{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T09:34:16Z","timestamp":1761989656314,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,2]],"date-time":"2022-03-02T00:00:00Z","timestamp":1646179200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["No. 2017YFC1600605"],"award-info":[{"award-number":["No. 2017YFC1600605"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 61673002, 61903009"],"award-info":[{"award-number":["No. 61673002, 61903009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Beijing Excellent Talent Training Support Project for Young Top-Notch Team","award":["No. 2018000026833TD01"],"award-info":[{"award-number":["No. 2018000026833TD01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The prediction of time series is of great significance for rational planning and risk prevention. However, time series data in various natural and artificial systems are nonstationary and complex, which makes them difficult to predict. An improved deep prediction method is proposed herein based on the dual variational mode decomposition of a nonstationary time series. First, criteria were determined based on information entropy and frequency statistics to determine the quantity of components in the variational mode decomposition, including the number of subsequences and the conditions for dual decomposition. Second, a deep prediction model was built for the subsequences obtained after the dual decomposition. Third, a general framework was proposed to integrate the data decomposition and deep prediction models. The method was verified on practical time series data with some contrast methods. The results show that it performed better than single deep network and traditional decomposition methods. The proposed method can effectively extract the characteristics of a nonstationary time series and obtain reliable prediction results.<\/jats:p>","DOI":"10.3390\/e24030360","type":"journal-article","created":{"date-parts":[[2022,3,2]],"date-time":"2022-03-02T08:37:16Z","timestamp":1646210236000},"page":"360","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Deep Prediction Model Based on Dual Decomposition with Entropy and Frequency Statistics for Nonstationary Time Series"],"prefix":"10.3390","volume":"24","author":[{"given":"Zhigang","family":"Shi","sequence":"first","affiliation":[{"name":"School of Artificial Intelligence, Beijing Technology and Business University, Beijing 100048, China"},{"name":"Beijing Laboratory for Intelligent Environmental Protection, Beijing Technology and Business University, Beijing 100048, China"},{"name":"State Environmental Protection Key Laboratory of Food Chain Pollution Control, Beijing Technology and Business University, Beijing 100048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8047-1010","authenticated-orcid":false,"given":"Yuting","family":"Bai","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Beijing Technology and Business University, Beijing 100048, China"},{"name":"Beijing Laboratory for Intelligent Environmental Protection, Beijing Technology and Business University, Beijing 100048, China"},{"name":"State Environmental Protection Key Laboratory of Food Chain Pollution Control, Beijing Technology and Business University, Beijing 100048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2230-0077","authenticated-orcid":false,"given":"Xuebo","family":"Jin","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Beijing Technology and Business University, Beijing 100048, China"},{"name":"Beijing Laboratory for Intelligent Environmental Protection, Beijing Technology and Business University, Beijing 100048, China"},{"name":"State Environmental Protection Key Laboratory of Food Chain Pollution Control, Beijing Technology and Business University, Beijing 100048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyi","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Beijing Technology and Business University, Beijing 100048, China"},{"name":"Beijing Laboratory for Intelligent Environmental Protection, Beijing Technology and Business University, Beijing 100048, China"},{"name":"State Environmental Protection Key Laboratory of Food Chain Pollution Control, Beijing Technology and Business University, Beijing 100048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tingli","family":"Su","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Beijing Technology and Business University, Beijing 100048, China"},{"name":"Beijing Laboratory