{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T14:07:29Z","timestamp":1760710049189,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2019,11,15]],"date-time":"2019-11-15T00:00:00Z","timestamp":1573776000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>To solve the soft sensor modeling (SSMI) problem in a nonlinear chemical process with dynamic time variation and multi-rate data, this paper proposes a dynamic SSMI method based on an autoregressive moving average (ARMA) model of weighted process data with discount (DSSMI-AMWPDD) and optimization methods. For the sustained influence of auxiliary variable data on the dominant variables, the ARMA model structure is adopted. To reduce the complexity of the model, the dynamic weighting model is combined with the ARMA model. To address the weights of auxiliary variable data with different sampling frequencies, a calculation method for AMWPDD is proposed using assumptions that are suitable for most sequential chemical processes. The proposed method can obtain a discount factor value (DFV) of auxiliary variable data, realizing the dynamic fusion of chemical process data. Particle swarm optimization (PSO) is employed to optimize the soft sensor model parameters. To address the poor convergence problem of PSO, \u03c9-dynamic PSO (\u03c9DPSO) is used to improve the PSO convergence via the dynamic fluctuation of the inertia weight. A continuous stirred tank reactor (CSTR) simulation experiment was performed. The results show that the proposed DSSMI-AMWPDD method can effectively improve the SSM prediction accuracy for a nonlinear time-varying chemical process. The AMWPDD proposed in this paper can reflect the dynamic change of chemical process and improve the accuracy of SSM data prediction. The \u03c9 dynamic PSO method proposed in this paper has faster convergence speed and higher convergence accuracy, thus, these models correlate with the concept of symmetry.<\/jats:p>","DOI":"10.3390\/sym11111414","type":"journal-article","created":{"date-parts":[[2019,11,15]],"date-time":"2019-11-15T11:24:32Z","timestamp":1573817072000},"page":"1414","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models"],"prefix":"10.3390","volume":"11","author":[{"given":"Longhao","family":"Li","sequence":"first","affiliation":[{"name":"College of Control Science and Engineering, China University of Petroleum (East China), Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yongshou","family":"Dai","sequence":"additional","affiliation":[{"name":"College of Control Science and Engineering, China University of Petroleum (East China), Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1252","DOI":"10.1016\/j.jprocont.2010.09.003","article-title":"A novel calibration approach of soft sensor based on multirate data fusion technology","volume":"20","author":"Wu","year":"2010","journal-title":"J. Process Control"},{"key":"ref_2","first-page":"28","article-title":"A deep learning based data driven soft sensor for bioprocesses","volume":"136","author":"Gopakumar","year":"2018","journal-title":"Biol. Eng. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1016\/j.compchemeng.2008.12.012","article-title":"Data-driven Soft Sensors in the process industry","volume":"33","author":"Kadlec","year":"2009","journal-title":"Comput. Chem. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"7394","DOI":"10.1021\/acs.iecr.5b04118","article-title":"A framework and modeling method of data-driven soft sensors based on semi-supervised gaussian regression","volume":"55","author":"Yan","year":"2016","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4226","DOI":"10.1109\/TIE.2016.2597764","article-title":"A data-driven hybrid arx and markov-chain modeling approach to process identification with time varying time delays","volume":"64","author":"Zhao","year":"2017","journal-title":"IEEE Trans. Ind. Electr."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.cjche.2018.06.012","article-title":"Prediction model of slurry ph based on mechanism and error compensation for mineral flotation process","volume":"26","author":"Wang","year":"2018","journal-title":"Chin. J. Chem. Eng."},{"key":"ref_7","first-page":"999","article-title":"Monitoring data quality control for a water distribution system using data self-recognition","volume":"57","author":"Liu","year":"2017","journal-title":"J. Tsinghua Univ. (Sci. Technol.)"},{"key":"ref_8","first-page":"1160","article-title":"Dynamic soft sensor modeling based on nonlinear slow feature analysis","volume":"33","author":"Di","year":"2016","journal-title":"Comput. Appl. Chem."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.1109\/TR.2015.2427156","article-title":"A novel dynamic-weighted probabilistic support vector regression-based ensemble for prognostics of time series data","volume":"64","author":"Liu","year":"2015","journal-title":"IEEE Trans. Reliab."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1016\/j.jfoodeng.2008.01.011","article-title":"Soft-sensor for on-line estimation of ethanol concentrations in wine stills","volume":"87","author":"Osorio","year":"2008","journal-title":"J. Food Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1550","DOI":"10.1109\/TCST.2013.2278412","article-title":"Novel bayesian framework for dynamic soft sensor based on support vector machine with finite impulse response","volume":"22","author":"Shang","year":"2014","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1021\/ie4020793","article-title":"An iterative two-level optimization method for the modeling of wiener structure nonlinear dynamic soft sensors","volume":"53","author":"Gao","year":"2014","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.jprocont.2017.08.014","article-title":"Modeling study of nonlinear dynamic soft sensors and robust parameter identification using swarm intelligent optimization CS-NLJ","volume":"58","author":"Wang","year":"2017","journal-title":"J. Process Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.cjche.2016.07.003","article-title":"A self-tuning control method for wiener nonlinear systems and its application to process control problems","volume":"25","author":"Yuan","year":"2017","journal-title":"Chin. J. Chem. