{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T16:04:36Z","timestamp":1784822676418,"version":"3.55.0"},"reference-count":45,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,22]],"date-time":"2024-01-22T00:00:00Z","timestamp":1705881600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science and Technology Support Plan Project of Guizhou","award":["QKHZC [2022] 267"],"award-info":[{"award-number":["QKHZC [2022] 267"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Large-scale and high-dimensional time series data are widely generated in modern applications such as intelligent transportation and environmental monitoring. However, such data contains much noise, outliers, and missing values due to interference during measurement or transmission. Directly forecasting such types of data (i.e., anomalous data) can be extremely challenging. The traditional method to deal with anomalies is to cut out the time series with anomalous value entries or replace the data. Both methods may lose important knowledge from the original data. In this paper, we propose a multidimensional time series forecasting framework that can better handle anomalous values: the robust temporal nonnegative matrix factorization forecasting model (RTNMFFM) for multi-dimensional time series. RTNMFFM integrates the autoregressive regularizer into nonnegative matrix factorization (NMF) with the application of the L2,1 norm in NMF. This approach improves robustness and alleviates overfitting compared to standard methods. In addition, to improve the accuracy of model forecasts on severely missing data, we propose a periodic smoothing penalty that keeps the sparse time slices as close as possible to the time slice with high confidence. Finally, we train the model using the alternating gradient descent algorithm. Numerous experiments demonstrate that RTNMFFM provides better robustness and better prediction accuracy.<\/jats:p>","DOI":"10.3390\/e26010092","type":"journal-article","created":{"date-parts":[[2024,1,23]],"date-time":"2024-01-23T08:28:28Z","timestamp":1705998508000},"page":"92","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Robust Multi-Dimensional Time Series Forecasting"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-4765-7531","authenticated-orcid":false,"given":"Chen","family":"Shen","sequence":"first","affiliation":[{"name":"State Key Laboratory of Public Big Data, College of Computer Science and Technology, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yong","family":"He","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Public Big Data, College of Computer Science and Technology, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jin","family":"Qin","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Public Big Data, College of Computer Science and Technology, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.enbuild.2019.05.021","article-title":"A methodology for energy multivariate time series forecasting in smart buildings based on feature selection","volume":"196","year":"2019","journal-title":"Energy Build."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.1007\/s11277-020-07108-5","article-title":"A survey on architecture, protocols and challenges in IoT","volume":"112","author":"Sobin","year":"2020","journal-title":"Wirel. 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