for Intelligent Environmental Protection, Beijing Technology and Business University, Beijing 100048, China"},{"name":"State Environmental Protection Key Laboratory of Food Chain Pollution Control, Beijing Technology and Business University, Beijing 100048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0074-3467","authenticated-orcid":false,"given":"Jianlei","family":"Kong","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Beijing Technology and Business University, Beijing 100048, China"},{"name":"Beijing Laboratory for Intelligent Environmental Protection, Beijing Technology and Business University, Beijing 100048, China"},{"name":"State Environmental Protection Key Laboratory of Food Chain Pollution Control, Beijing Technology and Business University, Beijing 100048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3368","DOI":"10.1080\/01431161.2019.1701724","article-title":"Spatio-temporal predictions of SST time series in China\u2019s offshore waters using a regional convolution long short-term memory (RC-LSTM) network","volume":"41","author":"Xu","year":"2020","journal-title":"Int. J. Remote Sens."},{"key":"ref_2","first-page":"317","article-title":"Short-term local prediction of wind speed and wind power based on singular spectrum analysis and locality-sensitive hashing","volume":"6","author":"Liu","year":"2018","journal-title":"Mod. Power Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.neucom.2018.01.038","article-title":"Predicting the direction of stock markets using optimized neural networks with Google Trends","volume":"285","author":"Hu","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bai, Y., Wang, X., Sun, Q., Jin, X., Wang, X., Su, T., and Kong, J. (2019). Spatio-Temporal Prediction for the Monitoring-Blind Area of Industrial Atmosphere Based on the Fusion Network. Int. J. Environ. Res. Public Health, 16.","DOI":"10.3390\/ijerph16203788"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bai, Y., Jin, X., Wang, X., Wang, X., and Xu, J. (2020). Dynamic Correlation Analysis Method of Air Pollutants in Spatio-Temporal Analysis. Int. J. Environ. Res. Public Health, 17.","DOI":"10.3390\/ijerph17010360"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yang, Y., Bai, Y., Wang, X., Wang, L., Jin, X., and Sun, Q. (2020). Group decision-making support for sustainable governance of algal bloom in urban lakes. Sustainability, 12.","DOI":"10.3390\/su12041494"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1016\/j.neucom.2007.07.018","article-title":"Soft-computing techniques and arma model for time series prediction","volume":"71","author":"Rojas","year":"2008","journal-title":"Neurocomputing"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.solener.2004.09.013","article-title":"Forecast of hourly average wind speed with ARMA models in Navarre (Spain)","volume":"79","author":"Torres","year":"2005","journal-title":"Sol. Energy"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3606","DOI":"10.1016\/j.apenergy.2010.05.012","article-title":"Day-ahead electricity price forecasting using wavelet transform combined with ARIMA and GARCH models","volume":"87","author":"Tan","year":"2010","journal-title":"Appl. Energy"},{"key":"ref_10","first-page":"701","article-title":"Chaotic time series prediction based on elm learning algorithm","volume":"44","author":"Bin","year":"2011","journal-title":"J. Tianjin Univ."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wang, F., Yu, Y., Zhang, Z., Li, J., Zhen, Z., and Li, K. (2018). Wavelet Decomposition and Convolutional LSTM Networks Based Improved Deep Learning Model for Solar Irradiance Forecasting. Appl. Sci., 8.","DOI":"10.3390\/app8081286"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"112254","DOI":"10.1016\/j.enconman.2019.112254","article-title":"A new prediction method based on VMD-PRBF-ARMA-E model considering wind speed characteristic","volume":"203","author":"Zhang","year":"2020","journal-title":"Energy Convers. Manag."