Quantification of Variability and Uncertainty Using Mixture Distributions: Evaluation of Sample Size, Mixing Weights, and Separation Between Components

Variability is the heterogeneity of values within a population. Uncertainty refers to lack of knowledge regarding the true value of a quantity. Mixture distributions have the potential to improve the goodness of fit to data sets not adequately described by a single parametric distribution. Uncertainty due to random sampling error in statistics of interests can be estimated based upon bootstrap simulation. In order to evaluate the robustness of using mixture distribution as a basis for estimating both variability and uncertainty, 108 synthetic data sets generated from selected population mixture log-normal distributions were investigated, and properties of variability and uncertainty estimates were evaluated with respect to variation in sample size, mixing weight, and separation between components of mixtures. Furthermore, mixture distributions were compared with single-component distributions. Findings include: (1). mixing weight influences the stability of variability and uncertainty estimates; (2). bootstrap simulation results tend to be more stable for larger sample sizes; (3). when two components are well separated, the stability of bootstrap simulation is improved; however, a larger degree of uncertainty arises regarding the percentiles coinciding with the separated region; (4). when two components are not well separated, a single distribution may often be a better choice because it has fewer parameters and better numerical stability; and (5). dependencies exist in sampling distributions of parameters of mixtures and are influenced by the amount of separation between the components. An emission factor case study based upon NO(x) emissions from coal-fired tangential boilers is used to illustrate the application of the approach.