{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T02:13:35Z","timestamp":1783995215041,"version":"3.55.0"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,17]],"date-time":"2025-01-17T00:00:00Z","timestamp":1737072000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ningbo Science and Technology Major Project","award":["2024Z259"],"award-info":[{"award-number":["2024Z259"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>Time-series data are widely applied in real-world scenarios, but the non-stationary nature of their statistical properties and joint distributions over time poses challenges for existing forecasting models. To tackle this challenge, this paper introduces a forecasting model called DFCNformer (De-stationary Fourier and Coefficient Network Transformer), designed to mitigate accuracy degradation caused by non-stationarity in time-series data. The model initially employs a stabilization strategy to unify the statistical characteristics of the input time series, restoring their original features at the output to enhance predictability. Then, a time-series decomposition method splits the data into seasonal and trend components. For the seasonal component, a Transformer-based encoder\u2013decoder architecture with De-stationary Fourier Attention (DSF Attention) captures temporal features, using differentiable attention weights to restore non-stationary information. For the trend component, a multilayer perceptron (MLP) is used for prediction, enhanced by a Dual Coefficient Network (Dual-CONET) that mitigates distributional shifts through learnable distribution coefficients. Ultimately, the forecasts of the seasonal and trend components are combined to generate the overall prediction. Experimental findings reveal that when the proposed model is tested on six public datasets, in comparison with five classic models it reduces the MSE by an average of 9.67%, with a maximum improvement of 40.23%.<\/jats:p>","DOI":"10.3390\/info16010062","type":"journal-article","created":{"date-parts":[[2025,1,17]],"date-time":"2025-01-17T11:24:56Z","timestamp":1737113096000},"page":"62","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["DFCNformer: A Transformer Framework for Non-Stationary Time-Series Forecasting Based on De-Stationary Fourier and Coefficient Network"],"prefix":"10.3390","volume":"16","author":[{"given":"Yuxin","family":"Jin","sequence":"first","affiliation":[{"name":"School of Computer Science and Technology (School of Artificial Intelligence), Zhejiang Sci-Tech University, Hangzhou 310018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuhan","family":"Mao","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Zhejiang Sci-Tech University, Hangzhou 310018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-6964-8143","authenticated-orcid":false,"given":"Genlang","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Computer and Data Engineering, Ningbo Tech University, Ningbo 315199, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wang, H., Li, J., and Chang, L. (2023). Predicting Time Series Energy Consumption Based on Transformer and LSTM. International Conference on 6GN for Future Wireless Networks, Springer Nature.","DOI":"10.1007\/978-3-031-53401-0_27"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"102146","DOI":"10.1016\/j.inffus.2023.102146","article-title":"Traffic flow matrix-based graph neural network with attention mechanism for traffic flow prediction","volume":"104","author":"Chen","year":"2024","journal-title":"Inf. Fusion"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1007\/s00530-024-01494-7","article-title":"TEST-Net: Transformer-enhanced Spatio-temporal network for infectious disease prediction","volume":"30","author":"Chen","year":"2024","journal-title":"Multimedia Syst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"He, K., Yang, Q., Ji, L., Pan, J., and Zou, Y. (2023). Financial time series forecasting with the deep learning ensemble model. Mathematics, 11.","DOI":"10.3390\/math11041054"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"104045","DOI":"10.1016\/j.jngse.2021.104045","article-title":"Machine learning-based production forecast for shale gas in unconventional reservoirs via integration of geological and operational factors","volume":"94","author":"Hui","year":"2021","journal-title":"J. Nat. Gas Sci. Eng."