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1108\/SR-05-2015-0073","article-title":"Time series estimation of gas sensor baseline drift using arma and kalman based models","volume":"36","author":"Zhang","year":"2016","journal-title":"Sens. Rev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"939","DOI":"10.3390\/en8020939","article-title":"Forecasting Fossil Fuel Energy Consumption for Power Generation Using QHSA-Based LSSVM Model","volume":"8","author":"Sun","year":"2015","journal-title":"Energies"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.neucom.2011.11.016","article-title":"The system identification and control of hammerstein system using non-uniform rational b-spline neural network and particle swarm optimization","volume":"82","author":"Hong","year":"2012","journal-title":"Neurocomputing"},{"key":"ref_18","unstructured":"Kennedy, J., and Eberhart, R.C. (December, January 27). Particle Swarm Optimization. Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1016\/j.asoc.2015.10.011","article-title":"A novel svm-knn-pso ensemble method for intrusion detection system","volume":"38","author":"Aburomman","year":"2015","journal-title":"Appl. Soft Comput."},{"key":"ref_20","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":"Neur. Prof. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1016\/j.cjche.2017.07.022","article-title":"Gas emission source term estimation with 1-step nonlinear partial swarm optimization\u2013Tikhonov regularization hybrid method","volume":"26","author":"Ma","year":"2018","journal-title":"Chin. J. Chem. Eng."},{"key":"ref_22","first-page":"163","article-title":"Prediction of flood season precipitation in southwest china based on improved pso-pls","volume":"2","author":"Wang","year":"2018","journal-title":"J. Trop. Meteorol."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Xiao, Y., Kang, N., Hong, Y., and Zhang, G. (2017). Misalignment Fault Diagnosis of DFWT Based on IEMD Energy Entropy and PSO-SVM. Entropy, 19.","DOI":"10.3390\/e19010006"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1016\/j.asoc.2015.10.048","article-title":"Improved accelerated PSO algorithm for mechanical engineering optimization problems","volume":"40","author":"Guedria","year":"2016","journal-title":"Appl. Sofe Comput."},{"key":"ref_25","first-page":"788","article-title":"Modeling of soft sensor for chemical process","volume":"64","author":"Cao","year":"2013","journal-title":"CIESC J."},{"key":"ref_26","first-page":"989","article-title":"Soft sensor modeling of moisture content in drying process based on LSSVM","volume":"2","author":"Zhang","year":"2009","journal-title":"Int. Conf. Electr. Meas. Instrum."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1007\/s00500-016-2351-3","article-title":"A new dynamic weighted majority control chart for data streams","volume":"22","author":"Mejri","year":"2016","journal-title":"Soft Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1342","DOI":"10.1287\/opre.1110.1028","article-title":"Safe dike heights at minimal costs: The nonhomogeneous case","volume":"60","author":"Brekelmans","year":"2012","journal-title":"Oper. Res."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3702","DOI":"10.1016\/j.csda.2005.07.003","article-title":"Multivariate discount weighted regression and local level models","volume":"50","author":"Triantafyllopoulos","year":"2006","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1002\/(SICI)1097-007X(199911\/12)27:6<605::AID-CTA86>3.0.CO;2-Z","article-title":"Least squares support vector machines","volume":"27","author":"Suykens","year":"2002","journal-title":"Int. J. Circ. Theor. Appl."},{"key":"ref_31","unstructured":"Zhao, X., Gao, Q., Tang, C., Liu, X., Song, J., and Zhou, C. (2016). Prediction of reservoir parameters of delta lithologic reservoirs based on support vector regression and well-steering. Oil Geophys. Prospect., 51."},{"key":"ref_32","first-page":"52","article-title":"Predicting non-Gaussian wind velocity using hybridizing intelligent optimization based LSSVM","volume":"36","author":"Li","year":"2017","journal-title":"J. Vib. Shock"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/S0925-2312(01)00644-0","article-title":"Weighted least squares support vector machines: Robustness and sparse approximation","volume":"48","author":"Suykens","year":"2002","journal-title":"Neurocomputing"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.ijsrc.2017.09.005","article-title":"Predicting characteristics of dune bedforms using PSO-LSSVM","volume":"32","author":"Roushangar","year":"2017","journal-title":"Int. J. Sediment Res."},{"key":"ref_35","first-page":"40","article-title":"Particle swarm optimization algorithm with sinusoidal changing inertia weight","volume":"48","author":"Jiang","year":"2012","journal-title":"Comput. Eng. Appl."},{"key":"ref_36","first-page":"1221","article-title":"Soft-sensing modeling of marine protease fermentation process based on improved PSO-RBFNN","volume":"69","author":"Zhu","year":"2018","journal-title":"CIESC J."},{"key":"ref_37","first-page":"1871","article-title":"Prediction of package volume based on improved PSO-BP","volume":"24","author":"Xu","year":"2018","journal-title":"Comput. Integr. Manuf. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.ces.2018.04.057","article-title":"Control of a nonlinear continuous stirred tank reactor via event triggered sliding modes","volume":"187","author":"Sinha","year":"2018","journal-title":"Chem. Eng. Sci."},{"key":"ref_39","unstructured":"Shao, W. (2016). Adaptive Soft Sensor Modeling Based on Local Learning, China U Petrol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1913","DOI":"10.1016\/j.jprocont.2012.09.006","article-title":"A bayesian approach to design of adaptive multi-model inferential sensors with application in oil sand industry","volume":"22","author":"Khatibisepehr","year":"2012","journal-title":"J. Process Control"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1414\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:34:53Z","timestamp":1760189693000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1414"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,15]]},"references-count":40,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2019,11]]}},"alternative-id":["sym11111414"],"URL":"https:\/\/doi.org\/10.3390\/sym11111414","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,11,15]]}}}