Understanding the sources that contribute to fine particulate matter (PM$_{2.5}$) is of crucial importance for designing and implementing targeted air pollution mitigation strategies. Determining what factors contribute to a pollutant's concentration goes under the name of source apportionment and it is a problem long studied by atmospheric scientists and statisticians alike. In this paper, we propose a Bayesian model for source apportionment, that advances the literature on source apportionment by allowing estimation of the number of sources and accounting for spatial and temporal dependence in the observed pollutants' concentrations. Taking as example observations of six species of fine particulate matter observed over the course of a year, we present a latent functional factor model that expresses the space-time varying observations of log concentrations of the six pollutant as a linear combination of space-time varying emissions produced by an unknown number of sources each multiplied by the corresponding source's relative contribution to the pollutant. Estimation of the number of sources is achieved by introducing source-specific shrinkage parameters. Application of the model to simulated data showcases its ability to retrieve the true number of sources and to reliably estimate the functional latent factors, whereas application to PM$_{2.5}$ speciation data in California identifies 3 major sources for the six PM$_{2.5}$ species.

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