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Xie, T., Zhang, G., Liu, H., Liu, F., and Du, P. (2018). A Hybrid Forecasting Method for Solar Output Power Based on Variational Mode Decomposition, Deep Belief Networks and Auto-Regressive Moving Average. Appl. Sci., 8.","DOI":"10.3390\/app8101901"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"8385416","DOI":"10.1155\/2020\/8385416","article-title":"Monthly Mean Meteorological Temperature Prediction Based on VMD-DSE and Volterra Adaptive Model","volume":"2020","author":"Li","year":"2020","journal-title":"Adv. Meteorol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1109\/TAES.1983.309419","article-title":"ARMA Time Series Modeling: An Effective Method","volume":"AES-19","author":"Cadzow","year":"1983","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1080\/2150704X.2017.1418992","article-title":"Short-term cloud coverage prediction using the ARIMA time series model","volume":"9","author":"Wang","year":"2018","journal-title":"Remote Sens. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1109\/TPWRS.2005.846044","article-title":"A garch forecasting model to predict day-ahead electricity prices","volume":"20","author":"Garcia","year":"2005","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Durbin, J., and Koopman, S.J. (2012). Time Series Analysis by State Space Methods, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199641178.001.0001"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Bai, Y.-T., Wang, X.-Y., Jin, X.-B., Zhao, Z.-Y., and Zhang, B.-H. (2020). A Neuron-Based Kalman Filter with Nonlinear Autoregressive Model. Sensors, 20.","DOI":"10.3390\/s20010299"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.neucom.2017.04.076","article-title":"Financial time series prediction using 2,1rf-elm","volume":"277","author":"Xue","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1623\/hysj.51.4.599","article-title":"Using support vector machines for long-term discharge prediction","volume":"51","author":"Lin","year":"2006","journal-title":"Hydrol. Sci. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1109\/TPWRS.2006.873409","article-title":"Day-ahead price forecasting of electricity markets by a new fuzzy neural network","volume":"21","author":"Amjady","year":"2006","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6781","DOI":"10.1007\/s00521-018-3488-z","article-title":"Dynamical regularized echo state network for time series prediction","volume":"31","author":"Yang","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6085","DOI":"10.1038\/s41598-018-24271-9","article-title":"Recurrent Neural Networks for Multivariate Time Series with Missing Values","volume":"8","author":"Che","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_25","unstructured":"Hochreiter, S., and Schmidhuber, J. (1996). LSTM can Solve Hard Long Time Lag Problems. Adv. Neural Inf. Process. Syst., 473\u2013479."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1016\/j.ejor.2017.11.054","article-title":"Deep learning with long short-term memory networks for financial market predictions","volume":"270","author":"Fischer","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Jin, X., Yang, N., Wang, X., Bai, Y., Su, T., and Kong, J. (2020). Hybrid Deep Learning Predictor for Smart Agriculture Sensing Based on Empirical Mode Decomposition and Gated Recurrent Unit Group Model. Sensors, 20.","DOI":"10.3390\/s20051334"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.neucom.2019.07.058","article-title":"A gated recurrent unit neural networks based wind speed error correction model for short-term wind power forecasting","volume":"365","author":"Ding","year":"2019","journal-title":"Neurocomputing"},{"key":"ref_29","first-page":"3","article-title":"Stl: A seasonal-trend decomposition procedure based on loess","volume":"6","author":"Cleveland","year":"1990","journal-title":"J. Off. Stat."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.neucom.2019.04.061","article-title":"Effective passenger flow forecasting using STL and ESN based on two improvement strategies","volume":"356","author":"Qin","year":"2019","journal-title":"Neurocomputing"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"142814","DOI":"10.1109\/ACCESS.2019.2944755","article-title":"The Forecasting of PM2.5 Using a Hybrid Model Based on Wavelet Transform and an Improved Deep Learning Algorithm","volume":"7","author":"Qiao","year":"2019","journal-title":"IEEE Access"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Gao, X., Li, X., Zhao, B., Ji, W., Jing, X., and He, Y. (2019). Short-Term Electricity Load Forecasting Model Based on EMD-GRU with Feature Selection. Energies, 12.","DOI":"10.3390\/en12061140"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Xu, Y., Zhang, J., Long, Z., and Chen, Y. (2018). A Novel Dual-Scale Deep Belief Network Method for Daily Urban Water Demand Forecasting. Energies, 