Air pollution has become a pressing challenge to global public health and environmental governance. Accurate analysis of pollutant concentrations critically depends on the completeness of monitoring data; however, the widespread presence of missing values in real-world datasets significantly compromises both assessment reliability and predictive performance. To address this issue, this study proposes a three-dimensional time-feature-space model, which integrates a Kolmogorov-Arnold Network (KAN) with a Transformer architecture, referred to as KT. The KAN module, equipped with an expert mixing mechanism, captures complex nonlinear temporal dynamics of pollutants, which are subsequently processed by the Transformer to model interdependencies among multiple pollutants via self-attention. A spatial feature selection strategy based on Spearman correlation is further employed to extract key spatiotemporal interactions through channel mixing or independent dynamic processing. Empirical evaluation on air quality monitoring data collected in Beijing from January 2023 to October 2024 shows that the proposed model reduces the mean absolute error (MAE) by 1.1% - 17.1% compared with several SOTA benchmark methods. These results clearly demonstrate the robustness and effectiveness of the proposed approach in estimating complex and incomplete air pollution data.
References
[1]
Cohen, A.J., Brauer, M., Burnett, R., Anderson, H.R., Frostad, J., Estep, K., et al. (2017) Estimates and 25-Year Trends of the Global Burden of Disease Attributable to Ambient Air Pollution: An Analysis of Data from the Global Burden of Diseases Study 2015. The Lancet, 389, 1907-1918. https://doi.org/10.1016/s0140-6736(17)30505-6
[2]
Chen, Z., Liu, P., Xia, X., Wang, L. and Li, X. (2022) The Underlying Mechanism of PM2.5-Induced Ischemic Stroke. Environmental Pollution, 310, Article 119827. https://doi.org/10.1016/j.envpol.2022.119827
[3]
Yu, Y., Yu, J.J.Q., Li, V.O.K. and Lam, J.C.K. (2017) Low-Rank Singular Value Thresholding for Recovering Missing Air Quality Data. 2017 IEEE International Conference on Big Data (Big Data), Boston, 11-14 December 2017, 508-513. https://doi.org/10.1109/bigdata.2017.8257965
[4]
Chan, C.K. and Yao, X. (2008) Air Pollution in Mega Cities in China. Atmospheric Environment, 42, 1-42. https://doi.org/10.1016/j.atmosenv.2007.09.003
[5]
Ma, J., Cheng, J.C.P., Ding, Y., Lin, C., Jiang, F., Wang, M., et al. (2020) Transfer Learning for Long-Interval Consecutive Missing Values Imputation without External Features in Air Pollution Time Series. Advanced Engineering Informatics, 44, Article 101092. https://doi.org/10.1016/j.aei.2020.101092
[6]
Kaiser, J. (2014) Dealing with Missing Values in Data. Journal of Systems Integration, 5, 42-51. https://doi.org/10.20470/jsi.v5i1.178
[7]
Luo, Y., Cai, X., Zhang, Y., Xu, J., et al. (2018) Multivariate Time Series Imputation with Generative Adversarial Networks. Proceedings of the 32nd International Conference on Neural Information Processing Systems, 3-8 December 2018, Red Hook, 1603-1614.
[8]
Bradbury, J., Merity, S., Xiong, C. and Socher, R. (2016) Quasi-Recurrent Neural Networks. arXiv:1611.01576.
[9]
Long, J., Shelhamer, E. and Darrell, T. (2015) Fully Convolutional Networks for Semantic Segmentation. 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, 7-12 June 2015, 3431-3440. https://doi.org/10.1109/CVPR.2015.7298965
[10]
Ma, J., Shou, Z., Zareian, A., Mansour, H., Vetro, A. and Chang, S.-F. (2019) CDSA: Cross-Dimensional Self-Attention for Multivariate, Geo-Tagged Time Series Imputation. arXiv:1905.09904.
[11]
Goodfellow, I.J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A. and Bengio, Y. (2014) Generative Adversarial Nets. In: Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N. and Weinberger, K., Eds., NIPS’14: Proceedings of the 28th International Conference on Neural Information Processing Systems, Vol. 2, MIT Press, 2672-2680.
[12]
Zeng, A., Chen, M., Zhang, L. and Xu, Q. (2023) Are Transformers Effective for Time Series Forecasting? Proceedings of the AAAI Conference on Artificial Intelligence, 37, 11121-11128. https://doi.org/10.1609/aaai.v37i9.26317
[13]
Kim, T., Kim, J., Tae, Y., Park, C., Choi, J.-H. and Choo, J. (2022) Reversible Instance Normalization for Accurate Time-Series Forecasting against Distribution Shift. International Conference on Learning Representations, Chicago, 25-29 October 2022. https://openreview.net/pdf?id=cGDAkQo1C0p
[14]
Liu, Z., Wang, Y., Vaidya, S., Ruehle, F., Halverson, J., Soljačić, M., Hou, T.Y. and Tegmark, M. (2024) Kan: Kolmogorov-Arnold Networks. arXiv:2404.19756.
