. For data sets containing one or more non-detected values, empirical bootstrap simulation was used to randomly sample detection limits for non-detected values and observations for sample values, and parametric distribution for variability were fit using MLE estimators for censored data. Goodness-of-fit for censored data was evaluated using the Kolmogorov-Smirnov test applied to a modified data set and by comparison of cumulative distributions of bootstrap confidence intervals and empirical data. The emission inventory 95 percent uncertainty ranges are as small as minus 25 to plus 42 percent for chromium for Houston to minus 75 to plus 224 percent for arsenic for Jacksonville. Uncertainty was dominated by only a few source categories. Recommendations are made for future improvements to the analysis.
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bootstrap simulation, maximum likelihood estimation, uncertainty, censored data, variabiliby, urban air toxic
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Degree
PhD
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Environmental Engineering
