Effect of adding variables on rotational ambiguity in positive matrix factorization solutions

A major problem that limit the use of the model-free analysis methods, like positive matrix factorization (PMF), for the resolution of multivariate data is that usually, there is rotational ambiguity in the results. Implementing adequate constraints may reduce such ambiguities. In some special cases, non-negativity constraints may serve as a sufficient condition to uniquely resolve profiles. In this study, effect of adding new variables on the extent of rotational ambiguity is investigated, and profiles with lower rotation have been obtained on the basis of data set structure. The displacement (DISP) method of error estimation (EE) for the factor elements incorporated into the US Environmental Protection Agency's version of positive matrix factorization (PMF) has been applied to capture the uncertainty of PMF analyses due to rotational ambiguity. To explore these considerations, results are presented for synthetic including submicron particle size distributions (12-470 nm) measured between January 2008 and December 2010 at the New York State Department of Environmental Conservation (NYS DEC) site in Rochester, NY. Generally, added informative variables can reduce the ambiguity in the profiles. The results of the PMF analysis showed the effect of adding new variables to feasible solutions led to significantly decreased (up to 44%) DISP intervals. More robust results with lower rotational uncertainty became possible and the results were more easily interpreted. This approach can be applied to any other field that utilize quantitative factor analysis methods.