},{"key":"ref_6","unstructured":"Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A., and Polosukhin, I. (2017, January 4\u20139). Attention is all you need. Proceedings of the 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1586","DOI":"10.1109\/TNNLS.2020.2985720","article-title":"LSTM-MSNet: Leveraging forecasts on sets of related time series with multiple seasonal patterns","volume":"32","author":"Bandara","year":"2020","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Wen, Q., Zhang, Z., Li, Y., and Sun, L. (2020, January 6\u201310). Fast RobustSTL: Efficient and robust seasonal-trend decomposition for time series with complex patterns. Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, Virtual Event.","DOI":"10.1145\/3394486.3403271"},{"key":"ref_9","first-page":"22419","article-title":"Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting","volume":"34","author":"Wu","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_10","first-page":"27268","article-title":"Fedformer: Frequency enhanced decomposed transformer for long-term series forecasting","volume":"162","author":"Zhou","year":"2022","journal-title":"Int. Conf. Mach. Learn."},{"key":"ref_11","unstructured":"Kim, T., Kim, J., Tae, Y., Park, C., Choi, J.H., and Choo, J. (2022, January 25\u201329). Reversible instance normalization for accurate time-series forecasting against distribution shift. Proceedings of the International Conference on Learning Representations, Virtual."},{"key":"ref_12","first-page":"9881","article-title":"Non-stationary transformers: Exploring the stationarity in time series forecasting","volume":"35","author":"Liu","year":"2022","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIM.2021.3123218","article-title":"A model for non-stationary time series and its applications in filtering and anomaly detection","volume":"70","author":"Wang","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ariyo, A.A., Adewumi, A.O., and Ayo, C.K. (2014, January 26\u201328). Stock price prediction using the ARIMA model. Proceedings of the 2014 UKSim-AMSS 16th International Conference on Computer Modelling and Simulation, Cambridge, UK.","DOI":"10.1109\/UKSim.2014.67"},{"key":"ref_15","unstructured":"Yu, R., Zheng, S., Anandkumar, A., and Yue, Y. (May, January 30). Long-term forecasting using tensor-train rnns. Proceedings of the 6th International Conference on Learning Representations (ICLR 2018), Vancouver, BC, Canada."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1016\/j.ijforecast.2019.07.001","article-title":"DeepAR: Probabilistic forecasting with autoregressive recurrent networks","volume":"36","author":"Salinas","year":"2020","journal-title":"Int. J. Forecast."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1007\/s12652-020-02761-x","article-title":"Improving time series forecasting using LSTM and attention models","volume":"13","author":"Abbasimehr","year":"2022","journal-title":"J. Ambient. Intell. Humaniz. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Casolaro, A., Capone, V., Iannuzzo, G., and Camastra, F. (2023). Deep learning for time series forecasting: Advances and open problems. Information, 14.","DOI":"10.3390\/info14110598"},{"key":"ref_19","unstructured":"Bai, S., Kolter, J.Z., and Koltun, V. (2018). An empirical evaluation of generic convolutional and recurrent networks for sequence modeling. arXiv."},{"key":"ref_20","first-page":"5816","article-title":"Scinet: Time series modeling and forecasting with sample convolution and interaction","volume":"35","author":"Liu","year":"2022","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Wan, R., Tian, C., Zhang, W., Deng, W., and Yang, F. (2022). A multivariate temporal convolutional attention network for time-series forecasting. Electronics, 11.","DOI":"10.3390\/electronics11101516"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.3390\/make5040068","article-title":"Predicting the long-term dependencies in time series using recurrent artificial neural networks","volume":"5","author":"Ubal","year":"2023","journal-title":"Mach. Learn. Knowl. Extr."},{"key":"ref_23","first-page":"11121","article-title":"Are transformers effective for time series forecasting?","volume":"37","author":"Zeng","year":"2023","journal-title":"Proc. AAAI Conf. Artif. Intell."},{"key":"ref_24","unstructured":"Kitaev, N., Kaiser, \u0141., and Levskaya, A. (2020). Reformer: The efficient transformer. arXiv."},{"key":"ref_25","first-page":"11106","article-title":"Informer: Beyond efficient transformer for long sequence time-series forecasting","volume":"35","author":"Zhou","year":"2021","journal-title":"Proc. AAAI Conf. Artif. Intell."},{"key":"ref_26","unstructured":"Zhang, Y., and Yan, J. (2023, January 1\u20135). Crossformer: Transformer utilizing cross-dimension dependency for multivariate time series forecasting. Proceedings of the Eleventh International Conference on Learning Representations, Kigali, Rwanda."