Impact of species uncertainty perturbation on the solution stability of positive matrix factorization of atmospheric particulate matter data

Statistical measures for evaluating the similarity of different source apportionment solutions are proposed. The sensitivity of positive matrix factorization to small perturbations in species measurement uncertainty estimates is examined using fine particulate matter measurements on organic carbon, elemental carbon, ions, and metals at the St. Louis-Midwest Supersite. A perturbed uncertainty matrix is created by multiplying each original uncertainty value by a random multiplier generated from a log-normal distribution with a mean of 1 and a standard deviation (and CV) equal to either 0.25, 0.50, or 0.75. The relative errors in reproducing the average contribution estimates from the perturbed data are generally highest for the gasoline exhaust, with the relative error (expressed as a percentage of the "true" value) exceeding 30% for all three perturbation scenarios. The most stable estimates of average source contribution were associated with secondary sulfate and secondary nitrate, with relative errors always less than 4%. Averaged over all 10 sources, the average values for our measure of relative error for the three scenarios are 8%, 14%, and 17%, respectively. Relative errors associated with day-today estimates of source contributions can be more than double the size of the relative errors associated with estimates of average source contributions, with errors for four of 10 source contributions exceeding 30% for the largest-perturbation scenario. The stability of source profile estimates in our simulation varies greatly between sources, with a mean correlation between perturbed gasoline exhaust profiles and the true profile equal to only 59% for the largest-perturbation scenario. The process used for evaluation is a tool that may be used to assess the stability of solutions in source apportionment studies.