{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T04:33:34Z","timestamp":1772771614382,"version":"3.50.1"},"reference-count":55,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T00:00:00Z","timestamp":1744243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Chongqing","award":["CSTB2022NSCQ-LZX0040"],"award-info":[{"award-number":["CSTB2022NSCQ-LZX0040"]}]},{"name":"Natural Science Foundation of Chongqing","award":["CSTB2023NSCQ-LZX0012"],"award-info":[{"award-number":["CSTB2023NSCQ-LZX0012"]}]},{"name":"Natural Science Foundation of Chongqing","award":["CSTB2023NSCQ-LZX0160"],"award-info":[{"award-number":["CSTB2023NSCQ-LZX0160"]}]},{"name":"Natural Science Foundation of Chongqing","award":["IVSTSKL-202302"],"award-info":[{"award-number":["IVSTSKL-202302"]}]},{"name":"Open Project of State Key Laboratory of Intelligent Vehicle Safety Technology","award":["CSTB2022NSCQ-LZX0040"],"award-info":[{"award-number":["CSTB2022NSCQ-LZX0040"]}]},{"name":"Open Project of State Key Laboratory of Intelligent Vehicle Safety Technology","award":["CSTB2023NSCQ-LZX0012"],"award-info":[{"award-number":["CSTB2023NSCQ-LZX0012"]}]},{"name":"Open Project of State Key Laboratory of Intelligent Vehicle Safety Technology","award":["CSTB2023NSCQ-LZX0160"],"award-info":[{"award-number":["CSTB2023NSCQ-LZX0160"]}]},{"name":"Open Project of State Key Laboratory of Intelligent Vehicle Safety Technology","award":["IVSTSKL-202302"],"award-info":[{"award-number":["IVSTSKL-202302"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Time-series forecasting is a cornerstone of decision making in domains such as finance, energy management, and meteorology, where precise predictions drive both economic and operational efficiency. However, traditional time-domain methods often struggle to capture the intricate symmetries and hierarchical dependencies inherent in complex multivariate time-series data. These methods frequently fail to distinguish between global trends and localized fluctuations, limiting their ability to model the multifaceted temporal dynamics that arise across different time scales. To address these challenges, we propose a novel dual-component framework that explicitly leverages the symmetry between long-term trends and short-term fluctuations. Inspired by the principles of signal decomposition, we partition time-series data into a low-frequency stabilization component and a high-frequency fluctuation component. The stabilization component captures inter-variable relationships and global frequency-domain component dependencies through Fourier-transformed frequency-domain representations, variable-oriented attention mechanisms, and dilated causal convolutions. Meanwhile, the fluctuation component models localized dynamics using a multi-granularity structure and time-step attention mechanisms to enhance the sensitivity and robustness to transient variations. By integrating these complementary perspectives, our approach provides a more holistic representation of time-series dynamics. Comprehensive experiments on benchmark datasets from electricity, transportation, and weather domains demonstrate that our method consistently outperforms state-of-the-art models, achieving superior accuracy. Beyond predictive performance, our framework offers a deeper interpretability of temporal behaviors, highlighting its potential for practical applications in complex systems. This work underscores the importance of symmetry-aware modeling in advancing time-series forecasting methodologies.<\/jats:p>","DOI":"10.3390\/sym17040577","type":"journal-article","created":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T08:46:20Z","timestamp":1744274780000},"page":"577","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Modeling Temporal Symmetry: Dual-Component Framework for Trends and Fluctuations in Time Series Forecasting"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-3747-7724","authenticated-orcid":false,"given":"Wei","family":"Ran","sequence":"first","affiliation":[{"name":"State Key Laboratory of Intelligent Vehicle Safety Technology, Chongqing 400023, China"},{"name":"Changan Automobile Company Limited, Chongqing 400023, China"}]},{"given":"Kanlun","family":"Tan","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Intelligent Vehicle Safety Technology, Chongqing 400023, China"},{"name":"Changan Automobile Company Limited, Chongqing 400023, China"}]},{"given":"Zhouyuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"National Center for Applied Mathematics, Chongqing Normal University, Chongqing 401331, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8261-3609","authenticated-orcid":false,"given":"Jiatian","family":"Pi","sequence":"additional","affiliation":[{"name":"National Center for Applied Mathematics, Chongqing Normal University, Chongqing 401331, China"}]},{"given":"Yichuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"National Center for Applied Mathematics, Chongqing Normal University, Chongqing 401331, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1007\/s11063-017-9627-1","article-title":"A hybrid model equipped with the minimum cycle decomposition concept for short-term forecasting of electrical load time series","volume":"46","author":"He","year":"2017","journal-title":"Neural Process. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s11063-015-9409-6","article-title":"A short-term traffic flow forecasting method based on the hybrid PSO-SVR","volume":"43","author":"Hu","year":"2016","journal-title":"Neural Process. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4407","DOI":"10.1007\/s11063-021-10606-7","article-title":"Short term solar power and temperature forecast using recurrent neural networks","volume":"53","author":"Gundu","year":"2021","journal-title":"Neural Process. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Alyousifi, Y., Othman, M., Sokkalingam, R., Faye, I., and Silva, P.C. (2020). Predicting daily air pollution index based on fuzzy time series markov chain model. Symmetry, 12.","DOI":"10.3390\/sym12020293"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s11063-021-10495-w","article-title":"Nonlinear neural network based forecasting model for predicting COVID-19 cases","volume":"55","author":"Namasudra","year":"2023","journal-title":"Neural Process. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yasar, H., and Kilimci, Z.H. (2020). US dollar\/Turkish lira exchange rate forecasting model based on deep learning methodologies and time series analysis. Symmetry, 12.","DOI":"10.3390\/sym12091553"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3357","DOI":"10.1007\/s11063-022-10767-z","article-title":"A new CNN-based model for financial time series: TAIEX and FTSE stocks forecasting","volume":"54","author":"Kirisci","year":"2022","journal-title":"Neural Process. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Cruz-N\u00e1jera, M.A., Trevi\u00f1o-Berrones, M.G., Ponce-Flores, M.P., Ter\u00e1n-Villanueva, J.D., Cast\u00e1n-Rocha, J.A., Ibarra-Mart\u00ednez, S., Santiago, A., and Laria-Menchaca, J. (2022). Short time series forecasting: Recommended methods and techniques. Symmetry, 14.","DOI":"10.3390\/sym14061231"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Wang, C., Zhang, Z., Wang, X., Liu, M., Chen, L., and Pi, J. (2024). Frequency-Enhanced Transformer with Symmetry-Based Lightweight Multi-Representation for Multivariate Time Series Forecasting. Symmetry, 16.","DOI":"10.3390\/sym16070797"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.ins.2019.01.076","article-title":"Evaluation of statistical and machine learning models for time series prediction: Identifying the state-of-the-art and the best conditions for the use of each model","volume":"484","author":"Parmezan","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bontempi, G., Ben Taieb, S., and Le Borgne, Y.A. (2013). Machine learning strategies for time series forecasting. Business Intelligence: Second European Summer School, eBISS 2012, Brussels, Belgium, July 15\u201321, 2012, Tutorial Lectures 2, Springer.","DOI":"10.1007\/978-3-642-36318-4_3"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1109\/MCI.2009.932254","article-title":"Time series prediction using support vector machines: A survey","volume":"4","author":"Sapankevych","year":"2009","journal-title":"IEEE Comput. Intell. Mag."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1111\/joes.12429","article-title":"Machine learning advances for time series forecasting","volume":"37","author":"Masini","year":"2023","journal-title":"J. Econ. Surv."},{"key":"ref_14","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_15","unstructured":"Oreshkin, B.N., Carpov, D., Chapados, N., and Bengio, Y. (2019). NBEATS: Neural basis expansion analysis for interpretable time series forecasting. arXiv."},{"key":"ref_16","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. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"7833","DOI":"10.1109\/JSEN.2019.2923982","article-title":"A review of deep learning models for time series prediction","volume":"21","author":"Han","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_18","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_19","doi-asserted-by":"crossref","unstructured":"Qin, Y., Song, D., Chen, H., Cheng, W., Jiang, G., and Cottrell, G. (2017). A dual-stage attention-based recurrent neural network for time series prediction. arXiv.","DOI":"10.24963\/ijcai.2017\/366"},{"key":"ref_20","unstructured":"Borovykh, A., Bohte, S., and Oosterlee, C.W. (2017). Conditional time series forecasting with convolutional neural networks. arXiv."},{"key":"ref_21","first-page":"4837","article-title":"Think globally, act locally: A deep neural network approach to high-dimensional time series forecasting","volume":"32","author":"Sen","year":"2019","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_22","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_23","first-page":"6000","article-title":"Attention is all you need","volume":"30","author":"Vaswani","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_24","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_25","unstructured":"Koopmans, L.H. (1995). The Spectral Analysis of Time Series, Elsevier."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Beran, J. (2017). Statistics for Long-Memory Processes, Routledge.","DOI":"10.1201\/9780203738481"},{"key":"ref_27","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_28","unstructured":"Zhou, T., Ma, Z., Wen, Q., Wang, X., Sun, L., and Jin, R. (2022, January 17\u201323). Fedformer: Frequency enhanced decomposed transformer for long-term series forecasting. Proceedings of the International Conference on Machine Learning, PMLR, Baltimore, MD, USA."},{"key":"ref_29","unstructured":"Xu, Z., Zeng, A., and Xu, Q. (2023). FITS: Modeling time series with 10k parameters. arXiv."},{"key":"ref_30","unstructured":"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."