},{"key":"ref_27","unstructured":"Zhang, X., Jin, X., Gopalswamy, K., Gupta, G., Park, Y., Shi, X., Wang, H., Maddix, D.C., and Wang, Y. (2022). First de-trend then attend: Rethinking attention for time-series forecasting. arXiv."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Ogasawara, E., Martinez, L.C., de Oliveira, D., Zimbrao, G., Pappa, G.L., and Mattoso, M. (2010, January 18\u201323). Adaptive normalization: A novel data normalization approach for non-stationary time series. Proceedings of the 2010 International Joint Conference on Neural Networks (IJCNN), Barcelona, Spain.","DOI":"10.1109\/IJCNN.2010.5596746"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"20200209","DOI":"10.1098\/rsta.2020.0209","article-title":"Time-series forecasting with deep learning: A survey","volume":"379","author":"Lim","year":"2021","journal-title":"Philos. Trans. R. Soc."},{"key":"ref_30","first-page":"5637","article-title":"Wilds: A benchmark of in-the-wild distribution shifts","volume":"139","author":"Koh","year":"2021","journal-title":"Int. Conf. Mach. Learn."},{"key":"ref_31","first-page":"7522","article-title":"Dish-ts: A general paradigm for alleviating distribution shift in time series forecasting","volume":"37","author":"Fan","year":"2023","journal-title":"Proc. AAAI Conf. Artif. Intell."},{"key":"ref_32","first-page":"14273","article-title":"Adaptive normalization for non-stationary time series forecasting: A temporal slice perspective","volume":"36","author":"Liu","year":"2024","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_33","unstructured":"Oreshkin, B.N., Carpov, D., Chapados, N., and Bengio, Y. (2019). N-BEATS: Neural basis expansion analysis for interpretable time series forecasting. arXiv."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3760","DOI":"10.1109\/TNNLS.2019.2944933","article-title":"Deep adaptive input normalization for time series forecasting","volume":"31","author":"Passalis","year":"2019","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"106196","DOI":"10.1016\/j.neunet.2024.106196","article-title":"TCDformer: A transformer framework for non-stationary time series forecasting based on trend and change-point detection","volume":"173","author":"Wan","year":"2024","journal-title":"Neural Netw."},{"key":"ref_36","unstructured":"Yi, K., Zhang, Q., Cao, L., Wang, S., Long, G., Hu, L., He, H., Niu, Z., Fan, W., and Xiong, H. (2023). A survey on deep learning based time series analysis with frequency transformation. arXiv."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Puech, T., Boussard, M., D\u2019Amato, A., and Millerand, G. (2020). A fully automated periodicity detection in time series. Advanced Analytics and Learning on Temporal Data: 4th ECML PKDD Workshop, AALTD 2019, W\u00fcrzburg, Germany, 20 September 2019, Springer International Publishing. Revised Selected Papers 4.","DOI":"10.1007\/978-3-030-39098-3_4"},{"key":"ref_38","unstructured":"Tang, P., and Zhang, W. (2024). PDMLP: Patch-based Decomposed MLP for Long-Term Time Series Forecastin. arXiv."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Lai, G., Chang, W.C., Yang, Y., and Liu, H. (2018, January 8\u201312). Modeling long-and short-term temporal patterns with deep neural networks. Proceedings of the 41st International ACM SIGIR Conference on Research & Development in Information Retrieval, Ann Arbor, MI, USA.","DOI":"10.1145\/3209978.3210006"},{"key":"ref_40","unstructured":"(2024, May 14). Traffic Dataset, Available online: http:\/\/pems.dot.ca.gov\/."},{"key":"ref_41","unstructured":"(2024, May 14). Weather Dataset. Available online: https:\/\/www.bgc-jena.mpg.de\/wetter\/."},{"key":"ref_42","unstructured":"(2024, May 14). ILI Dataset, Available online: https:\/\/gis.cdc.gov\/grasp\/fluview\/fluportaldashboard.html."},{"key":"ref_43","unstructured":"(2024, May 14). Citypower Dataset. Available online: https:\/\/kaggle.com\/datasets\/fedesoriano\/electric-power-consumption."}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/16\/1\/62\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:30:43Z","timestamp":1759919443000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/16\/1\/62"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,17]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["info16010062"],"URL":"https:\/\/doi.org\/10.3390\/info16010062","relation":{},"ISSN":["2078-2489"],"issn-type":[{"value":"2078-2489","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,17]]}}}