[15]
Ravanelli, M., Brakel, P., Omologo, M. and Bengio, Y. (2018) Light Gated Recurrent Units for Speech Recognition. IEEE Transactions on Emerging Topics in Computational Intelligence, 2, 92-102. https://doi.org/10.1109/tetci.2017.2762739
[16]
Du, S., Li, T. and Horng, S. (2018) Time Series Forecasting Using Sequence-To-Sequence Deep Learning Framework. 2018 9th International Symposium on Parallel Architectures, Algorithms and Programming (PAAP), Taipei, 26-28 December 2018, 171-176. https://doi.org/10.1109/paap.2018.00037
[17]
Nur’Adzan, N.A., Jaya, M.I., Faizal, M. and Zamri, N.A. (2025) A Systematic Review of Recurrent Neural Network Adoption in Missing Data Imputation. International Journal of Computing and Digital Systems, 17, 1-17. https://doi.org/10.12785/ijcds/1571041166
[18]
Yu, Y., Li, V.O.K., Lam, J.C.K., Chan, K. and Zhang, Q. (2025) CTDI: CNN-Transformer-Based Spatial-Temporal Missing Air Pollution Data Imputation. IEEE Transactions on Big Data, 11, 2443-2456. https://doi.org/10.1109/tbdata.2025.3533882
[19]
Cini, A., Marisca, I. and Alippi, C. (2021) Filling the Gaps: Multivariate Time Series Imputation by Graph Neural Networks. arXiv:2108.00298.
[20]
Kim, S., Lee, T. and Lee, J. (2025) TMF-GNN: Temporal Matrix Factorization-Based Graph Neural Network for Multivariate Time Series Forecasting with Missing Values. Expert Systems with Applications, 275, Article 127001. https://doi.org/10.1016/j.eswa.2025.127001
[21]
Yldz, A.Y., Koç, E. and Koç, A. (2022) Multivariate Time Series Imputation with Transformers. IEEE Signal Processing Letters, 29, 2517-2521. https://doi.org/10.1109/lsp.2022.3224880
[22]
Nie, T., Qin, G., Ma, W., Mei, Y. and Sun, J. (2024) ImputeFormer: Low Rankness-Induced Transformers for Generalizable Spatiotemporal Imputation. Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Barcelona, 25-29 August 2024, 2260-2271. https://doi.org/10.1145/3637528.3671751
[23]
Wu, H., Hu, T., Liu, Y., Zhou, H., Wang, J. and Long, M. (2022) TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis. arXiv:2210.02186.
[24]
Tan, S., Wang, Y., Yuan, Q., Zheng, L., Li, T., Shen, H., et al. (2022) Reconstructing Global PM2.5 Monitoring Dataset from Openaq Using a Two-Step Spatio-Temporal Model Based on SES-IDW and LSTM. Environmental Research Letters, 17, Article 034014. https://doi.org/10.1088/1748-9326/ac52c9
[25]
Huang, S., Zhao, Z., Li, C. and Bai, L. (2025) Timekan: Kan-Based Frequency Decomposition Learning Architecture for Long-Term Time Series Forecasting. arXiv:2502.06910.
[26]
Xu, K., Chen, L. and Wang, S. (2024) Are Kan Effective for Identifying and Tracking Concept Drift in Time Series? arXiv:2410.10041.
[27]
Han, X., Zhang, X., Wu, Y., Zhang, Z. and Wu, Z. (2024) Kan4tsf: Are Kan and Kan-Based Models Effective for Time Series Forecasting? arXiv:2408.11306.
[28]
Zhou, H., Zhang, S., Peng, J., Zhang, S., Li, J., Xiong, H., et al. (2021) Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting. Proceedings of the AAAI Conference on Artificial Intelligence, 35, 11106-11115. https://doi.org/10.1609/aaai.v35i12.17325
[29]
Liu, Y., Hu, T., Zhang, H., Wu, H., Wang, S., Ma, L. and Long, M. (2024) iTransformer: Inverted Transformers Are Effective for Time Series Forecasting. arXiv:2310.06625. https://arxiv.org/abs/2310.06625
[30]
Du, W., Côté, D. and Liu, Y. (2023) SAITS: Self-Attention-Based Imputation for Time Series. Expert Systems with Applications, 219, Article 119619. https://doi.org/10.1016/j.eswa.2023.119619
[31]
Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, Ł. and Polosukhin, I. (2017) Attention Is All You Need. Advances in Neural Information Processing Systems, 30. https://proceedings.neurips.cc/paper_files/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf
[32]
Fortuin, V., Baranchuk, D., Rätsch, G. and Mandt, S. (2020) GP-VAE: Deep Probabilistic Time Series Imputation. International Conference on Artificial Intelligence and Statistics, 1651-1661. https://proceedings.mlr.press/v108/fortuin20a.html
[33]
Cao, W., Wang, D., Li, J., Zhou, H., Li, L. and Li, Y. (2018) Brits: Bidirectional Recurrent Imputation for Time Series. In: Bengio, S., Wallach, H., Larochelle, H., Grauman, K., Ce-sa-Bianchi, N. and Garnett, R., Eds., Advances in Neural Information Processing Systems, 31.
[34]
Yoon, J., Zame, W.R. and van der Schaar, M. (2019) Estimating Missing Data in Temporal Data Streams Using Multi-Directional Recurrent Neural Networks. IEEE Transactions on Biomedical Engineering, 66, 1477-1490. https://doi.org/10.1109/tbme.2018.2874712
[35]
Zhan, T., He, Y., Deng, Y., Li, Z., Du, W. and Wen, Q. (2025) Time Evidence Fusion Network: Multi-Source View in Long-Term Time Series Forecasting. IEEE Transactions on Pattern Analysis and Machine Intelligence, 47, 11220-11233. https://doi.org/10.1109/tpami.2025.3596905
[36]
Wang, S., Li, J., Shi, X., Ye, Z., Mo, B., Lin, W., Ju, S., Chu, Z. and Jin, M. (2024) Timemixer++: A General Time Series Pattern Machine for Universal Predictive Analysis. arXiv:2410.16032.