Identification in source apportionment using geometry

Abstract: Source apportionment analysis, which aims to quantify the attribution of observed concentrations of multiple air pollutants to specific sources, can be formulated as a non-negative matrix factorization (NMF) problem. However, NMF is non-unique and typically relies on unverifiable assumptions such as sparsity and uninterpretable scalings. In this manuscript, we establish identifiability of the source attribution percentage matrix under much weaker and more realistic conditions. We introduce the population-level estimand for this matrix, and show that it is scale-invariant and identifiable even when the NMF factors are not. Viewing the data as a point cloud in a conical hull, we show that a geometric estimator of the source attribution percentage matrix is consistent without any sparsity or parametric distributional assumptions, and while accommodating spatio-temporal dependence. Numerical experiments corroborate the theory.