A graphical diagnostic method for assessing the rotation in factor analytical models of atmospheric pollution

Factor analytic tools such as principal component analysis (PCA) and positive matrix factorization (PMF), suffer from rotational ambiguity in the results: different solutions (factors) provide equally good fits to the measured data. The PMF model imposes non-negativity of both source profiles and source contributions in order to reduce the rotational problem. Such constraints are generally insufficient to ensure a unique solution. In the Unmix approach, edges of the multidimensional distribution of source contributions define the variable relationships in the factors. The present work extends this idea into an easy-to-use graphical procedure called G space plotting for PMF modeling. Scatter plots are created of pairs of source contribution factors. When factors are plotted in this way, unrealistic rotations appear as oblique edges that define the distribution of points away from one (or both) of the coordinate axes. With a correct rotation, the limiting edges usually coincide with the axes or lay parallel with them. Inspection of the plots helps one in choosing a realistic rotation.