{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T18:12:13Z","timestamp":1790359933452,"version":"4.1.0"},"reference-count":45,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T00:00:00Z","timestamp":1754956800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Systems"],"abstract":"<jats:p>Multivariate time series forecasting requires modeling complex and evolving spatio-temporal dependencies as well as frequency-domain patterns; however, the existing Transformer-based approaches often struggle to effectively capture dynamic inter-series correlations and disentangle relevant spectral components, leading to limited forecasting accuracy and robustness under non-stationary conditions. To address these challenges, we propose TSGformer, a Transformer-based architecture that integrates multi-scale adaptive graph learning, adaptive spectral decomposition, and cross-scale interactive fusion modules to jointly model temporal, spatial, and spectral dynamics in multivariate time series data. Specifically, TSGformer constructs dynamic graphs at multiple temporal scales to adaptively learn evolving inter-variable relationships, applies an adaptive spectral enhancement module to emphasize critical frequency components while suppressing noise, and employs interactive convolution blocks to fuse multi-domain features effectively. Extensive experiments across eight benchmark datasets show that TSGformer achieves the best results on five datasets, with an MSE of 0.354 on Exchange, improving upon the best baselisnes by 2.4%. Ablation studies further verify the effectiveness of each proposed component, and visualization analyses reveal that TSGformer captures meaningful dynamic correlations aligned with real-world patterns.<\/jats:p>","DOI":"10.3390\/systems13080688","type":"journal-article","created":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T16:30:36Z","timestamp":1755016236000},"page":"688","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["TSGformer: A Unified Temporal\u2013Spatial Graph Transformer with Adaptive Cross-Scale Modeling for Multivariate Time Series"],"prefix":"10.3390","volume":"13","author":[{"given":"Yan","family":"Chen","sequence":"first","affiliation":[{"name":"College of Air Transportation, Shanghai University of Engineering Science, Shanghai 201600, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7912-0643","authenticated-orcid":false,"given":"Cheng","family":"Li","sequence":"additional","affiliation":[{"name":"College of Air Transportation, Shanghai University of Engineering Science, Shanghai 201600, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoli","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Air Transportation, Shanghai University of Engineering Science, Shanghai 201600, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1177\/10946705241232169","article-title":"Hierarchical Time Series Forecasting in Emergency Medical Services","volume":"28","author":"Hyndman","year":"2025","journal-title":"J. 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