Identification and consistent estimation in source apportionment using geometry

Abstract

Source apportionment, the attribution of observed multipollutant concentrations to underlying sources, can be cast as a non-negative matrix factorization (NMF) problem. Because NMF is non-unique, source apportionment imposes additional, often unverifiable, constraints such as sparsity. Geometric approaches offer an alternative route to identification, but many of them still rely on source profiles with arbitrary scalings, make strong structural assumptions including exact separability, and lack a statistical framework for consistent estimation. In this manuscript, we address these limitations. We introduce the source attribution matrix that is scale-invariant and establish its identifiability under a stochastic framework that replaces hard separability constraints with soft probabilistic relaxations. We then present a scalable geometric algorithm to estimate the source attribution matrix and prove its consistency. To our knowledge, this is the first consistency result for estimating the source attribution matrix that requires no exact sparsity, makes no parametric distributional assumptions, and accommodates spatio-temporal dependence in data. Numerical experiments confirm the theory.