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Lee-Thorp, J., Ainslie, J., Eckstein, I., and Ontanon, S. (2021). Fnet: Mixing tokens with fourier transforms. arXiv.","DOI":"10.18653\/v1\/2022.naacl-main.319"},{"key":"ref_32","first-page":"12677","article-title":"Film: Frequency improved legendre memory model for long-term time series forecasting","volume":"35","author":"Zhou","year":"2022","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_33","unstructured":"Li, S., Jin, X., Xuan, Y., Zhou, X., Chen, W., Wang, Y.X., and Yan, X. (2019, January 8\u201314). Enhancing the locality and breaking the memory bottleneck of transformer on time series forecasting. Proceedings of the 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, BC, Canada."},{"key":"ref_34","unstructured":"Liu, S., Yu, H., Liao, C., Li, J., Lin, W., Liu, A.X., and Dustdar, S. (2021, January 4). Pyraformer: Low-complexity pyramidal attention for long-range time series modeling and forecasting. Proceedings of the International Conference on Learning Representations, Vienna, Austria."},{"key":"ref_35","unstructured":"Liu, Y., Hu, T., Zhang, H., Wu, H., Wang, S., Ma, L., and Long, M. (2023). itransformer: Inverted transformers are effective for time series forecasting. arXiv."},{"key":"ref_36","unstructured":"Dosovitskiy, A. (2020). An image is worth 16x16 words: Transformers for image recognition at scale. arXiv."},{"key":"ref_37","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_38","unstructured":"Wang, Y., Wu, H., Dong, J., Liu, Y., Qiu, Y., Zhang, H., Wang, J., and Long, M. (2024). Timexer: Empowering transformers for time series forecasting with exogenous variables. arXiv."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1111\/j.1553-2712.1998.tb02493.x","article-title":"Time series analysis using autoregressive integrated moving average (ARIMA) models","volume":"5","author":"Nelson","year":"1998","journal-title":"Acad. Emerg. Med."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/0304-3932(92)90016-U","article-title":"Modeling long-run behavior with the fractional ARIMA model","volume":"29","author":"Sowell","year":"1992","journal-title":"J. Monet. Econ."},{"key":"ref_41","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_42","unstructured":"Kalekar, P.S. (2025, April 03). Time Series Forecasting Using Holt-Winters Exponential Smoothing. Kanwal Rekhi School of Information Technology, Mumbai, India. Available online: https:\/\/c.mql5.com\/forextsd\/forum\/69\/exponentialsmoothing.pdf."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","article-title":"Long short-term memory","volume":"9","author":"Hochreiter","year":"1997","journal-title":"Neural Comput."},{"key":"ref_44","unstructured":"Wang, H., Peng, J., Huang, F., Wang, J., Chen, J., and Xiao, Y. (2022, January 25\u201329). Micn: Multi-scale local and global context modeling for long-term series forecasting. Proceedings of the Eleventh International Conference on Learning Representations, Virtual."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"109726","DOI":"10.1016\/j.asoc.2022.109726","article-title":"A new decomposition ensemble model for stock price forecasting based on system clustering and particle swarm optimization","volume":"130","author":"Guo","year":"2022","journal-title":"Appl. Soft Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"109714","DOI":"10.1016\/j.asoc.2022.109714","article-title":"A decomposition-based memetic neural architecture search algorithm for univariate time series forecasting","volume":"130","author":"Li","year":"2022","journal-title":"Appl. Soft Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"110794","DOI":"10.1016\/j.knosys.2023.110794","article-title":"DFNet: Decomposition fusion model for long sequence time-series forecasting","volume":"277","author":"Zhang","year":"2023","journal-title":"Knowl. Based Syst."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s11263-015-0816-y","article-title":"ImageNet large scale visual recognition challenge","volume":"115","author":"Russakovsky","year":"2015","journal-title":"Int. J. Comput. Vis."},{"key":"ref_49","unstructured":"Ba, J.L., Kiros, J.R., and Hinton, G.E. (2016). Layer normalization. arXiv."},{"key":"ref_50","first-page":"17","article-title":"Instance normalization: The missing ingredient for fast stylization","volume":"Volume 11217","author":"Ulyanov","year":"2018","journal-title":"Proceedings of the Computer Vision\u2013ECCV 2018: 15th European Conference"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Gehring, J., Auli, M., Grangier, D., and Dauphin, Y.N. (2016). A convolutional encoder model for neural machine translation. arXiv.","DOI":"10.18653\/v1\/P17-1012"},{"key":"ref_52","unstructured":"Van Den Oord, A., Dieleman, S., Zen, H., Simonyan, K., Vinyals, O., Graves, A., Kalchbrenner, N., Senior, A., and Kavukcuoglu, K. (2016). Wavenet: A generative model for raw audio. arXiv."},{"key":"ref_53","first-page":"8026","article-title":"An imperative style, high-performance deep learning library","volume":"32","author":"Paszke","year":"1912","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_54","unstructured":"Kingma, D.P., and Ba, J. (2015, January 7\u20139). A method for stochastic optimization. Proceedings of the International Conference on Learning Representations (ICLR), San Diego, CA, USA."},{"key":"ref_55","unstructured":"Garza, A., and Mergenthaler-Canseco, M. (2023). TimeGPT-1. arXiv."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/577\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:12:17Z","timestamp":1760029937000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/577"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,10]]},"references-count":55,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["sym17040577"],"URL":"https:\/\/doi.org\/10.3390\/sym17040577","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,10]]}}}