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

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Abstract

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.

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Acknowledgements

We are most grateful to the AE and two referees for their valuable comments and constructive suggestions, which have led to a substantial improvement of this paper. YC Xia’s research is partially supported by MOE Tier 1 Grant: R-155-000-193-114, and MOE Grant of Singapore: MOE2014-T2-1-072, and National Natural Science Foundation of China, 11771066.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China

    Mei Yan & Yingcun Xia

  2. School of Mathematics, Statistics and Physics, Newcastle University, Newcastle, NE1 7RU, UK

    Xiaojie Yang

  3. Department of Statistics and Applied Probability, National University of Singapore, Singapore, 117546, Singapore

    Weiqiang Hang & Yingcun Xia

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  1. Mei Yan
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  2. Xiaojie Yang
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  3. Weiqiang Hang
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  4. Yingcun Xia
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Correspondence to Yingcun Xia.

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Yan, M., Yang, X., Hang, W. et al. Determining the number of factors for non-negative matrix and its application in source apportionment of air pollution in Singapore. Stoch Environ Res Risk Assess 33, 1175–1186 (2019). https://doi.org/10.1007/s00477-019-01677-z

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