11.","DOI":"10.3390\/en11051068"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Wang, K., Niu, D., Sun, L., Zhen, H., and Xu, X. (2019). Wind power short-term forecasting hybrid model based on ceemd-se method. Processes, 7.","DOI":"10.3390\/pr7110843"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Jin, X., Yang, N., Wang, X., Bai, Y., Su, T., and Kong, J. (2019). Integrated Predictor Based on Decomposition Mechanism for PM2.5 Long-Term Prediction. Appl. Sci., 9.","DOI":"10.3390\/app9214533"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Jin, X.B., Yang, N.X., Wang, X.Y., Bai, Y.T., Su, T.L., and Kong, J.L. (2020). Deep hybrid model based on emd with classification by frequency characteristics for long-term air quality prediction. Mathematics, 8.","DOI":"10.3390\/math8020214"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"04020008","DOI":"10.1061\/(ASCE)HE.1943-5584.0001902","article-title":"Annual Streamflow Time Series Prediction Using Extreme Learning Machine Based on Gravitational Search Algorithm and Variational Mode Decomposition","volume":"25","author":"Niu","year":"2020","journal-title":"J. Hydrol. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.apenergy.2016.12.130","article-title":"Electricity price forecasting by a hybrid model, combining wavelet transform, ARMA and kernel-based extreme learning machine methods","volume":"190","author":"Yang","year":"2017","journal-title":"Appl. Energy"},{"key":"ref_39","first-page":"102","article-title":"Study of the correlation coefficients in mathematical statistics","volume":"39","author":"Zhang","year":"2009","journal-title":"Math. Pract. Theory"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"106134","DOI":"10.1016\/j.compag.2021.106134","article-title":"Multi-stream Hybrid Architecture Based on Cross-level Fusion Strategy for Fine-grained Crop Species Recognition in Precision Agriculture","volume":"185","author":"Kong","year":"2021","journal-title":"Comput. Electron. Agric."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1194565","DOI":"10.1155\/2021\/1194565","article-title":"Deep-stacking network approach by multisource data mining for hazardous risk identification in IoT-based intelligent food management systems","volume":"2021","author":"Kong","year":"2021","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Jin, X.B., Zheng, W.Z., Kong, J.L., Wang, X.Y., Bai, Y.T., Su, T.L., and Lin, S. (2021). Deep-learning Forecasting Method for Electric Power Load Via Attention-based encoder-decoder With Bayesian Optimization. Energies, 14.","DOI":"10.3390\/en14061596"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Jin, X.B., Zheng, W.Z., Kong, J.L., Wang, X.Y., Zuo, M., Zhang, Q.C., and Lin, S. (2021). Deep-Learning Temporal Predictor via Bi-directional Self-attentive Encoder-decoder framework for IOT-based Environmental Sensing in Intelligent Greenhouse. Agriculture, 11.","DOI":"10.3390\/agriculture11080802"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Zheng, Y.Y., Kong, J.L., Jin, X.B., Wang, X.Y., Su, T.L., and Zuo, M. (2019). Crop Deep: The Crop Vision Dataset for Deep-learning-based Classification and Detection in Precision Agriculture. Sensors, 19.","DOI":"10.3390\/s19051058"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Jin, X.B., Gong, W.T., Kong, J.L., Bai, Y.T., and Su, T.L. (2022). PFVAE: A Planar Flow-Based Variational Auto-Encoder Prediction Model for Time Series Data. Mathematics, 10.","DOI":"10.3390\/math10040610"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Jin, X.B., Gong, W.T., Kong, J.L., Bai, Y.T., and Su, T.L. (2022). A Variational Bayesian deep network with data self-screening layer for massive time-series data forecasting. Entropy, 24.","DOI":"10.3390\/e24030335"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Jin, X.B., Zhang, J.S., Kong, J.L., Su, T.L., and Bai, Y.T. (2022). A Reversible Automatic Selection Normalization (RASN) Deep Network for Predicting in the Smart Agriculture System. Agronomy, 2.","DOI":"10.3390\/agronomy12030591"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/360\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:30:36Z","timestamp":1760135436000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/360"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,2]]},"references-count":47,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["e24030360"],"URL":"https:\/\/doi.org\/10.3390\/e24030360","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,3,2]]}}}