Determining the number of factors for non-negative matrix and its application in source apportionment of air pollution in Singapore

The non-negative matrix factorization has been used in many disciplines of research, where the number of factors plays a crucial role. However, a fully data-driven method for determining the number is yet not available in the literature. Based on the fact that the most appropriate number of factors should generate the best prediction, in this paper we propose a selection method using a two-step delete-one-out approach, called twice cross-validation. This method is easy to implement and is fully data-driven. It also works when constraints are imposed on the factorization including the sparsity. Intensive simulations and real data analyses suggest that the proposed method performs well in most cases and can select the number of factors correctly when the number of factors is much less than the dimension of variables and the sample size is reasonably large. As an important application, the proposed method is used for source apportionment of air pollution in Singapore, and provides physically reasonable source profiles.